Claudia Fonte Sánchez

AND Pierre Gabriel

Stéphane Mischler

Stéphane Mischler On

Claudia Fonte Sanchez

S T Éphane

Mischler C Fonte Sanchez

On the Krein-Rutman theorem and beyond

1. Introduction 1.1. Framework and the main result. In this work, we revisit the Krein-Rutman theory for semigroups of positive operators in a Banach lattice framework and we provide some very general, efficient and handy results with constructive estimates about -the existence of a solution to the first eigentriplet problem; -the geometry of the principal eigenvalue problem; -the asymptotic stability of the first eigenvector with possible constructive rate of convergence. This abstract theory is motivated and illustrated by several examples of differential, intro-differential and integral operators. In particular, we revisit the first eigenvalue problem and the asymptotic stability of the first eigenvector for -some parabolic equations in a bounded domain and in the whole space; -some transport equations in a bounded or unbounded domain, including some growth-fragmentation models and some kinetic models; -the kinetic Fokker-Planck equation in bounded domain; -some mutation-selection models. The results we establish on these examples are more general and more accurate that what we can find in the literature. Our approach is in the same time able to tackle some critical cases, but also it is very natural and makes possible to bring out the main important properties for each example and to get rid of many technical issues. The present work is motivated by new problems and ideas presented in the lectures on the Krein-Rutman theorem by P.-L. Lions at Collège de France [START_REF] Lions | Premières valeurs et fonctions propres. Cours du Collège de France 2020-2021[END_REF] and by the recent contributions by Bansaye et al [START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF] and by Cañizo and Mischler [START_REF] Cañizo | Harris-type results on geometric and subgeometric convergence to equilibrium for stochastic semigroups[END_REF] developing Harris techniques. Bringing and developing these ideas and techniques together with the more classical spectral analysis approach developed or synthesized in previous contributions by Krein and Rutman [START_REF] Kreȋn | Linear operators leaving invariant a cone in a Banach space[END_REF], by Arendt et al [START_REF] Arendt | One-parameter semigroups of positive operators[END_REF], by Mischler and Scher [START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF], by Bátkai et al [START_REF] Bátkai | Positive operator semigroups[END_REF] and many others, we are then able to significantly generalize and improve the Krein-Rutman theory for positive semigroups. The abstract results are developed in the framework of a quite general Banach lattice X, that is a Banach space (X, • ) endowed with a compatible order relation ≥ and thus with associated positive cone X + := {f ∈ X; f ≥ 0}, which satisfies either X = Y ′ or X ′ = Y for another dual Banach lattice Y . The precise (and standard) framework will be presented in Section 2.1, and some additional properties will be added when needed (these ones always hold in usual Banach lattices used in PDE and stochastic processes theory). On the other hand, all the applications we will presented are made in the following examples of usual Banach lattices :

• X := C 0 (E), the space of continuous functions which tend to 0 at infinity (when E is not a compact set) endowed with the uniform norm, or X := C 0,m (E) its weighted variant;

• X := L p (E) = L p (E, E , µ), the Lebesgue space of functions associated to the Borel σ-algebra E , a positive σ-finite measure µ and an exponent p ∈ [1, ∞], or X := L p m (E) its weighted variant; • X := M 1 (E) = (C 0 (E)) ′ , the space of Radon measures defined as the dual space of C 0 (E), or X := M 1 m (E) its weighted variant. In all the above examples, E denotes a σ-compact metric space, and we write E = ∪E R , with E R ⊂ E R+1 , E R compact.

We next consider a positive one-parameter semigroup of operators S = S L on X (we will indifferently writes S t = S(t) = S L (t) for t ≥ 0), and we denote by L its generator, by D(L) ⊂ X the domain of L, by ρ(L) ⊂ C the resolvent set of L and by Σ(L) = C\ρ(L) the spectrum of L. We also denote by S * and L * the corresponding semigroup and generator on the dual space Y , and we refer to Section 2.1 for more notations.

As announced, we may split the issue into several pieces concerning the stationary and the evolution associated problems.

• Existence. We are first interested in the existence part of the first or principal eigentriplet problem, namely we wish to bring out very general conditions under which (S1) there exists a solution (λ 1 , f 1 , φ 1 ) ∈ R × X × Y to the eigentriplet problem

Lf 1 = λ 1 f 1 , f 1 ≥ 0, f 1 = 0, (1.1) L * φ 1 = λ 1 φ 1 , φ 1 ≥ 0, φ 1 = 0, (1.2)
and furthermore λ 1 coincides with the spectral bound, namely (1.3)

λ 1 = s(L) := sup{ℜeλ; λ ∈ Σ(L)} = inf{κ ∈ R; ∆ κ ⊂ ρ(L)},
where ∆ α is the open half plan ∆ α := {z ∈ C; ℜez > α}.

We emphasize on the fact that this problem is named as the principal eigenvalue problem because λ 1 ∈ Σ(L) ⊂ {z ∈ C, ℜe(z) ≤ λ 1 }.

• Geometry. A second issue is about an accurate analysis of the principal eigentriplet solution and of the geometry of the (principal part of the) spectrum.

On the one hand, concerning the eigentriplet solution, we investigate conditions such that (S2) f 1 is strictly positive (we refer to Section 4.1 for a definition) and f 1 is the unique (up to normalization) positive eigenvector for L, φ 1 is strictly positive and φ 1 is the unique (up to normalization) positive eigenvector for L * , and finally λ 1 is geometrically and algebraically simple for both L and L * . We then may make the (usual) normalization choice (1.4)

f 1 = 1, f 1 , φ 1 = 1 or φ 1 = 1, f 1 , φ 1 = 1 .
We are next interested by describing the boundary point spectrum Σ + P (L) := Σ P (L) ∩ Σ + (L), where we define the boundary spectrum Σ + (L) := s(L) + iR and Σ P (L) as the point spectrum (or set of eigenvalues). More precisely, we exhibit some conditions such that (S3 1 ) Σ + P (L)λ 1 is a (discrete) additive subgroup of iR; (S3 2 ) Σ + P (L) is trivial, namely (1.5) Σ + P (L) = {λ 1 }; or (S3 3 ) Σ + P (L) is trivial and Σ(L) enjoys a spectral gap property (on its principal part), namely (1.6) ∃ κ < λ 1 ; Σ(L) ∩ ∆ κ = {λ 1 }.

In the last situation (1.6), a band separates the spectral value λ 1 to the remainder of the spectrum, while there is no spectral gap when (1.5) holds but (1.6) does not.

The importance of such a eigentriplet comes from the fact that we may associate the Malthusian function F 1 (t) := e λ1t f 1 , which is a particular solution to the evolution equation (with maximal growth) and a natural candidate to capture the main asymptotic feature of generic semigroup flow.

• Asymptotic stability. In order to formulate our third main issue, namely the asymptotic stability of F 1 , we introduce the rescaled operators L = Lλ 1 and L * = L *λ 1 , so that Lf 1 = 0, L * φ 1 = 0, or in other words, f 1 is a stationary state of the semigroup S = S L and φ 1 is a stationary state of the semigroup S * = S L * , and thus a conservation law for S:

S(t)f 1 = f 1 , S * (t)φ 1 = φ 1 , S(t)f, φ 1 = f, φ 1 ,
for any t ≥ 0 and any f ∈ X. Because of the property of the eigentriplet and of the normalization assumption (1.4), we may reduce the issue to considering the case f ∈ X satisfies φ 1 , f = 0 when (S3 2 ) or (S3 3 ) holds and more generally f ∈ Y ⊥ 0 when (S3 1 ) holds, where Y 0 stands for the eigenspace associated to the eigenvalues belonging to Σ + P (L). Depending of the hypotheses we made on L and S L , we are able to establish some (E1) mean ergodic property, namely 1 T T 0 S t f dt → 0 as T → ∞;

(E2) ergodic property, namely S t f → 0 as t → ∞;

(E3) quantitative asymptotic stability, which may be geometric (or exponential) in the spectral gap (1.6) case, namely (E3 1 ) S(t)f ≤ C e -εt f , ∀ t ≥ 0, ∀ f ∈ X, f, φ 1 = 0, for possible constructive constants ε > 0 and C ≥ 1, or under the weaker condition (1.5) only subgeometric, namely

(E3 2 ) S(t)f 1 ≤ Θ(t) f 2 , ∀ t ≥ 0, ∀ f ∈ X, f, φ 1 = 0, where • 2 = • X ,
• 1 is a weaker norm and Θ : R + → R + is a constructive decay function satisfying Θ(t) ց 0 when t ր ∞.

We aim now to allude some general hypotheses on the semigroup S L or its generator L such that the above three main issues may be tackled. Additionally to the yet mentioned fact that S L is positive (which is almost equivalent to the fact that its resolvent is a positive operator, that L enjoys a weak maximum principle or that L enjoys Kato's inequality) our hypotheses are mainly of two kinds : -strict positivity conditions; -regularity conditions; and these ones may be formulated at the stationary level directly on the generator L or its resolvent R L or they may be formulated at the evolution level on the semigroup of operators S L . Of course, in order to establish constructive results these hypotheses will have to be formulated in a quantitative way. The strict positivity we will introduce and use are of different kinds: -strong maximum principle on the generator, or equivalently irreducibility of the semigroup; -reverse Kato's inequality for the generator or aperiodicity condition of the semigroup; -Doblin-Harris condition on the semigroup, which may be formulated as

(1.7) S T f ≥ g 0 ψ 0 , f , ∀ f ∈ X + ,
for some for some T > 0 and convenient g 0 ∈ X + \{0}, ψ 0 ∈ Y + \{0}.

Less systematically but in a crucial way, we will make use of somehow related -barrier functions and positive subeigenfunctions, which for the last one typically writes

(1.8) ∃ κ 0 ∈ R, ∃ φ 0 ∈ Y + \{0}, L * φ 0 ≥ κ 0 φ 0 .
On the other hand, some regularity is needed on the dominant part of the semigroup. In order to briefly explain the issue, we assume that L = A + B with A ∈ B(X) and B is the generator of a positive semigroup S B . In such a context, we may write the resolvent factorization identity

R L = R B + R B AR L
on the resolvent R L of L and R B of B, and its iterated version (1.9)

R L = V + WR L , V := R B + • • • + R B (AR B ) N -1 , W := (R B A) N .
At the level of the generator, our regularity assumption then typically writes

(1.10) sup z∈∆κ V(z) B(X) < ∞, sup z∈∆κ W(z) B(X,X1) < ∞,
for some κ ∈ R and X 1 ⊂ X, which is nothing but the classical Voigt's power compact condition when X 1 ⊂ X with compact embedding. Similarly, at the level of the semigroup, we may write the associated Duhamel formula S L = S B + (S B A) * S L , (we refer to Section 3.1 for a precise definition) and its iterated version (1.11)

S L = V + W * S L , V := N -1 ℓ=0 S B * (AS B ) ( * ℓ) , W := (S B A) ( * N ) ,
with N ≥ 1. At the level of the semigroup, our regularity assumption then typically writes (1.12) sup t≥0

V (t)e -κt B(X) < ∞, sup t≥0 W (t)e -κt B(X,X1) < ∞,
for some κ ∈ R and X 1 ⊂ X in the dissipative framework and a variant of these estimates in a weak dissipative framework. The crucial information is κ < κ 0 (dissipative framework) or κ = κ 0 (more involved weak dissipative framework).

We are now in position to state in a very informal way our main result at the level of the abstract Banach lattice framework.

Theorem 1.1 (rough version). Let us consider a Banach lattice X picked up in the examples listed above and a positive semigroup S L on X which enjoys the above splitting structure (1.9), (1.10), (1.11), (1.12).

(1) Conclusion (S1) holds under the localization of the principal spectrum assumption κ < κ 0 and a weak compactness assumption on the regular part W or W in the splitting.

(2) Under an additional strong maximum principle the conclusion (S2) holds. When X ⊂ L 1 loc , we additionally conclude that (S3 1 ) and (E1) hold. In order to make one step further, we have the three next possibilities (3) Under an additional inverse Kato's condition or an aperiodicity property, the conclusion (S3 2 ) holds, as well as (E2) when X ⊂ L 1 loc . (4) Alternatively, under an additional strong compactness assumption on the regular part W of the semigroup, the quantitative exponential asymptotic stability (E3 1 ) holds without constructive constants, and thus also the spectral gap conclusion (S3 3 ) holds (in a not constructive way).

(5) Alternatively, under the additional Doblin-Harris condition (1.7) and an appropriate regularity estimate on the regular part of the splitting W , the quantitative asymptotic stability (E3) holds for both the geometric and subgeometric framework with now constructive constants.

More general and precise statements will be presented in Sections 2, 3, 4, 5 and 6, where in particular some variants in a weak dissipative framework (κ = κ 0 ) will be presented. It is worth emphasizing that the assumptions in ( 4) and ( 5) may be optimal in the sense that reciprocal implications are likely to be true. We do not follow that line of investigation but rather refer to [START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF] where such kind of results are established. 1.2. Discussion about Theorem 1.1. We discuss several works related to the main Theorem 1.1 as well as the hypotheses and the techniques used during the proof. 1.2.1. The Krein-Rutman work and related approaches. For a strictly positive matrix in a finite dimensional space, Perron [START_REF] Perron | Zur Theorie der Matrices[END_REF] and Frobenius [START_REF] Frobenius | Ueber matrizen aus nicht negativen elementen[END_REF] establish at the beginning of the 20th Century that the eigenvalue with largest real part is unique, real and simple. In their pioneer work, Krein and Rutman establish in [START_REF] Kreȋn | Linear operators leaving invariant a cone in a Banach space[END_REF] for the very first time possible infinite dimensional functional space versions of the Perron-Frobenius theorem.

Theorem 1.2 (Krein-Rutman). Consider a Banach lattice with positive cone X + and strictly positive cone X ++ := intX + = ∅. Consider a linear and compact operator R : X → X such that R : X + → X + and R : X + \{0} → X ++ . Then there exists a unique eigentriplet (µ 1 , f 1 , φ 1 ) such that

µ 1 > 0, f 1 ∈ X ++ , f 1 = µ 1 Rf 1 , φ 1 ∈ X ′ ++ , φ 1 = µ 1 R * φ 1 .
The non-emptiness of X ++ and the strict positivity assumption R : X + \{0} → X ++ can be relaxed, to the price of loosing the uniqueness and strict positivity properties of the eigenvectors. For a bounded operator R on X, we denote by r(R) the spectral radius r(R) := sup{|λ|; λ ∈ Σ(R)} ≤ R .

Theorem 1.3 (Krein-Rutman). Consider a Banach lattice with positive cone X + and a linear and compact operator R : X → X such that R : X + → X + and r(R) > 0. Then there exists an eigentriplet (µ 1 , f 1 , φ 1 ) with

µ 1 = r(R), f 1 ∈ X + \ {0}, f 1 = µ 1 Rf 1 , φ 1 ∈ X ′ + \ {0}, φ 1 = µ 1 R * φ 1 .
In Theorems 1.2 and 1.3, the operator R corresponds to a resolvent operator R := (κ -L) -1 for κ > 0 large enough, so that when it applies, we deduce in particular that the first eigenvalue problem (1.1)-(1.2) has a solution with λ 1 = κ-µ 1 . The two conditions intX + = ∅ and R : X + \{0} → X ++ are very strong. The first one essentially imposes to work in the space of continuous function and the second one to work in a bounded domain. The result is however suitable and directly applicable (and somehow restricted) to an elliptic operator with smooth coefficients set in a bounded domain with suitable boundary conditions or to a Fredholm integral operator with positive kernel also set in a bounded domain. In the elliptic context, the property R : X + \{0} → X ++ is nothing but the strong maximum principle while the compactness property of R comes from the elliptic regularity. We refer to Section 2.3 for further discussions. The weaker condition r(R) > 0 is less restrictive and is in particular always satisfied for irreducible operators, by virtue of de Pagter's theorem [START_REF] De Pagter | Irreducible compact operators[END_REF]. In the same framework, Theorems 1.2 and 1.3 have been next slightly extended by Bonsall [START_REF] Bonsall | Linear operators in complete positive cones[END_REF], Schaefer [START_REF] Schaefer | Halbgeordnete lokalkonvexe Vektorräume[END_REF], Karlin [START_REF] Karlin | Positive operators[END_REF] or Nussbaum [START_REF] Nussbaum | Eigenvectors of nonlinear positive operators and the linear Kreȋn-Rutman theorem[END_REF] for instance. We also refer to the book by Dautray and Lions [START_REF] Dautray | Mathematical analysis and numerical methods for science and technology[END_REF] for a clear and comprehensible presentation and several possible versions. In his paper [START_REF] Birkhoff | Extensions of Jentzsch's theorem[END_REF], G. Birkhoff derived the Perron-Frobenius theorem by proving a contraction principle in Hilbert's projective metric for positive matrices. His result actually applies to any "uniformly positive bounded" linear operators of a Banach lattice, such as integral operators with positive kernels, and also provides geometric stability estimates. A closely related result was proved by E. Hopf [START_REF] Hopf | An inequality for positive linear integral operators[END_REF], and this Birkhoff-Hopf contraction theorem was subsequently generalized and sharpened, and its proof simplified, by several authors, see in particular [START_REF] Bauer | An elementary proof of the Hopf inequality for positive operators[END_REF][START_REF] Brooks | The contraction mapping principle and some applications, volume 9 of Electronic Journal of Differential Equations[END_REF][START_REF] Bushell | Hilbert's metric and positive contraction mappings in a Banach space[END_REF][START_REF] Eveson | Applications of the Birkhoff-Hopf theorem to the spectral theory of positive linear operators[END_REF][START_REF] Eveson | An elementary proof of the Birkhoff-Hopf theorem[END_REF][START_REF] Kohlberg | The contraction mapping approach to the Perron-Frobenius theory: why Hilbert's metric?[END_REF][START_REF] Nussbaum | Hilbert's projective metric and iterated nonlinear maps[END_REF][START_REF] Ostrowski | Positive matrices and functional analysis[END_REF]. This approach of the Krein-Rutman theorem requires some "uniform positivity and boundedness" of the operator, which is quite restrictive, but it nevertheless allows to recover, through an approximation procedure, the standard result of Theorem 1.2, see [START_REF] Brooks | The contraction mapping principle and some applications, volume 9 of Electronic Journal of Differential Equations[END_REF]Thm. 6.18]. The contraction in Hilbert's projective metric has the advantage to be applicable in partially order linear vector spaces without any topological structure [START_REF] Eveson | An elementary proof of the Birkhoff-Hopf theorem[END_REF], and to nonlinear maps [START_REF] Nussbaum | Hilbert's projective metric and iterated nonlinear maps[END_REF].

1.2.2. Spectral analysis approach. In his paper [START_REF] Phillips | Semi-groups of positive contraction operators[END_REF], R.S. Phillips formalized the notion of positive semigroup acting on a Banach lattice paving the way to a new field of research. In the precursory work [START_REF] Vidav | Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator[END_REF] by Vidav and next in a series of papers by Greiner and co-authors [START_REF] Greiner | A typical Perron-Frobenius theorem with applications to an age-dependent population equation[END_REF][START_REF] Greiner | On the spectral bound of the generator of semigroups of positive operators[END_REF][START_REF] Arendt | One-parameter semigroups of positive operators[END_REF], Webb [START_REF] Webb | Theory of nonlinear age-dependent population dynamics[END_REF][START_REF] Webb | An operator-theoretic formulation of asynchronous exponential growth[END_REF] and Bürger [START_REF] Bürger | Perturbations of positive semigroups and applications to population genetics[END_REF] (see also [15, C-III, Cor. 2.12, Thm. 3.12], [152, Thm. VI.1.12, Cor. VI. 1.13] or more recently Theorem 14.17 in the very pedagogical book [START_REF] Bátkai | Positive operator semigroups[END_REF]) significant generalizations of the Krein-Rutman theory were established leading to, roughly speaking, the following result.

Theorem 1.4. Consider a positive semigroup S L on a (suitable) Banach lattice X which is irreducible and such that s(L) > -∞ is a pole, then

• s(L) is a first-order pole with one-dimensional and strictly positive residue, so that in particular there exists a solution (λ 1 , f 1 , φ 1 ) to the eigentriplet problem; • There exists α ∈ R such that Σ + (L) = {s(L) + iαZ} consists of first-order poles with onedimensional residue.

• A practical way for verifying that s(L) > -∞ is a pole consists in assuming that L enjoys the splitting structure L = A + B, as described above, with s(B) < s(L) and A is B power compact, that is to say W is compact, on ∆ s(B) . Assuming furthermore that S L is quasi-compact then • S L is exponential asymptotically stable in Span{φ 1 } ⊥ (without constructive constants).

The most important improvements here are the fact that the condition intX + = ∅ and the strong compactness of the resolvent operator R L are removed, and also that the exponential asymptotically stability is established. The hypotheses seem stronger to those stated in Theorem 1.1- [START_REF] Abdallah | Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system[END_REF], where only weak compactness is required what is not the case here. It is however worth emphasizing that in an AL-space and an AM-space (what includes the examples C 0 (E) and L 1 (E)) a power weak compactness implies a power strong compactness (see [START_REF] Bürger | Perturbations of positive semigroups and applications to population genetics[END_REF]Rk. 2.1] and [337, Cor. 1 of Thm. II.9.9]). The hypotheses and conclusions are similar to those stated in Theorem 1.1-(4). The proof is based on the one hand on the Banach lattices theory as formalized for instance by Schaefer [START_REF] Schaefer | Banach lattices and positive operators[END_REF] (see also [START_REF] Arendt | Kato's equality and spectral decomposition for positive C 0 -groups[END_REF][START_REF] Arendt | Generators of positive semigroups[END_REF][START_REF] Arendt | Kato's inequality: a characterisation of generators of positive semigroups[END_REF][START_REF] Arendt | One-parameter semigroups of positive operators[END_REF] for significant developments) using notions as ideals and quasi-interior points. On the other hand, it takes advantage on the perturbation techniques initiated by Phillips in [START_REF] Phillips | Perturbation theory for semi-groups of linear operators[END_REF] and developed further by Jörgens [START_REF] Jörgens | An asymptotic expansion in the theory of neutron transport[END_REF], Vidav [START_REF] Ribarič | Analytic properties of the inverse A(z) -1 of an analytic linear operator valued function A(z)[END_REF][START_REF] Vidav | Spectra of perturbed semigroups with applications to transport theory[END_REF] and Voigt [START_REF] Voigt | A perturbation theorem for the essential spectral radius of strongly continuous semigroups[END_REF] leading to the notions of power compact resolvent and quasi-compact semigroup, essential spectrum and Calkin algebra. The above theorem in particular applies to a positive and irreducible semigroup which is eventually norm continuous and its generator has compact resolvent (see for instance Corollary VI.1.13 in [START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF] and for the definition of an eventually norm continuous). In that case indeed, one can show that s(L) > -∞, Σ + (L) is bounded and consists of poles, so that Σ + (L) = {s(L)} and the essential growth bound ω ess (S) associated to the essential spectrum (see for instance [START_REF] Bátkai | Positive operator semigroups[END_REF]Sec 14.1] for a definition) satisfies ω ess (S) < ω(S) = s(L). The theorem was motivated and successfully applied to Boltzmann like transport operator [START_REF] Vidav | Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator[END_REF], cell division operator [START_REF] Diekmann | On the stability of the cell size distribution[END_REF], age structured equation [START_REF] Webb | An operator-theoretic formulation of asynchronous exponential growth[END_REF] and selection-mutation dynamics [START_REF] Bürger | Perturbations of positive semigroups and applications to population genetics[END_REF]. We also refer to [START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF]Ch. VI] and [START_REF] Bátkai | Positive operator semigroups[END_REF] for other numerous applications. Although very general and quite efficient, we formulate several criticisms about the above result.

-The exponential convergence result is definitively not constructive and that approach is not able to say anything about the weak dissipative case (a framework we will introduce latter, see in particular Section 3.3).

-We may observe that Theorem 1.4 is not so popular in the probability and the PDE communities and still many works in these domains refer to the original Krein and Rutman theorem even when some additional (approximation) arguments are needed rather than applying directly Theorem 1. [START_REF] Agoshkov | Some trace and extension theorems for functions from W 1 x i ,p (D) in the case of a bounded domain[END_REF]. By the way, we did not find in the literature where Theorem 1.4 is stated in such an handy way (the closer formulation is probably [152, Thm. VI. 1.12] which is given without proof).

-The proof of Theorem 1.4 that we may find in the above quoted references is written in a very specific and abstract language which make it quite obscure.

In [START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Mischler | Erratum: Spectral analysis of semigroups and growth-fragmentation equations[END_REF], one of the authors proposes the following variant.

Theorem 1.5. Consider a positive semigroup S L which satisfies (1.8) with κ 0 ∈ R, it is irreducible and its generator enjoys the splitting structure (1.9)-(1.10) for some κ < κ 0 and X 1 ⊂ X with compact imbedding. Assuming furthermore that (1.13) ∃ α > 0, sup z∈∆κ z α W(z) B(X) < ∞, the quantitative exponential asymptotic stability (E3 1 ) holds (without constructive constants).

The proof of Theorem 1.5 is based on a partial (but principal) spectral mapping and Weyl's theorem (in the spirit of Voigt [START_REF] Voigt | A perturbation theorem for the essential spectral radius of strongly continuous semigroups[END_REF]) coupled with a simple analysis of the first eigenelement problem based on the irreducibility of the semigroup, but which is really simpler than the deep result on irreducible semigroup stated in Theorem 1.4. On the other hand, that approach is unable to tackle the situation when Σ + P (L) is not a singleton. One of the main features in Theorem 1.5

and the other results established in [START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Mischler | Erratum: Spectral analysis of semigroups and growth-fragmentation equations[END_REF] is the clear identification of the simple localisation of the principal spectrum condition with (1.8).

Dynamical and probabilistic approach.

It is well known from the mean ergodicity theory of Von Neumann and Birkhoff introduced in the 1930s in [START_REF] Neumann | Proof of the quasi-ergodic hypothesis[END_REF][START_REF] Birkhoff | Proof of the ergodic theorem[END_REF] that for a bounded semigroup a possible stationary state (and thus a first eigenvector associated to the first eigenvalue λ 1 = 0) can be obtain through a dynamical approach by establishing that the Cesàro mean of the semigroup appropriately converges. A classical reference is [START_REF] Krengel | Ergodic theorems[END_REF], see also [START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF]Sec. V.4] for a short presentation.

The existence of invariant measures for Markov chains/processes can be derived through a contraction approach by using coupling arguments reminiscent from the ideas of Doeblin [START_REF] Doblin | Éléments d'une théorie générale des chaînes simples constantes de Markoff[END_REF] and Harris [START_REF] Harris | The existence of stationary measures for certain Markov processes[END_REF]. This yields a simplified Krein-Rutman theorem in the Banach lattice of finite measures for Markov operators, providing the existence of f 1 whilst λ 1 = 0 and φ 1 = 1 are known by definition. Doeblin's condition is a handy criterion which ensures contraction in total variation norm, and hence existence, uniqueness, and geometric stability of the invariant measure, see for instance [START_REF] Gabriel | Measure solutions to the conservative renewal equation[END_REF][START_REF] Cañizo | Harris-type results on geometric and subgeometric convergence to equilibrium for stochastic semigroups[END_REF] for this very classical and easy result. It turns out that this contraction is related to the contraction in Hilbert's metric, see [START_REF] Gaubert | Dobrushin's ergodicity coefficient for Markov operators on cones[END_REF]. The drawback of Doeblin's condition is that it is quite demanding and typically requires the state space to be bounded. Harris's idea allows an extension to the unbounded setting by localizing Doeblin's condition in a "small set" which is visited infinitely often. The return to small sets is often obtained by using a Lyapunov function. When the Lyapunov function is strong enough for ensuring exponential return, contraction in weighted total variation norm can be established and geometric stability of the invariant measure is inferred [START_REF] Meyn | Stability of Markovian processes I: criteria for discrete-time Chains[END_REF][START_REF] Meyn | Stability of Markovian processes II: continuous-time processes and sampled chains[END_REF][START_REF] Meyn | Stability of Markovian processes III: Foster-Lyapunov criteria for continuoustime processes[END_REF][START_REF] Meyn | Markov chains and stochastic stability[END_REF][START_REF] Hairer | Yet another look at Harris' ergodic theorem for Markov chains[END_REF][START_REF] Cañizo | Harris-type results on geometric and subgeometric convergence to equilibrium for stochastic semigroups[END_REF], leading to the following result (which is made constructive in the two last references).

Theorem 1.6. Consider a positive semigroup S on the Banach space X = M 1 m (E) for some weight function m : E → [1, ∞). Suppose that S is conservative, in the sense that (1) S * t 1 = 1 for all t ≥ 0, and assume that, for some subset K ⊂ E on which m is bounded and some time T > 0, (2) S * T m ≤ αm + θ1 K , for some α ∈ (0, 1) and θ > 0;

(3) S T f ≥ f, 1 K g 0 , for all f ∈ X + and some g 0 ∈ X + such that g 0 , 1 K > 0. Then there exists a unique probability measure f 1 ∈ M 1 m such that (λ 1 = 0, f 1 , φ 1 = 1) is solution to the first eigentriplet problem, and the quantitative exponential stability (E3 1 ) holds with constructive constants. Moreover, some reciprocal implication holds true.

When only a weak version of the above Lyapunov condition (2) is available, an extension of the theory to a weakly dissipative framework is possible and has been developed in [START_REF] Tuominen | Subgeometric rates of convergence of f -ergodic Markov chains[END_REF][START_REF] Douc | Practical drift conditions for subgeometric rates of convergence[END_REF][START_REF] Douc | Subgeometric rates of convergence of f -ergodic strong Markov processes[END_REF][START_REF] Hairer | Convergence of Markov processes[END_REF][START_REF] Cañizo | Harris-type results on geometric and subgeometric convergence to equilibrium for stochastic semigroups[END_REF] leading to existence, uniqueness, but only sub-geometric stability of the invariant measure. We also mention that ergodicity properties of Feynman-Kac semigroups were investigated in [START_REF] Del Moral | On the stability of interacting processes with applications to filtering and genetic algorithms[END_REF][START_REF] Del Moral | On the stability of nonlinear Feynman-Kac semigroups[END_REF] and [START_REF] Kontoyiannis | Spectral theory and limit theorems for geometrically ergodic Markov processes[END_REF][START_REF] Kontoyiannis | Large deviations asymptotics and the spectral theory of multiplicatively regular Markov processes[END_REF].

Using a condition proposed in [131, Condition Z], the Doeblin-Harris method was extended to non-conservative semigroups in [START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF][START_REF] Champagnat | Exponential convergence to quasi-stationary distribution and Q-process[END_REF][START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF][START_REF] Champagnat | Practical criteria for R-positive recurrence of unbounded semigroups[END_REF][START_REF] Cloez | On an irreducibility type condition for the ergodicity of nonconservative semigroups[END_REF]. In [START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF] necessary and sufficient conditions for the geometric stability of (λ 1 , f 1 , φ 1 ) in weighted total variation norm are obtained. To our knowledge, no extension to the above mentioned weakly dissipative setting is available. The following result is an immediate consequence of [START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF]Thm. 2.1].

Theorem 1.7. Consider the same situation as in Theorem 1.6 but relax the conservativeness assumption (1) by the assumption that there exists a function φ 0 : E → (0, ∞), bounded from above and below by positive constants on K, such that φ 0 ≤ m on E, and satisfying (1a) S * T φ 0 ≥ βφ 0 , for some β > 0; (1b) 1 K S * t φ 0 ≤ C g 0 , 1 K S * t φ 0 , for all t ≥ 0 and some C > 0; and replace the condition α ∈ (0, 1) by α ∈ (0, β) in the assumption [START_REF] Agoshkov | Problema sledov funktsiȋ iz prostranstv H 1 p (Ω × D) i issledovanie nekotorykh variatsionnykh kraevykh zadach perenosa[END_REF]. Then, there exists a unique solution (λ 1 , f 1 , φ 1 ) to the first eigentriplet problem and the quantitative exponential stability (E3 1 ) holds with constructive constants. Moreover, some reciprocal implication holds true.

Positivity conditions required for the Doeblin-Harris approach are less restrictive than for Birkhoff contraction. Conversely, unlike contraction in Hilbert's metric, Doeblin-Harris method strongly uses the linearity of the operators, and may thus not be easily extendable to nonlinear operator. However, since it is based on contraction arguments, it can be extended to time-inhomogeneous semigroups [START_REF] Bansaye | Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions[END_REF]. Finally, the existence of a first eigenmeasure in a non-conservative setting were established in [START_REF] Collet | Quasi-stationary distributions for structured birth and death processes with mutations[END_REF][START_REF] Collet | Stochastic models for a chemostat and long-time behavior[END_REF] through a Lyapunov function property, a suitable renormalization and a fixed point argument.

The key point in this approach is that it provides a constructive rate of convergence while its drawback is that it is somehow restricted to a M 1 m (or L 1 m ) framework and that some of the conditions (typically (1b) in Theorem 1.7) are not fully intuitive and may be hard to verify in the applications.

PDE approaches.

At least as far as the existence issue is concerned, one of the most common way in PDE papers in order to tackle the existence part of the first eigentriplet problem consists in approximating (by regularization of the coefficients, add of a small viscosity, discretization) the eigentriplet problem, then use the most classical Perron-Frobenius Theorem [START_REF] Perron | Zur Theorie der Matrices[END_REF][START_REF] Frobenius | Ueber matrizen aus nicht negativen elementen[END_REF] or Krein-Rutman Theorem [START_REF] Kreȋn | Linear operators leaving invariant a cone in a Banach space[END_REF][START_REF] Dautray | Mathematical analysis and numerical methods for science and technology[END_REF] and next to derive appropriate estimates and pass to the limit through a "stability argument".

Recently, in order to circumvent the above approximation step, a new abstract and general version of the existence part of the Krein-Rutman theory has been developed by Lions in [START_REF] Lions | Premières valeurs et fonctions propres. Cours du Collège de France 2020-2021[END_REF] which, as for the early works [START_REF] Lions | Two remarks on Monge-Ampère equations[END_REF][START_REF] Lions | Bifurcation and optimal stochastic control[END_REF], is also adapted to nonlinear operators and it includes the following statement (in the linear operators framework).

Theorem 1.8. Consider a Banach lattice with positive cone X + and a linear and bounded operator R : X → X such that (i) R : X + → X + ; (ii) ∃ g 2 ∈ X + \{0}, ∃ C 2 > 0 such that Rg 2 ≤ C 2 g 2 , and set K 2 := {g ∈ X + ; ∃C, g ≤ Cg 2 }; (iii) µ 1 := sup J < +∞, where J := {µ ≥ 0; ∃h ∈ K 2 , h ≥ µRh + g 2 };

(iv) any sequence (g n ) of almost first eigenvectors is relatively (possibly weakly) compact, where we say that (g n ) is a sequence of almost first eigenvectors if g n = µ n Rg n + ε n , (g n ) is bounded, µ n ր µ 1 and ε n → 0. Then there exists f 1 ∈ K 2 such that f 1 = µ 1 Rf 1 and f 1 = 1.

The statement and proof of Theorem 1.8 somehow generalize the existence part of the Krein-Rutman theorem presented in Theorem 1.4 because the required splitting structure and associated power compactness are replaced by the very natural stability principle (iv). Applications to elliptic operator with strong or critical confinement property in the whole space R d setting are also presented in [START_REF] Lions | Premières valeurs et fonctions propres. Cours du Collège de France 2020-2021[END_REF].

Let us also mention the huge literature on the characterization of the first eigenvalue by a minmax formula. As explained with more details below, this approach has first been introduced in the Courant-Fischer min-max theorem [START_REF] Fischer | Über quadratische Formen mit reellen Koeffizienten[END_REF][START_REF] Courant | Über die Eigenwerte bei den Differentialgleichungen der mathematischen Physik[END_REF][START_REF] Courant | Methods of mathematical physics[END_REF] providing a variational characterization of eigenvalues in an abstract Hilbert setting for self-adjoint elliptic operators. Inspired next by pointwise minmax formula established for simple self-adjoint operators [START_REF] Duffin | Lower bounds for eigenvalues[END_REF][START_REF] Protter | Lower bounds for the first eigenvalue of elliptic equations[END_REF][START_REF] Hersch | Sur la fréquence fondamentale d'une membrane vibrante: évaluations par défaut et principe de maximum[END_REF] using a technique which goes back to Picard [START_REF] Picard | Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives[END_REF], it has been next generalized to non self-adjoint elliptic operators in [START_REF] Protter | On the spectrum of general second order operators[END_REF][START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF] among others. More recently, the same approach has been generalized to non elliptic operators, see for instance [START_REF] Coville | On generalized principal eigenvalues of nonlocal operators with a drift[END_REF] and the references therein.

On the other hand, and beyond the eigentriplet problem, the convergence towards the first eigenfunction may be proved using the general relative entropy (GRE) method which has been applied to a large class of evolution PDE in [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF] which principle is as follows. Assume that (λ 1 , f 1 , φ 1 ) ∈ R × X × X ′ is a solution to the first eigenvalue problem, that λ 1 = 0 (a case to which one can always reduces from the general case by a mere change of operator and unknown), that X, X ′ ⊂ L 1 loc (O) and then define the generalized relative entropy

J (f ) := O j(f /f 1 ) f 1 φ 1 dz
for any given convex function j : R → R + . For any solution f (t) ∈ X to the (appropriate) evolution PDE, one may establish (at least formal) the identity (1.14) J (f (t)) + t 0 D J (f (s)) ds = J (f (0)), ∀ t ≥ 0, where D J ≥ 0 is the associated generalized dissipation of relative entropy, so that J is a Lyapunov functional (it is decreasing along the flow associated to the evolution PDE). Under suitable positivity hypothesis, one has D J (f ) = 0 if and only if f ∈ Vect(f 1 ), and then one may deduce from (1.14) and some lower semicontinuity assumption on the operator D J that f (t) → cf 1 as t → ∞ (without rate and with c ∈ R). The GRE method is of course connected to j-divergence in information theory and statistics [START_REF] Csiszár | Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten[END_REF][START_REF] Csiszár | A class of measures of informativity of observation channels[END_REF][START_REF] Burbea | On the convexity of some divergence measures based on entropy functions[END_REF][START_REF] Menéndez | h, Φ)-entropy differential metric[END_REF] and to j-entropy in probability and PDE theory [START_REF] Chafaï | Entropies, convexity, and functional inequalities: on Φ-entropies and Φ-Sobolev inequalities[END_REF][START_REF] Gentil | The Lévy-Fokker-Planck equation: Φ-entropies and convergence to equilibrium[END_REF], where however here it is crucial to identify the associated operator D J and that this last one enjoys suitable properties.

1.2.5. Hypotheses and proof. We now briefly discuss the strategy of the proof of Theorem 1.1 and how it is connected to the above material. Additional comments will be made in the corresponding Sections 2 to 6. As already said, the first eigenvalue problem is mainly split into three steps: existence, geometry and asymptotic stability. From a general point of view, our approach is more general than the initial Krein-Rutman theorem as well as less abstract than the usual semigroup school approach. We believe it is more intuitive and handy for the possible applications since it is presented as a series of estimates to be checked and the necessary assumptions are made clearer at each step.

• Concerning the existence of a solution to the first eigentriplet problem, our result improves the previous known results because [START_REF] Abdallah | Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system[END_REF] only weak compactness property is needed (while Theorems 1.4 & 1.5 require strong compactness assumptions), [START_REF] Agoshkov | Problema sledov funktsiȋ iz prostranstv H 1 p (Ω × D) i issledovanie nekotorykh variatsionnykh kraevykh zadach perenosa[END_REF] it is more flexible than Theorems 1.4, 1.5, 1.7 & 1.8 (the two first ones being restricted to the generator of a strongly continuous semigroup, the third one being restricted to a M 1 m framework and involving the tricky condition (1b) and the last one being somehow restricted to a weighted L ∞ framework), (3) it applies to weakly dissipative cases (so that no spectral gap is needed). We present two different proofs: one based on a stationary problem approach and another one based on a dynamical problem approach (with which we are able to tackle the weakly dissipative case). Our stationary problem approach mixes in a first step the (clearly formulated) approximation argument of [START_REF] Bátkai | Positive operator semigroups[END_REF]proof of Thm. 12.15] together with the stability argument of [START_REF] Lions | Premières valeurs et fonctions propres. Cours du Collège de France 2020-2021[END_REF], where it is worth emphasizing that the condition κ < κ 0 in Theorem 1.1 is nothing but a practical (and possibly constructive) condition ensuring that assumption s(B) < s(L) holds in Theorem 1.4. On a second step, we exhibit several practical situations where the required stability condition is fulfilled recovering as a particular case the existence part in Theorems 1.4 & 1.7. We would like to point out here that the splitting hypothesizes (1.11)-(1.12) on the semigroup is a generalization of the Lyapunov condition [START_REF] Agoshkov | Problema sledov funktsiȋ iz prostranstv H 1 p (Ω × D) i issledovanie nekotorykh variatsionnykh kraevykh zadach perenosa[END_REF] in Theorem 1.6 on the semigroup which in turn generalizes the classical Lyapunov condition on the generator, namely for instance L * ψ 2 ≤ κψ 1 + Kψ 0 with ψ i ∈ X ′ , ψ 1 , ψ 2 ≥ ψ 0 together with ψ 2 ≤ ψ 1 (super Lyapunov condition), ψ 2 = ψ 1 (standard Lyapunov condition), ψ 2 ≥ ψ 1 (weak Lyapunov condition). We refer to [START_REF] Cañizo | Harris-type results on geometric and subgeometric convergence to equilibrium for stochastic semigroups[END_REF] and to Sections 2 and 3 for further discussions on that question. On the other hand, our dynamical approach mixes the splitting method yet alluded above together with some argument picked up from Von Neumann & Birkoff mean ergodic theory in the spirit of but in a more elaborate way than in [START_REF] Cañizo | Harris-type results on geometric and subgeometric convergence to equilibrium for stochastic semigroups[END_REF]Sec. 6].

• The proof about the geometry of the principal eigenvalue problem in Theorem 1.1 is a refinement of many arguments already developed in the literature. More precisely, the uniqueness of the first eigentriplet (λ 1 , f 1 , φ 1 ) and the strict positivity of the eigenvectors is established by taking up again in a more general setting some arguments developed in [START_REF] Perthame | Exponential decay for the fragmentation or cell-division equation[END_REF][START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Kavian | The Fokker-Planck equation with subcritical confinement force[END_REF]. The subgroup structure of the boundary point spectrum Σ + P (L) is next established under suitable (but not very restrictive) geometrical properties on the Banach lattice X, these ones being always true for the usual examples we have in mind and thath we have already listed above. The proof mainly mimics the usual proof (as for instance presented in [START_REF] Bátkai | Positive operator semigroups[END_REF]Sec. 14.3]) but it is less abstract and more general.

Especially, the proof does not refer to the notions of ideals, quasi-interior points or Calkin algebra nor uses the Kakutani lattice isomorphism theorem but rather uses the simpler notion of strict positivity (defined by duality) and some convient structural properties of the signum operator. In order to go one step further and to prove the triviality property Σ + P (L) = {λ 1 }, we propose one quite original approach (which we believe to be new at this level of generality) based on an inverse Kato's inequality condition of L (by refining some arguments picked up from [START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Kavian | The Fokker-Planck equation with subcritical confinement force[END_REF]) and some more standard ones based on an aperiodicity condition on the semigroup S L , on a localisation of the point spectrum condition or on a quasi-compacteness condition on the semigroup S L .

• Finally, the proof on the asymptotic stability of the first eigenvector picks up and mixes some spectral analysis, dynamical system, entropy method and Doblin-Haris coupling arguments. On a first step, we mainly rewrite some very classical dynamical system results mixed together with some arguments coming from the General Relative Entropy method in order to get our mean ergodicity and ergodicity results which are really general and very little demanding about the trajectories. We also rewrite the most classical result about the exponential asymptotic stability (without constructive constants) of the first eigenfunction proposing a very simple (and self-contained) proof which does not make any references to abstract notions as Calkin algebra, essential spectrum or essential growth bound. Last, we adapt the Doblin-Harris approach as qualitatively formulated in [START_REF] Hairer | Yet another look at Harris' ergodic theorem for Markov chains[END_REF][START_REF] Cañizo | Harris-type results on geometric and subgeometric convergence to equilibrium for stochastic semigroups[END_REF][START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF] in order to get the quantitative asymptotic stability of the first eigenfunction with constructive constants.

Some examples of applications.

The abstract Krein-Rutman theory developed in these notes and alluded above have been cooked up in order to answer to the first eigenvalue problem for PDEs. We show its efficiency by applying it to several examples of evolution PDEs. These examples must be thus considered both as a motivation and an illustration of simultaneously developed abstract theory. 1.3.1. Parabolic equations. In Part 7, we are interested by parabolic equations in divergence form

∂ t f = ∂ i (a ij ∂ j f ) + ∂ i (β i f ) + b j ∂ j f + cf in (0, ∞) × Ω,
on the function f = f (t, x), t ≥ 0, x ∈ Ω, with general conditions on the coefficients a ij , β i , b j , c and in both the case of a bounded domain Ω ⊂ R d (and we then complement the equation with a Dirichlet boundary condition) and the case Ω = R d . The importance of parabolic equations for Physics, Chemistry, Biology and Economy modeling is well known and we do not discuss it here. We consider the four following casses.

• For a bounded domain Ω ⊂ R d , we consider a general elliptic operator in divergence form

Lf := ∂ i (a ij ∂ j f ) + b i ∂ i f + ∂ i (β i f ) + cf, f ∈ H 1 0
(Ω), under the very general assumption about the regularity of the coefficients a ij ∈ L ∞ (Ω), a ij ≥ νδ ij , for some ν > 0, b i , β j ∈ L r (Ω), c ∈ L r/2 (Ω), r > d.

• In the case when Ω = R d , we focus first our analysis by considering

Lf := ∆f + b • ∇f + cf, f ∈ H 1 (R d ), with drift b ∈ L ∞ loc (R d ), potential c ∈ L 2 loc (R d
) and a confinement condition that (roughly speaking) we impose through the properties c → -∞ as |x| → ∞ and b is dominated by c at the infinity. A typical case is given by c ∼ -|x| γ and b ∼ x|x| β-1 as |x| → ∞, with γ > max(0, β -1).

• Still in the case when Ω = R d , we next consider the similar problem

Lf := ∆f + b • ∇f + rcf, f ∈ H 1 (R d ),
with now c ∈ C 0 (R d ), b ∈ C 0 (R d ) and r ∈ R + a parameter. That hypotheses correspond to a critical confinement case and we further assume that r > 0 is large enough. • In the case when Ω = R d again, we finally consider the elliptic operator

Lf := ∆f + b • ∇f + cf, with the drift confinement b = ∇U, U (x) = 1 β x γ , γ > 0,
and with c dominated by b at the infinity. We further assume c ≥ divb when γ ∈ (0, 1]. It is worth emphasizing that this corresponds to a perturbation of the classical Fokker-Planck operator associated to the potential U .

For each of these operators we are able to complete the existence, geometric and stability program as stated in Theorem 1.1, with constructive estimates on the first eigentriplet solution and more or less explicit rate of convergence to the first eigenfunction. Few suitable additional assumptions on the coefficients and on the regularity of Ω as well as the precise results will be discussed in the corresponding sections.

The first eigenvalue problem in the three first situations has been studied in [252, 8th and 9th courses] which inspired our study and to which we refer for motivations and possible extensions. Since mainly the existence issue is considered in [START_REF] Lions | Premières valeurs et fonctions propres. Cours du Collège de France 2020-2021[END_REF], our results supplement the previous analysis by tackling the geometry of the principal spectrum and the exponential asymptotic stability of the first eigenfunction. On the other hand, the fourth situation in the conservative case (c = divb) is very classical and we refer to [START_REF] Bakry | A simple proof of the Poincaré inequality for a large class of probability measures including the log-concave case[END_REF][START_REF] Bakry | Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré[END_REF][START_REF] Gualdani | Factorization of non-symmetric operators and exponential Htheorem[END_REF][START_REF] Mischler | Exponential stability of slowly decaying solutions to the kinetic-Fokker-Planck equation[END_REF][START_REF] Kavian | The Fokker-Planck equation with subcritical confinement force[END_REF] and the references therein. We believe that the extension to a non conservative case as considered here is new.

Of course, when the operator L is the Laplace operator or more generally is a self-adjoint elliptic operator, there exists a huge literature about the analysis of its spectrum and in particular about its first eigenvalue problem because among other things this is related to the ground state problem in quantum mechanic. We do not have the precise historical reference where similar results to the ones developed here are established for the first time. We may for instance refer to the contributions by Poincaré [START_REF] Poincare | Sur les Equations aux Derivees Partielles de la Physique Mathematique[END_REF] and by Courant and Hilbert [START_REF] Courant | Über die Eigenwerte bei den Differentialgleichungen der mathematischen Physik[END_REF][START_REF] Courant | Methods of mathematical physics[END_REF]. We also refer to the textbook [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Thm 8.38] for the quite general and modern proof which mixes minimisation technique, strong maximum principle and Hilbert structure arguments. It is worth mentioning that in earlier works, the Krein-Rutman theorem has been proved using elementary ODE method when considering the Sturm-Liouville operator (in dimension d = 1), see for instance [START_REF] Bôcher | The theorems of oscillation of Sturm and Klein[END_REF]. Still for a self-adjoint elliptic operator, the Courant-Fischer min-max theorem [START_REF] Fischer | Über quadratische Formen mit reellen Koeffizienten[END_REF][START_REF] Courant | Über die Eigenwerte bei den Differentialgleichungen der mathematischen Physik[END_REF] gives a variational characterization of eigenvalues through Rayleigh quotient [START_REF] Rayleigh | The Theory of Sound[END_REF] and the Weyl theorem [START_REF] Weyl | Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen[END_REF][START_REF] Weyl | Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)[END_REF][START_REF] Molčanov | On conditions for discreteness of the spectrum of self-adjoint differential equations of the second order[END_REF][START_REF] Persson | Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator[END_REF] provides some information about the distribution of the eigenvalues. More specifically, some constructive lower bound on the best constant in Poincaré inequality and thus on the first eigenvalue may be obtain through the Faber-Krahn [START_REF] Faber | Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt[END_REF][START_REF] Krahn | Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises[END_REF] isoperimetric inequality as presented in [START_REF] Carron | Inégalités isopérimétriques de Faber-Krahn et conséquences[END_REF], see also Polya-Svzego [START_REF] Pólya | Isoperimetric Inequalities in Mathematical Physics[END_REF][START_REF] Pólya | Two more inequalities between physical and geometrical quantities[END_REF] and Payne-Weinberger [START_REF] Payne | New bounds for solutions of second order elliptic partial differential equations[END_REF][START_REF] Payne | Some isoperimetric inequalities for membrane frequencies and torsional rigidity[END_REF]. Other results on that direction but based on the Lyapunov condition are obtained in [START_REF] Bakry | A simple proof of the Poincaré inequality for a large class of probability measures including the log-concave case[END_REF][START_REF] Bakry | Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré[END_REF] and we also refer to [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF] and the references therein.

On the other hand, in the case of an elliptic operator which is not self-adjoint the first result on the principal eigenvalue problem seems to be Protter, Weinberger [START_REF] Protter | On the spectrum of general second order operators[END_REF]Rk. 2] who consider the case of smooth domain and coefficients (without precise statement about the regularity) and use minmax formula and the Krein-Rutman Theorem 1.2, see also [START_REF] Peetre | Sur la positivité de la fonction de Green[END_REF]. Next, Chicco [START_REF] Chicco | Principio di massimo generalizzato e valutazione del primo autovalore per problemi ellittici del secondo ordine di tipo variazionale[END_REF][START_REF] Chicco | Some properties of the first eigenvalue and the first eigenfunction of linear second order elliptic partial differential equations in divergence form[END_REF] establishes the existence, uniqueness and some monotony properties of the first eigenvalue-eigenfunction in the weak solutions framework of Stampacchia [START_REF] Stampacchia | Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus[END_REF][START_REF] Stampacchia | Èquations elliptiques du second ordre à coefficients discontinus[END_REF] with mild regularity assumptions on the coefficients and which corresponds to the framework we will consider here (when we will consider the case of a bounded domain). These work has been followed by several papers by Donsker and Varadhan [START_REF] Donsker | On the principal eigenvalue of second-order elliptic differential operators[END_REF][START_REF] Donsker | On a variational formula for the principal eigenvalue for operators with maximum principle[END_REF] and next by the famous work of Berestycki, Nirenberg, Varadhan [START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF] opening a new field of research. These works are mainly based on strong maximum principle technique, see [START_REF] Protter | Maximum principles in differential equations[END_REF]. We also mention the recent works by Champagnat and Villemonais [START_REF] Champagnat | Practical criteria for R-positive recurrence of unbounded semigroups[END_REF][START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF] where similar results to ours for smooth enough coefficients are established using a variant of the probabilistic Doblin-Harris argument as already mentioned in Section 1.2.3. We also emphasize that in the conservative case, the long time behavior problem has been widely studied and some constructive estimates has been obtained in [START_REF] Bakry | Diffusions hypercontractives[END_REF][START_REF] Beckner | A generalized Poincaré inequality for Gaussian measures[END_REF][START_REF] Holley | Logarithmic Sobolev inequalities and stochastic Ising models[END_REF][START_REF] Toscani | Entropy production and the rate of convergence to equilibrium for the Fokker-Planck equation[END_REF][START_REF] Toscani | On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds[END_REF] by the mean of log-Sobolev inequality, in [START_REF] Bakry | A simple proof of the Poincaré inequality for a large class of probability measures including the log-concave case[END_REF][START_REF] Bakry | Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré[END_REF][START_REF] Röckner | Weak Poincaré inequalities and L 2 -convergence rates of Markov semigroups[END_REF][START_REF] Kavian | The Fokker-Planck equation with subcritical confinement force[END_REF] by the mean of Poincaré inequality and in [START_REF] Gualdani | Factorization of non-symmetric operators and exponential Htheorem[END_REF][START_REF] Kavian | The Fokker-Planck equation with subcritical confinement force[END_REF] by the mean of semigroup arguments. for some kernels r O : Σ -× O → R + , r Σ : Σ -× Σ + → R + . All the (quite usual) notations will be explained at the begin of Part 8. It is worth emphasizing here that this framework in particular covers the cases of the renewal equation, the growth-fragmentation equation and the kinetic linear Boltzmann equation on which we will come back below. This framework is motivated by and generalizes the transport theory developed in [START_REF] Bardos | Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d'approximation; application à l'équation de transport[END_REF][START_REF] Beals | Abstract time-dependent transport equations[END_REF][START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF][START_REF] Boyer | Trace theorems and spatial continuity properties for the solutions of the transport equation[END_REF][START_REF] Crippa | Initial-boundary value problems for continuity equations with BV coefficients[END_REF].

In a first step, we consider a very general vector field a by assuming that it satisfies the usual Sobolev regularity condition of DiPerna-Lions transport theory [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF]. We also make general assumptions on R O and R Σ , but a very strong and somehow restrictive positivity condition on K . Such an equation can be motivated by the abstract transport theory developed [START_REF] Beals | Abstract time-dependent transport equations[END_REF] as well as non-local reaction-diffusion models [START_REF] Cloez | On an irreducibility type condition for the ergodicity of nonconservative semigroups[END_REF][START_REF] Coville | On generalized principal eigenvalues of nonlocal operators with a drift[END_REF][START_REF] Li | On eigenvalue problems arising from nonlocal diffusion models[END_REF] and selection-mutation models in changing environment [START_REF] Forien | Ancestral lineages in mutation selection equilibria with moving optimum[END_REF][START_REF] Henry | Time reversal of spinal processes for linear and non-linear branching processes near stationarity[END_REF]. Under these general conditions and additional ones we will detail later, we are able to solve the existence and geometrical part of the first eigenvalue problem and to prove an ergodicity result (without rate of convergence) generalizing some similar results obtained in [START_REF] Cloez | On an irreducibility type condition for the ergodicity of nonconservative semigroups[END_REF][START_REF] Coville | On generalized principal eigenvalues of nonlocal operators with a drift[END_REF][START_REF] Li | On eigenvalue problems arising from nonlocal diffusion models[END_REF].

Because of the strong positivity condition made on K , the above mentioned result does not apply to the growth-fragmentation equation and the kinetic linear Boltzmann equation. We thus consider separately these important particular cases in the two next parts. Other singular jump kernels lacking strong positivity can appear in other models, for instance in neurosciences [START_REF] Dumont | The mean-field equation of a leaky integrate-and-fire neural network: measure solutions and steady states[END_REF], and must also be treated through a specific study.

Another related model is the age structured (or renewal) equation

∂ t f + ∂ y f = -Kf in (0, ∞) × (0, ∞), f (t, 0) = (Rf (t, •))(y) = ∞ 0 r(y * )f (t, y * )dy * .
It corresponds to the case D = 1, O = (0, ∞), a = 1, R Σ = 0, Σ -= {0} and K = 0 in the transport equation (1.16). The age structured equation is very popular because it is useful for describing dynamic of populations [START_REF] Sharpe | A Problem in Age-Distribution[END_REF]19,[START_REF] Cushing | An introduction to structured population dynamics[END_REF][START_REF] Webb | Theory of nonlinear age-dependent population dynamics[END_REF][START_REF] Metz | The dynamics of physiologically structured populations[END_REF] and simple neuronal dynamic [START_REF] Murray | Mathematical biology[END_REF][START_REF] Pakdaman | Dynamics of a structured neuron population[END_REF].

The long time behaviour can be analyzed though Laplace transform technique [START_REF] Feller | On the integral equation of renewal theory[END_REF][START_REF] Feller | An introduction to probability theory and its applications[END_REF][START_REF] Iannelli | Mathematical Theory of Age-Structured Population Dynamics[END_REF], relative entropy method [START_REF] Mischler | Stability in a nonlinear population maturation model[END_REF][START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF][START_REF] Gwiazda | Invariants and exponential rate of convergence to steady state in the renewal equation[END_REF], spectral analysis tool [START_REF] Webb | A semigroup proof of the Sharpe-Lotka theorem[END_REF][START_REF] Greiner | A typical Perron-Frobenius theorem with applications to an age-dependent population equation[END_REF][START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Mischler | Relaxation in time elapsed neuron network models in the weak connectivity regime[END_REF][START_REF] Mischler | Weak and strong connectivity regimes for a general time elapsed neuron network model[END_REF] and Doblin approach [START_REF] Bansaye | Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions[END_REF][START_REF] Cañizo | Asymptotic behaviour of neuron population models structured by elapsed-time[END_REF][START_REF] Gabriel | Measure solutions to the conservative renewal equation[END_REF]. Because K = 0, our previous result on the first eigenvalue problem does not apply. We just briefly observe that the method can be applied on the dual equation, thus guaranteeing the existence of (λ 1 , φ 1 ), and then that the validity of Doblin's condition ensures the existence and uniqueness of the triplet (λ 1 , f 1 , φ 1 ), its positivity, and the exponential ergodicity.

1.3.3. Growth-fragmentation equation. In Section 9, we consider the growth-fragmentation equation ∂ t f = Lf = Gf + F f posed on R + , with the growth operator Gf = -∂ x (af ) and the fragmentation operator

(F f )(x) = ∞ x k(y, x)f (y) dy -K(x)f (x), K(x) := x 0 k(y, x) y x dy.
Since the work of Diekmann, Heijmans and Thieme [START_REF] Diekmann | On the stability of the cell size distribution[END_REF], many authors studied this equation by using various methods. We can mention, among many others, [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF][START_REF] Perthame | Exponential decay for the fragmentation or cell-division equation[END_REF][START_REF] Laurençot | Exponential decay for the growth-fragmentation/cell-division equation[END_REF][START_REF] Jauffret | Eigenelements of a general aggregation-fragmentation model[END_REF][START_REF] Cáceres | Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations[END_REF][START_REF] Balagué | Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates[END_REF][START_REF] Bernard | Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts[END_REF] for studies based on suitable weak distance, entropy and functional inequalities, [START_REF] Metz | The dynamics of physiologically structured populations[END_REF][START_REF] Pichór | Continuous Markov semigroups and stability of transport equations[END_REF][START_REF] Cáceres | Rate of convergence to self-similarity for the fragmentation equation in L 1 spaces[END_REF][START_REF] Banasiak | Asynchronous exponential growth of a general structured population model[END_REF][START_REF] Bernard | Asymptotic behavior of the growth-fragmentation equation with bounded fragmentation rate[END_REF][START_REF] Bernard | Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate[END_REF][START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Mokhtar-Kharroubi | On spectral gaps of growth-fragmentation semigroups in higher moment spaces[END_REF] in the framework of positive semigroups, [START_REF] Bertoin | A probabilistic approach to spectral analysis of growth-fragmentation equations[END_REF][START_REF] Bertoin | The strong Malthusian behavior of growth-fragmentation processes[END_REF] for a probabilistic approach via the Feynman-Kac formula, [START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF][START_REF] Cañizo | Spectral gap for the growth-fragmentation equation via Harris's theorem[END_REF][START_REF] Gabriel | Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation[END_REF][START_REF] Villemonais | A quasi-stationary approach to the long-term asymptotics of the growthfragmentation equation[END_REF] for Harris's method, and also [START_REF] Franco | Modelling physiologically structured populations: renewal equations and partial differential equations[END_REF] for a recent new approach based on the reformulation of the equation as an abstract renewal problem. Our aim here is not to treat the most general cases of coefficients, but rather to illustrate the variety of the possible behaviors of the equation together with the efficiency and flexibility of the method developed in the first sections. We thus focus on a specific case of fragmentation operator, namely the equal mitosis kernel k(x, y) = 2K(x)δ x/2 (dy) = 4K(x)δ 2y (dx), so that the equation writes

∂ t f (t, x) = -∂ x a(x)f (t, x) -K(x)f (t, x) + 4K(2x)f (t, 2x).
In particular, we are interested in the case when the growth rate a is such that a(2x) = 2a(x) for all sizes x, for which the boundary spectrum is not trivial and the solutions then exhibit persistent oscillations in time. When this condition is not satisfied, we recover the more usual exponential convergence to the first eigenfunction.

We also aim at studying the variant of this equation where a variability v is introduced as a growth speed parameter which is inherent to any individual, in the spirit of [START_REF] Mischler | Stability in a nonlinear population maturation model[END_REF][START_REF] Rotenberg | Transport theory for growing cell populations[END_REF] where such a variable is added in the renewal equation. More precisely we consider the growth-fragmentation equation with variability v ∈ [1, 2] and the equal mitosis division kernel which reads

∂ t f (t, x, v) = -v∂ x a(x)f (t, x, v) -K(x)f (t, x, v) + 4 2 1 K(2x)℘(v, v * )f (t, 2x, v * )dv * .
This model was introduced in [START_REF] Doumic | Statistical estimation of a growth-fragmentation model observed on a genealogical tree[END_REF], and then also considered in [START_REF] Olivier | How does variability in cell aging and growth rates influence the malthus parameter?[END_REF]. We prove that, unlike the case without variability, it exhibits exponential relaxation to the first eigenfunction even when a(2x) = 2a(x) for all x.

∂ t f + v • ∇ x f -∇ x Φ(x) • ∇ v f = K [f ] -Kf in (0, ∞) × O, on the function f = f (t, x, v), t ≥ 0, (x, v) ∈ O = Ω × V, Ω ⊂ R d , V ⊂ R d , d ≥ 1.
We assume that K : O → R + , that K is a linear integral operator defined by

K [g] := R d rk(x, v, v * ) g(v * ) dv * ,
for some real number r > 0 and some kernel k : Ω × V × V → R + , and that Φ is a space confining potential Φ : Ω → R. We restrict our analysis to the case V := R d and Ω is either the torus Ω := T d (and we assume Φ = 0) or it is the whole space Ω := R d (and we assume that Φ is a power function). This equation is vey famous because it provide a model for neutron transport theory in nuclear reactors [START_REF] Case | Linear transport theory[END_REF][START_REF] Bell | Nuclear reactor theory[END_REF] and for cells migration in a chemotactic gradient [START_REF] Alt | Orientation of cells migrating in a chemotactic gradient[END_REF]. We refer to [START_REF] Bardos | Diffusion approximation and computation of the critical size[END_REF][START_REF] Golse | Generalized solutions of the radiative transfer equations in a singular case[END_REF][START_REF] Bardos | The nonaccretive radiative transfer equations: existence of solutions and Rosseland approximation[END_REF][START_REF] Bardos | The Rosseland approximation for the radiative transfer equations[END_REF][START_REF] Mokhtar-Kharroubi | Mathematical topics in neutron transport theory[END_REF] for a mathematical analysis of the neutron transport equation and its diffusive approximation and to [START_REF] Hillen | Existence of weak solutions for a hyperbolic model of chemosensitive movement[END_REF][START_REF] Chalub | Kinetic models for chemotaxis and their drift-diffusion limits[END_REF] for the same concerning kinetic models for chemotaxis.

Because the linear integral operator K is local in the position variable, this problem does not fall in the class of transport equation covered by the Krein-Rutman theorem established in Part 8 and a specific analysis is necessary. Under suitable positivity and regularity conditions on the kernel, we are able to complete the existence, geometric and stability program as stated in Theorem 1.1, with constructive estimates in the torus case, generalizing and improving previous works [START_REF] Birkhoff | Positivity and criticality[END_REF][START_REF] Vidav | Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator[END_REF][START_REF] Vidav | Spectra of perturbed semigroups with applications to transport theory[END_REF][START_REF] Greiner | Spectral properties and asymptotic behavior of the linear transport equation[END_REF][START_REF] Lods | On linear kinetic equations involving unbounded cross-sections[END_REF][START_REF] Mokhtar-Kharroubi | Time asymptotic behaviour and compactness in transport theory[END_REF][START_REF] Mokhtar-Kharroubi | Optimal spectral theory of the linear Boltzmann equation[END_REF][START_REF] Mokhtar-Kharroubi | Spectral properties of a class of positive semigroups on Banach lattices and streaming operators[END_REF][START_REF] Mokhtar-Kharroubi | On L 1 exponential trend to equilibrium for conservative linear kinetic equations on the torus[END_REF] where spectral analysis arguments are used and [START_REF] Champagnat | Exponential convergence to quasi-stationary distribution and Q-process[END_REF] based on a probability approach. It is worth emphasizing that these works are concerning the same equation in a bounded domain with no-flow boundary condition. Most of the literature is about the conservative case (when λ 1 = 0 and φ 1 = 1) which has been tackled by the mean of spectral analysis method [START_REF] Bernard | On the exponential decay to equilibrium of the degenerate linear Boltzmann equation[END_REF][START_REF] Mokhtar-Kharroubi | On L 1 exponential trend to equilibrium for conservative linear kinetic equations on the torus[END_REF][START_REF] Mokhtar-Kharroubi | Existence of invariant densities and time asymptotics of conservative linear kinetic equations on the torus without spectral gaps[END_REF], of entropy method [START_REF] Desvillettes | Hypocoercivity: the example of linear transport[END_REF][START_REF] Bernard | Optimal estimate of the spectral gap for the degenerate Goldstein-Taylor model[END_REF], of geometric control method [START_REF] Han-Kwan | Geometric analysis of the linear Boltzmann equation I. Trend to equilibrium[END_REF][START_REF] Dietert | Quantitative geometric control in linear kinetic theory[END_REF], by hypocoercivity method [START_REF] Hérau | Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation[END_REF][START_REF] Dolbeault | Hypocoercivity for linear kinetic equations conserving mass[END_REF][START_REF] Evans | Hypocoercivity in phi-entropy for the linear relaxation Boltzmann equation on the torus[END_REF] or by Harris coupling approach [START_REF] Cañizo | Hypocoercivity of linear kinetic equations via Harris's theorem[END_REF].

∂ t f + v • ∇ y f = ∆ v f + b • ∇ v f + cf in (0, ∞) × O, on the function f = f (t, x, v), t ≥ 0, (x, v) ∈ O := Ω × R d , Ω ⊂ R d is a bounded domain, b : O → R d is a
given vector field and c : O → R is a given function. In contrast with the previous part, collisions are typically modeled by a Fokker-Planck operator ∆ v f + div v (vf ) which takes into account a thermal bath of (Gaussian) white-noise, see Kolmogorov [START_REF] Kolmogorov | Zufällige Bewegungen (zur Theorie der Brownschen Bewegung)[END_REF], instead of the integral collisional operator K [f ] -Kf in the linear Boltzmann equation (1.16). The above equation is complemented with the Maxwell boundary condition

γ -f = α(x)D x γ + f + β(x)Γ x γ + f,
where γ ± f stand for the outgoing and incoming trace functions, α and β are accommodation coefficients, D x is a boundary diffusive operator and Γ x is the specular reflection operator. All these classical objects will be precisely defined in Part 11. We refer to [START_REF] Hörmander | Hypoelliptic second order differential equations[END_REF][START_REF] Neunzert | On the Vlasov-Fokker-Planck equation[END_REF][START_REF] Degond | Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in 1 and 2 space dimensions[END_REF][START_REF] Bouchut | Existence and uniqueness of a global smooth solution for the Vlasov-Poisson-Fokker-Planck system in three dimensions[END_REF][START_REF] Carrillo | Global weak solutions for the initial-boundary-value problems to the Vlasov-Poisson-Fokker-Planck system[END_REF][START_REF] Bouchut | Hypoelliptic regularity in kinetic equations[END_REF][START_REF] Mischler | Kinetic equations with Maxwell boundary conditions[END_REF]353,[START_REF] Zhu | Velocity averaging and Hölder regularity for kinetic Fokker-Planck equations with general transport operators and rough coefficients[END_REF] for a mathematical analysis of the kinetic Fokker-Planck equation or related problems. Under suitable boundedness and regularity conditions on the coefficients we are able to complete the existence, geometric and stability (without constructive estimates ) program as stated in Theorem 1.1, generalizing the previous works [START_REF] Lelièvre | Quasi-stationary distribution for the Langevin process in cylindrical domains, Part I: Existence, uniqueness and long-time convergence[END_REF][START_REF] Guillin | Quasi-stationary distribution for strongly feller markov processes by lyapunov functions and applications to hypoelliptic hamiltonian systems[END_REF] (partially based on [START_REF] Rey-Bellet | Ergodic properties of Markov processes[END_REF][START_REF] Herzog | A practical criterion for positivity of transition densities[END_REF][START_REF] Lelièvre | A probabilistic study of the kinetic Fokker-Planck equation in cylindrical domains[END_REF]) where similar results are established for the same kind of equation in a bounded domain with noflow boundary condition. From a technical point of view, our proof is based on trace results as those developed in [START_REF] Lods | On linear kinetic equations involving unbounded cross-sections[END_REF], boundary estimates picked up from [START_REF] Arkeryd | On diffuse reflection at the boundary for the Boltzmann equation and related equations[END_REF][START_REF] Lods | On linear kinetic equations involving unbounded cross-sections[END_REF][START_REF] Bernou | Hypocoercivity for kinetic linear equations in bounded domains with general Maxwell boundary condition[END_REF] and regularity estimates recently obtained in [START_REF] Hérau | Anisotropic hypoelliptic estimates for Landau-type operators[END_REF][START_REF] Golse | Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation[END_REF][START_REF] Guerand | Quantitative De Giorgi methods in kinetic theory[END_REF]. We also emphasize that in the conservative case, many works have been done related to hypocoercivity and constructive rate of convergence to the steady state in [START_REF] Desvillettes | On the trend to global equilibrium in spatially inhomogeneous entropydissipating systems: the linear Fokker-Planck equation[END_REF][START_REF] Guo | The Landau equation in a periodic box[END_REF][START_REF] Eckmann | Spectral properties of hypoelliptic operators[END_REF][START_REF] Hérau | Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential[END_REF][START_REF] Helffer | Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians[END_REF]353] or more recently in [START_REF] Dolbeault | Hypocoercivity for linear kinetic equations conserving mass[END_REF][START_REF] Mischler | Exponential stability of slowly decaying solutions to the kinetic-Fokker-Planck equation[END_REF][START_REF] Cao | The kinetic Fokker-Planck equation with weak confinement force[END_REF][START_REF] Bouin | Hypocoercivity without confinement[END_REF][START_REF] Albritton | Variational methods for the kinetic fokker-planck equation[END_REF].

1.3.6. Mutation-selection equation. Last, in Section 12, we consider the mutation-selection evolution equation

∂ t f = Lf = J * f -W (x)f in (0, ∞) × R d ,
This nonlocal-diffusion equation appears for instance in the modeling of genetic variability in evolutionary biology. In this context, f = f (t, x) represents the density of a population, at time t ≥ 0, of phenotypical trait x on the multi-dimensional phenotypic trait space R d . The rate of change in f per generation is given by the convolution term with kernel J which models the mutations, and the fitness function -W which stands for the difference between birth and death. This model has been widely used in the literature; we refer, for example, to the works of Kimura [START_REF] Kimura | A stochastic model concerning the maintenance of genetic variability in quantitative characters[END_REF], Lande [START_REF] Lande | The maintenance of genetic variability by mutation in a polygenic character with linked loci[END_REF], Fleming [START_REF] Fleming | Equilibrium distributions of continuous polygenic traits[END_REF] and Bürger [START_REF] Bürger | The mathematical theory of selection, recombination, and mutation[END_REF] as examples of biological applications. On the mathematical analysis point of view, the Krein-Rutman problem was investigated by Bürger in [START_REF] Bürger | Perturbations of positive semigroups and applications to population genetics[END_REF][START_REF] Bürger | Stationary distributions under mutation-selection balance: Structure and properties[END_REF] and more recently by Coville and co-authors [START_REF] Coville | On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators[END_REF][START_REF] Li | On eigenvalue problems arising from nonlocal diffusion models[END_REF], as well as by Alfaro and co-authors in [START_REF] Alfaro | Confining integro-differential equations originating from evolutionary biology: ground states and long time dynamics[END_REF] where a quantified spectral gap is obtained for symmetric kernels J. A main difference of this equation compared to more classical "local" diffusion models, where the convolution is replaced by a Laplacian, is that the first eigenvector f 1 might be a measure with atoms [START_REF] Bürger | Perturbations of positive semigroups and applications to population genetics[END_REF][START_REF] Bürger | Stationary distributions under mutation-selection balance: Structure and properties[END_REF][START_REF] Coville | Singular measure as principal eigenfunctions of some nonlocal operators[END_REF]. Some conditions are then needed relating W and J for guaranteeing that the first eigenvector is an eigenfunction [START_REF] Alfaro | Confining integro-differential equations originating from evolutionary biology: ground states and long time dynamics[END_REF][START_REF] Bürger | Perturbations of positive semigroups and applications to population genetics[END_REF][START_REF] Li | On eigenvalue problems arising from nonlocal diffusion models[END_REF]. All the above mentioned results deal with kernels J which are densities, namely absolutely continuous with respect to the Lebesgue measure. In our study, we allow the convolution kernel to have a singular part. In Section 12.1 we extend the results of the literature to the case of a small enough singular part. In Section 12.2 we consider a specific kernel which is purely singular, supported by the canonical axes of R d , and we extend the recent result of Velleret [START_REF] Velleret | Exponential quasi-ergodicity for processes with discontinuous trajectories[END_REF] to more general confining functions W . 1.4. Organization of the paper. The paper is organized in two main parts: the sections 2 to 6 are dedicated to the development of the abstract results about the Krein-Rutman problem, and the last sections 7 to 12 aim at illustrating the applicability of these results to various linear positivity preserving PDEs. More precisely, with start with the existence part of the Krein-Rutman theorem, namely the conclusion (S1). This question is addressed through a stationary approach in Section 2 and through a dynamical approach in Section 3. Section 4 is devoted to the stronger conclusion of uniqueness of the first eigentriplet in the sense of (S2), as well as to the mean ergodic property (E1). In Section 5, we are interested in the geometry of the boundary point spectrum, deriving conditions that guarantee (S3 1 ), (S3 2 ) or (S3 3 ), as well as in the ergodic properties (E2) and (E3 1 ). Finally, in Section 6, we tackle the problem of quantifying the conclusions (S3 3 ) and (E3) by using constructive contraction arguments of the Doeblin-Harris type. The purpose of the last six sections is to apply the theory developed in the first sections to the examples of PDEs presented in Section 1.3: some parabolic equations (Section 7), transport equations with integral terms (Section 8) and in particular growth-fragmentation equations (Section 9) and kinetic equations (Section 10), kinetic Fokker-Planck equations (Section 11), and purely integral mutation-selection equations (Section 12).

Existence through a stationary problem approach

In this part we provide a general existence result for the first eigentriplet problem by considering a family of approximating stationary problems and using a stability argument. We start by presenting the basic material about the Banach lattice framework and conclude with a comparison with several previous works.

2.1.

The Banach lattice framework. We start recalling the Banach lattice framework by stating (most of the time without proof) some well-known facts that one can find in reference monographs as [69, Chapitre II: Espaces de Riesz] or [START_REF] Schaefer | Banach lattices and positive operators[END_REF][START_REF] Arendt | One-parameter semigroups of positive operators[END_REF][START_REF] Banasiak | Perturbations of positive semigroups with applications[END_REF][START_REF] Bátkai | Positive operator semigroups[END_REF].

Banach lattice. A real Banach lattice is a real Banach space (X, • ) endowed with a partial order denoted by ≥ (or ≤) such that the following holds: (1) The set X + := {f ∈ X; f ≥ 0} is a nonempty convex cone (compatibility of the order with the vector space structure).

(2) For any f ∈ X, there exist some unique positive part f + ∈ X + and negative part f -∈ X + such that f = f +f -which are minimal: f = gh, g, h ≥ 0 imply g ≥ f + and h ≥ f -(generation and properness of the positive cone). We set |f | := f + + f -∈ X + the absolute value of f ∈ X.

(3) For any f, g ∈ X, |f | ≤ |g| implies f ≤ g (compatibility of norm and order structures).

Under these assumptions, one can show that -The convex cone X + is closed, pointed X + ∩ (-X + ) = {0} and generating X = X + -X + .

-The lattice operations

f → f + , f → f -and f → |f | are continuous (1-Lipschitz).
-The order intervals {h ∈ X; g ≤ h ≤ f } are closed and bounded for any given f, g ∈ X, f ≥ g. It is worth emphasizing that one commonly defines the supremum and infimum operations by

f ∨ g := g + (f -g) + ≥ f, g, f ∧ g := g -(g -f ) + ≤ f, g,
for any f, g ∈ X, and these operations can be used as an alternative way for defining a Banach lattice (the lattice structure refers indeed to these supremum and infimum operations). We may note the following elementary formulas (2.1)

f + ∧ f -= 0, |f | = f , ∀ f ∈ X. We write f ⊥ g when |f | ∧ |g| = 0 or equivalently when |f | + |g| = |f | ∨ |g|.
In that case, we have

|f | + |g| = |f + g|.
Dual Banach lattice. On the dual space X ′ , we may naturally associate a dual order ≥ (or ≤) by writing for ϕ ∈

X ′ ϕ ≥ 0 (or ϕ ∈ X ′ + ) iff ∀ f ∈ X + ϕ, f ≥ 0. For ϕ ∈ X ′ ,
there exist some unique ϕ ± ∈ X ′ + such that ϕ = ϕ +ϕ -which also satisfy (and are defined by)

∀ f ∈ X + , ϕ ± , f = sup 0≤g≤f ±ϕ, g .
One can show that the above conditions (1), ( 2) and (3) of a Banach lattice are fulfilled, and thus

X ′ = (X ′ , • X ′ , ≥
) is a Banach lattice. We observe that for any f ∈ X + there exists

f * ∈ X ′ + such that (2.2) f * , f = f 2 = f * 2 X ′ ,
as a classical corollary of the Hahn-Banach dominated extension theorem. Moreover, for any f ∈ X,

(2.3) f ≥ 0 iff ϕ, f ≥ 0, ∀ ϕ ∈ X ′ + ,
as an immediate application of the Hahn-Banach separation theorem. In other words, the restriction to X of the dual order in X ′′ associated to the order defined (by duality) on X ′ is nothing but the initial order, in particular the positive cone X ′ + is weakly * closed. The functional framework : The duality bracket. We consider two Banach lattices X, Y such that X = Y ′ with Y separable or such that Y = X ′ . We emphasize on the facts that

for f ∈ X : f ∈ X + iff f, ϕ ≥ 0, ∀ ϕ ∈ Y + , (2.4) for ϕ ∈ Y : ϕ ∈ Y + iff f, ϕ ≥ 0, ∀ f ∈ X + , (2.5)
which are immediate consequences of (2.3) and of the definition of the dual order.

Examples. For the space C 0 (E), the order is defined by f ≥ 0 iff f (x) ≥ 0 for any x ∈ E. For a space L p (E, E , µ), 1 ≤ p ≤ ∞, the order is defined by f ≥ 0 iff f (x) ≥ 0 for µ-a.e. x ∈ E. For the space M 1 (E), the order is defined by f ≥ 0 iff in the Hahn decomposition f = f +f -, there holds f -= 0, or equivalently, by duality: f ≥ 0 iff f, ϕ ≥ 0 for any ϕ ∈ C 0 (E), ϕ ≥ 0. Because confinement will play a major role in our analysis, we will use some weighted version of the above space associated to a weight (continuous or Borel measurable) function m : E → (0, ∞) that we introduce now. We recall that E always denotes a σ-compact metric space, and we write

E = ∪E R , with E R ⊂ E R+1 , E R compact. In that context, we write x n → ∞ if for any R ≥ 1 there exists n R such that x n / ∈ E R for any n ≥ n R . • We denote by C m,0 (E) the space C m,0 (E) := {ϕ ∈ C(E); |ϕ(x)|/m(x) → 0 as x → ∞} endowed with the norm ϕ Cm,0 := ϕ/m C0 . • We denote by M 1 m (E) := (C m,0 (E)) ′ the associated space of Radon measures. • We denote by L p m (E) = L p m (E, E , µ) the space L p m (E) := {f ∈ L 1 loc (E); f L p m := f m L p < ∞}. It is worth emphasizing that L p m (E, E , µ) = L p (E, E , m p µ) when p ∈ [1, ∞).
Positive operator. We denote by B(X) the set of linear and bounded operators on X. We also denote by K (X) the subspace of compact operators. We say that a bounded operator A ∈ B(X) is positive, and we write A ≥ 0, if

Af ∈ X + , ∀ f ∈ X + .
We will also sometimes abuse notations by writing A ∈ B(X + ) for meaning that A ≥ 0. For a positive operator A ∈ B(X), we have

(2.6) |Af | ≤ A|f |, ∀ f ∈ X, and A = sup 0≤f ∈BX Af ,
where B X is the unit closed ball. More generally, we have

(2.7) (Af ) ∨ (Ag) ≤ A(f ∨ g), ∀ f, g ∈ X.
For X and Y in duality, and A ∈ B(X) and A * ∈ B(Y ) in duality, in the sense that

Af, φ = f, A * φ , ∀ f ∈ X, φ ∈ Y, there holds (2.8) A ≥ 0 iff A * ≥ 0.
Let us present the elementary and classical but instructive proof of the direct implication, the reciprocal way being similar. We assume thus A ≥ 0. We take ϕ ∈ Y + and we define ψ := A * ϕ.

We then take f ∈ X + and we define g := Af , so that g ≥ 0 by assumption. We compute

ψ, f = A * ϕ, f = ϕ, Af = ϕ, g ≥ 0.
Since f ∈ X + is arbitrary, we get ψ ∈ Y + , and thus A * ≥ 0.

Semigroup, generator and spectrum.

In this work, a semigroup S = S(t) = (S t ) on X will always denote a semigroup of linear and bounded operators on a Banach lattice X which trajectories are -either strongly continuous, namely, the mapping t → S t f is continuous for the norm of X for any fixed f ∈ X; -either weakly * continuous, namely X = Y ′ for some separable Banach lattice Y such that

∀ f ∈ X, ∀ φ ∈ Y , t → S t f, φ X,Y is continuous and ∀ t ≥ 0, ∀ φ ∈ Y , f → S t f, φ X,Y is continuous.
That is in particular the case when there exists a strongly continuous semigroup P on Y such that S t = P * t for any t ≥ 0. For a semigroup S, we denote by L its generator and D(L) the associated domain, and thus we sometimes write S = S L . We also denote the iterated domain defined recursively by

D(L k ) := {f ∈ D(L k-1 ), Lf ∈ D(L k-1 )} for any k ≥ 2 and D(L ∞ ) := k≥1 D(L k ). We recall that D(L) is dense in X and the graph of L is closed in X × X. We define the growth bound (2.9) ω = ω(S) := lim sup t→∞ 1 t log S(t) ∈ R ∪ {-∞}, so that (2.10) ∀ ω ′ > ω, ∃ M ≥ 1, S(t) B(X) ≤ M e ω ′ t , ∀ t ≥ 0,
and ω is the infimum of ω ′ ∈ R such that (2.10) holds. We say that S is a semigroup of contractions when S satisfies (2.10) with M = 1 and ω ′ = 0. The resolvent set ρ(L) is the set of z ∈ C such that if z -L : D(L) → X is bijective and its inverse belongs to B(X). We define the resolvent operator by

(2.11) R(z) = R L (z) := (z -L) -1 , ∀ z ∈ ρ(L),
and the spectrum by Σ(L) := C\ρ(L). Denoting the half complex plane of abscissa α ∈ R by (2.12)

∆ α := {z ∈ C; ℜe(z) > α},
we have ρ(L) ⊃ ∆ ω and, for any z ∈ ∆ ω , there holds

(2.13) R(z) = ∞ 0 S(t)e -zt dt.
Positive semigroup. We say that a semigroup (S t ) on a Banach lattice X is positive if

S t ≥ 0, ∀ t ≥ 0.
Lemma 2.1. For a semigroup S on a Banach lattice X, there is equivalence between (a) S is positive; (b) the associate resolvent operator R is positive: R(κ) ≥ 0 for all κ > ω (or for all sufficiently large κ).

∃ λ 1 ∈ R, ∃ f 1 ∈ X + \{0}, Lf 1 = λ 1 f 1 .
We will also discuss the existence part for the dual problem at the end of the section. We first assume (H1) ∃ κ 1 ∈ R such that λ -L is invertible and (λ -L) -1 : X + → X + for any λ ≥ κ 1 . Note that an operator L satisfying (H1) is sometimes called a resolvent positive operator after the paper of Arendt [START_REF] Arendt | Resolvent positive operators[END_REF]. We then set (2.15)

I := {κ ∈ R; λ -L is invertible, (λ -L) -1 ≥ 0 for any λ ≥ κ},
which is a non empty and non upper bounded interval due to (H1). We finally set (2.16)

λ 1 := inf I ∈ [-∞, κ 1 ].
For the sake of completeness, we recall now some general facts about I and λ 1 when L is the generator of a positive semigroup. We also refer to [152, Sec. 1.b, Chap. VI] or [START_REF] Bátkai | Positive operator semigroups[END_REF]Chapter 12] and the references therein for more details.

Lemma 2.2. When L is the generator of a positive semigroup S = S L , then (i) (H1) automatically holds with any κ 1 > ω(S), so that λ 1 ≤ ω(S);

(ii) Σ(L) ∩ ∆ λ1 = ∅ and the representation formula (2.13) holds true for any z ∈ ∆ λ1 ;

(iii) it may happen that λ 1 = -∞.

The important property (ii) is probably due to [START_REF] Greiner | On the spectral bound of the generator of semigroups of positive operators[END_REF].

Proof of Lemma 2.2. The claim (i) is an immediate consequence of the representation formula (2.13) for any κ 1 > ω(S) and the positivity of S(t) for any t ≥ 0 (that is nothing but Lemma 2.1). We prove (ii). Take λ > λ 1 . From the classical identity

S(t)e -λt -I = (L -λ) t 0 S(s)e -λs ds, ∀ t ≥ 0,
and the positivity property of S, we have

0 ≤ V (t) := t 0 S(s)e -λs ds = R(λ) -R(λ)S(t)e -λt ≤ R(λ),
for any t ≥ 0. From that estimate, we get V (t) ≤ R(λ) . For any z ∈ ∆ λ , an integration by part yields

t 0 e -zs S(s) ds = e -(z-λ)t V (t) + (z -λ) t 0 e -(z-λ)s V (s) ds.
The estimate on V makes possible to pass to the limit t → ∞ in the above identity, and we deduce

U(z) := ∞ 0 e -zs S(s) ds = (z -λ) ∞ 0 e -(z-λ)s V (s) ds ∈ B(X).
In that situation, one classically knows that z ∈ ρ(L) and (z -L) -1 = U(z). We have thus established Σ(L) ∩ ∆ λ = ∅ and we conclude the proof of (ii) by observing that (2.13) is then nothing but the above formula.

(iii) On L p (0, 1), 1 ≤ p < ∞, the translation semigroup defined for a > 0 by

S(t)f (x) := f (x + at)1 x+at≤1 , ∀ t ≥ 0, x ∈ (0, 1),
is strongly continuous and positive. Since S(t) ≡ 0 for any t ≥ 1/a, we have ω(S) = -∞, and thus λ 1 = -∞ because of (i).

For further discussion, we give some probably classical results about the condition (H1) and some equivalent definitions of the set I.

Lemma 2.3. The operator L satisfies (H1) if and only if the operator L * satisfies (H1). Furthermore, under condition (H1) for L (or L * ), we have (2.17)

I = I i , ∀ i = 2, 3, 4,
with

I 2 := {κ ∈ R; λ -L is invertible for any λ ≥ κ}, I 3 := {κ ∈ R; λ -L * is invertible, (λ -L * ) -1 ≥ 0 for any λ ≥ κ}, I 4 := {κ ∈ R; λ -L * is invertible for any λ ≥ κ}.
Proof of Lemma 2.3. The equivalence of condition (H1) for the operators L and L * is an immediate consequence of the identity ρ(L) = ρ(L * ) (see for instance [START_REF] Kato | Perturbation theory for linear operators[END_REF]Thm. III.6.22]) and the fact that (λ -L) -1 : (2.8). As a consequence, we have

X + → X + iff (λ -L * ) -1 : Y + → Y + as recalled in
I = I 3 and I 2 = I 4 .
We obviously have I 2 ⊂ I and let us show the reverse inclusion. We denote R = R L . On the one hand, for any z 0 ∈ ρ(L) and any z ∈ C, |z -

z 0 | < R(z 0 ) -1 , we know that (2.18) R(z) = R(z 0 ) ∞ k=0 (z 0 -z) k R(z 0 ) k ,
which gives a proof of the fact that resolvent set ρ(L) is open and that R is an holomorphic function on ρ(L). Formula (2.18) also ensures that for λ 0 , λ ∈ R, the condition R(λ 0 ) ≥ 0 implies that R(λ) ≥ 0 provided that λ 0λ > 0 is small enough and thus R(λ) ≥ 0 for any λ in the non upper bounded connected component of the set ρ(L) ∩ R thanks to a continuation argument. In particular, I is an open set and I = I 2 .

We next assume (H2) ∃ κ 0 ∈ R such that inf I ≥ κ 0 . We do not further consider in these notes the case when inf I = -∞ and moreover we will particularly focus on the possibility to exhibit constructive lower bound κ 0 . We point out several conditions under which (H2) is satisfied.

Lemma 2.4. Condition (H2) holds under one of the four following conditions

(i) ∃ κ 0 ∈ R, ∃ φ 0 ∈ Y + \{0} such that L * φ 0 ≥ κ 0 φ 0 , which means ∀ f ∈ D(L) ∩ X + , φ 0 , (κ 0 -L)f ≤ 0; (ii) ∃ κ 0 ∈ R, ∃ f 0 ∈ X + \{0} such that Lf 0 ≥ κ 0 f 0 , which means ∀ φ ∈ D(L * ) ∩ Y + , (κ 0 -L * )φ, f 0 ≤ 0; (iii) L * is the generator of a positive semigroup S * = (S * t ) and ∃ κ 0 ∈ R, ∃ φ 0 ∈ Y + \{0}, ∃ T > 0 such that S * T φ 0 ≥ e κ0T φ 0 ; (iv) L is the generator of a positive semigroup S = (S t ) and ∃ κ 0 ∈ R, ∃ f 0 ∈ X + \{0}, ∃ T > 0 such that S T f 0 ≥ e κ0T f 0 .
Proof of Lemma 2.4. In the three cases, we claim that κ 0 / ∈ I and thus inf I ≥ κ 0 . We argue by contradiction, assuming λ 1 < κ 0 , so that κ 0 ∈ I = I i for any i = 2, 3, 4. We assume (i). For any g ∈ X + , we define f := (κ 0 -L) -1 g ∈ X + and we compute 0 ≤ φ 0 , g = φ 0 , (κ 0 -L)f ≤ 0.

That implies φ 0 , g = 0 for any g ≥ 0, so that φ 0 = 0 and a contradiction. We assume (ii). For any ψ ∈ Y + , we define φ := (κ 0 -L * ) -1 ψ ∈ Y + and we compute

0 ≤ ψ, f 0 = (κ 0 -L * )φ, f 0 ≤ 0.
That implies ψ, f 0 = 0 for any ψ ≥ 0, so that f 0 = 0 and a contradiction. We assume first that (iii) holds for any T > 0. For any f ∈ D(L) ∩ X + \{0}, we compute

φ 0 , (κ 0 -L)f = - d dt φ 0 , e -κ0t S t f ≤ 0,
which is precisely (i). We assume now that (iii) holds. If κ 0 ∈ I, for any g ∈ X + , we may define f = (κ 0 -L) -1 g ∈ X + ∩ D(L) and from condition (iii), we have

0 ≤ e -nκ0T S nT f -f, φ 0 = (L -κ 0 ) nT 0 e -κ0t S t f dt, φ 0 ,
for any n ∈ N. From the very definition of f , we also have

(L -κ 0 ) nT 0 e -κ0t S t f dt = nT 0 e -κ0t S t (L -κ 0 )f dt = - nT 0 e -κ0t S t g dt ≤ 0.
The two pieces of information together imply nT 0 e -κ0t S t g dt, φ 0 = 0.

Passing to the limit n → ∞ thanks to Lemma 2.2-(ii) and using (2.11)-(2.13), we obtain

0 = ∞ 0 e -κ0t S t g dt, φ 0 = f, φ 0 = g, (κ 0 -L * ) -1 φ 0 .
That implies (κ 0 -L * ) -1 φ 0 = 0 since g is arbitrary, what is not possible since φ 0 = 0. The proof of (H2) under assumption (iv) is similar and thus skipped.

Remark 2.5. (1) In practice, we may build f 0 or φ 0 through an explicit computation or use a barrier fonction and strong maximum principle techniques. We refer to Lemma 4.12 for a possible general result in that direction.

(2) When (ii) holds with f 0 ∈ X + \{0} ∩ D(L) and L is the generator of a positive semigroup S, then (iv) holds for any T > 0. In that case, we may indeed compute

S T e -κ0T f 0 -f 0 = T 0 S t e -κ0t (L -κ 0 )f 0 ds ≥ 0.
Lemma 2.6. Under conditions (H1) and (H2), there hold

(2.19) λ 1 ∈ [κ 0 , κ 1 ]
and

(2.20) ∃ λ n ց λ 1 , ∃ fn ∈ D(L) ∩ X + , ε n := λ n fn -L fn ≥ 0, fn = 1, ε n → 0.
Proof of Lemma 2.6. We obviously have λ 1 ≤ κ 1 from assumption (H1) and λ 1 ≥ κ 0 by assumption (H2), so that (2.19) is proved. Consider now a sequence (λ n ) n≥2 such that λ n ց λ 1 as n → ∞. We eventually have R(λ n ) → ∞ as n → ∞, where we denote by R = R L the resolvent of L. On the contrary, we would have R(λ n ′ ) ≤ M for some subsequence λ n ′ ց λ 1 and some constant M > 0. Because of (2.18) this implies that (λ n ′ε, λ n ′ ) ⊂ I for any n ′ and some ε > 0, and this is in contradiction with the definition of λ 1 . The blow up R(λ n ) → ∞ means that

∃ f n ∈ D(L), ∃ g n ∈ X, R(λ n )g n = f n , f n → ∞, g n ≤ 1.

By splitting g

n = g + n -g - n , we get f n = R(λ n )g + n -R(λ n )g - n with g ± n ≤ 1 and ( R(λ n )g + n → ∞ or R(λ n )g - n → ∞).
Changing notations, we have thus

∃ f n ≥ 0, ∃ g n ≥ 0, R(λ n )g n = f n , f n → ∞, g n ≤ 1.
We get (2.20) by defining fn := f n / f n and ε n := g n / f n .

We learn a very similar proof in [START_REF] Lions | Premières valeurs et fonctions propres. Cours du Collège de France 2020-2021[END_REF], from which our own proof is adapted. The same type of arguments can also be found in [START_REF] Bátkai | Positive operator semigroups[END_REF]proof of Theorem 12.15].

We finally assume that (H3) for any sequence ( fn ) of X such that (2.20) holds, there exist f 1 ∈ X + \{0} and a subsequence ( fn ′ ) such that fn ′ ⇀ f 1 for the weak convergence or the weak * convergence. We discuss several situations in which assumption (H3) holds. We start with a very classical framework formalized for instance by Voigt [START_REF] Voigt | A perturbation theorem for the essential spectral radius of strongly continuous semigroups[END_REF], see also Karlin [228,Cor. 1] or Sasser [START_REF] Sasser | Quasi-positive operators[END_REF] for earlier similar situations and results, which is however somehow restrictive since it is based on a strong compactness property assumed at the level of the associated semigroup of operators.

Lemma 2.7. We assume that L generates a positive semigroup S, that (H2) holds for a constant κ 0 ∈ R and that there exists T > 0 such that the splitting

(2.21) S T = V T + K T ,
holds with V T B(X) ≤ e κT , κ < κ 0 , and K T ∈ K (X). Then condition (H3) holds for the primal and the dual problems. Since e λnT ≥ e κ0T > e κT , the operator e λnT -V T is invertible with inverse Q(λ n ) := (e λnT -V T ) -1 uniformly bounded and converging in B(X) to Q(λ 1 ) = (e λ1T -V T ) -1 . We thus have

fn = w n + v n , w n := Q(λ n )K T fn , v n := -Q(λ n )e λnT εn ,
with v n X → 0 and (w n ) relatively compact in X. There exist thus a subsequence ( fn k ) and g ∈ X such that K T fn k → g and next

w n k -Q(λ 1 )g = (Q(λ n k ) -Q(λ 1 ))K T fn k + Q(λ 1 )(K T fn k -g) → 0.
We deduce that fn k → f 1 strongly in X. Because of the positivity and normalized properties of fn , we get f 1 ∈ X + , f 1 X = 1, and we conclude that (H3) holds for the primal problem

Observing that the dual semigroup

S * satisfies S * T = V * T + K * T with V * T B(Y ) ≤ e κT and K * T ∈ K (Y )
, the same proof implies that (H3) holds for the dual problem.

In the six next lemmas, we will assume that (H1)-(H2) holds associated to some constants κ i ∈ R, κ 0 < κ 1 , and we always make the following splitting structure hypothesis (HS1) there exists a splitting L = A + B such that Bα is invertible for any α ≥ κ 0 and

(2.22) V(α) := N -1 i=0 (R B (α)A) i R B (α), W(α) := (R B (α)A) N ,
are bounded in B(X) uniformly with respect to α ≥ κ 0 and for some N ≥ 1, where we recall that R B (α) := (α -B) -1 is the resolvent of B.

We first present a result also based on a strong compactness property which is assumed to hold however at the level of the resolvent operator. We will be able to use that result in most of the applications.

Lemma 2.8. (1) We assume (H1)-(H2)-(HS1) and there exists N ≥ 1 such that (2.23) W(α) is strongly compact locally uniformly on α ≥ κ 0 , in the sense that if α n → α, α n ≥ κ 0 , and (g n ) is a bounded sequence in X, then there exist f ∈ X and a subsequence (

g n k ) such that W(α n k )g n k → f strongly in X.
Then condition (H3) holds.

(2) We assume (H1)-(H2) and (HS1) where R B (α) is bounded uniformly in α ≥ κ 0 , A ∈ B(X) and W(α) ∈ K (X) for any fixed α ≥ κ 0 and some N ≥ 1. Then condition (H3) holds both for the primal and the dual problems.

Remark 2.9.

(1) The property (2.23) holds if we assume W(α) : X → X 1 is bounded uniformly in α ≥ κ 0 and X 1 ⊂ X with strong compact embedding.

(2) The property (2.23) holds if we assume (H1)-(H2)-(HS1) together with the facts that R B (α) and R B (α)A are bounded uniformly in α ≥ κ 0 and W(α) ∈ K (X) for any fixed α ≥ κ 0 . Consider indeed α n → α, α n ≥ κ 0 , and (g n ) a bounded sequence in X. On the one hand, there exist f ∈ X and a subsequence (g n k ) such that W(α)g n k → f strongly in X, because W(α) ∈ K (X). On the other hand, using the resolvent identity

R B (λ) -R B (µ) = (µ -λ)R B (λ)R B (µ), we have W(α) -W(α n ) = (α n -α) N j=1 (R B (α)A) N -j R B (α)(R B (α n )A) j → 0, so that W(α n k )g n k → f strongly in
or equivalently fn = R B (λ n )A fn + R B (λ n )ε n .
Iterating that last identity and using the notations (2.22), we get

(2.25) fn = w n + v n , w n := W(λ n ) fn , v n := V(λ n )ε n .
We observe that (w n ) is strongly relatively compact from (2.23) and fn X = 1, so that there exist a subsequence (w n k ) and f 1 ∈ X such that w n k → f 1 strongly in X. Since v n → 0 strongly in X, we deduce that fn k → f 1 strongly in X. We conclude that condition (H3) holds as in the proof of Lemma 2.7.

We next assume [START_REF] Agoshkov | Problema sledov funktsiȋ iz prostranstv H 1 p (Ω × D) i issledovanie nekotorykh variatsionnykh kraevykh zadach perenosa[END_REF]. As observed in Remark 2.9-(2), the property (2.23) holds and thus also the condition (H3) for the primal problem. We claim that the same locally uniform strong compactness property (2.23) holds for the dual problem at order N + 1 and thus condition (H3) holds for the dual problem. We may indeed use Remark 2.9-(2) since then R B * (α) and A * R B * (α) are bounded uniformly in α ≥ κ 0 and

(A * R B * (α)) N +1 = A * W(α) * R B * (α) ∈ K (Y ), ∀ α ≥ κ 0 ,
as a product of two bounded operator with a compact operator.

Remark 2.10. Instead of (HS1) in Lemma 2.8, one can assume that there exists a splitting

L = A + B and N ≥ 1 such that B -α is invertible for any α ≥ κ 0 and R B (α) := (α -B) -1 , V(α) := N -1 i=0 (AR B (α)) i , W(α) := (AR B (α)) N
are respectively bounded in B(X) uniformly with respect to α ≥ κ 0 and strongly compact locally uniformly on α ≥ κ 0 . Starting indeed again from (2.24) and defining h n := (λ n -B) fn , we may write

h n = AR B (λ n )h n + ε n . Observing that h n X ≥ R B (λ n ) -1 B(X) ≥ c > 0
by assumption, we deduce that ĥn := h n / h n X satisfies ĥn = wn + vn , wn := W(λ n ) ĥn , vn := V(λ n )ε n , with ĥn = 1 and εn := ε n / h n X → 0. Similarly as in the proof of Lemma 2.8, we conclude to the existence of subsequence ( ĥn k ) and

h 1 ∈ X + \ {0} such that ĥn k → h 1 strongly in X. Defining f 1 := R B (λ 1 )h 1 / R B (λ 1 )h 1 ,
we have again fn k → f 1 strongly in X and next that condition (H3) holds.

As we see now, strong compactness is not really necessary.

Lemma 2.11. We assume (H1)-(H2)-(HS1) and there exists N ≥ 1 such that W(α) : X → X 1 ⊂ X is positive and uniformly bounded in α ≥ κ 0 and, denoting X 0 := X, we assume that for any R 1 ≥ R 0 > 0 the set

(2.26) C = C R0,R1 := {g ∈ X + ; g X0 ≥ R 0 , g X1 ≤ R 1 }
is relatively sequentially compact for the weak topology σ(X, Y ) and 0 / ∈ C, where the closure is taken in the sense of the weak topology σ(X, Y ). Then condition (H3) holds.

Remark 2.12. When X 1 ⊂ X 0 with strongly compact embedding the above set C clearly satisfies the required conditions. In particular, Lemma 2.11 generalizes the result stated in Remark 2.9-(1).

Proof of Lemma 2.11. We go back to the proof of Lemma 2.8 and we start with (2.25). We recall that fn X0 = 1 and v n X0 → 0 from (2.20) and that w n ≥ 0 because W(λ n ) is a positive operator. We also observe that

w n X1 ≤ C W fn X0 = C W and
w n X0 ≥ 1v n X0 ≥ 1/2 for any n ≥ n * , with n * ≥ 1 large enough, so that w n ∈ C := C 1/2,CW for any n ≥ n * . By the compactness properties of C, there exist a subsequence (w n k ) and f 1 ∈ X + \{0} such that w n k ⇀ f 1 weakly σ(X, Y ). Since v n → 0 strongly in X, we deduce that fn k ⇀ f 1 weakly σ(X, Y ) and that ends the proof of (H3).

We present a typical concrete application of the preceding result. Lemma 2.13. We assume

X = L p (E, E , µ), p ∈ [1, ∞), (H1)-(H2)-(HS1) with A ≥ 0, R B (α) ≥ 0 for α ≥ κ 0 ,
and there exists N ≥ 1 such that

(2.27) W(α) : X → X 1 is uniformly bounded in α ≥ κ 0 , for a subspace X 1 ⊂ X such that {g p ; g ≥ 0, g X1 ≤ R 1 } is a weakly compact subset of L 1 (E) for any R 1 > 0.
Then condition (H3) holds.

Remark 2.14. (1) A typical example in the above statement is X 1 := L q ∩ L p m for some exponent q > p and some weight function

m : E → [1, ∞) such that m(x) → ∞ as x → ∞.
(2) The same result holds under the condition that if (u n ) is a nonnegative and bounded sequence in L p then the nonnegative sequence w n := W(λ n )u n is such that w p n is weakly compact in L 1 . Proof of Lemma 2.13. For 0 < R 0 < R 1 , we define C by (2.26) with X 0 := L p . From the weak compactness property made on X 1 , we observe that For g ∈ C, we may then write

α(R) := sup g∈C g1 E c R L p → 0, as R → ∞,
R 0 ≤ g L p ≤ g ∧ M 1 ER L p + g1 E c R L p + g1 g≥M L p and thus M 1-1/p g1 ER 1/p L 1 ≥ g ∧ M 1 ER L p ≥ R 0 -α(R) -β(M ) ≥ R 0 /2,
for some R, M > 0 large enough. On the one hand, from the reflexivity of L p or the Dunford-Pettis theorem, the set C is relatively sequentially compact for the weak topology σ(L p , L p ′ ). On the other hand, because 1 ER ∈ L p ′ the last estimate implies that any element g * ∈ C, where the closure is taken in the sense of the weak topology σ(L p , L p ′ ), satisfies

g * , 1 ER = g * 1 ER L 1 ≥ M 1-p (R 0 /2) p > 0,
and in particular 0 / ∈ C. We deduce that (H3) holds as a consequence of Lemma 2.11.

We present a second kind of result where some weak compactness is involved.

Lemma 2.15. We assume (H1)-(H2)-(HS1) and there exists N ≥ 1 such that

(2.28) W(α) : X 0 → X ⊂ X 0 is uniformly bounded in α ≥ κ 0
and, denoting X 1 := X, the set C defined by (2.26) satisfies the same properties as the ones stated in Lemma 2.11. Then condition (H3) holds.

Remark 2.16. If we replace the norm • X0 by a seminorm f X0 := |f |, ϕ 0 , ϕ 0 ∈ Y + , and we define C accordingly by (2.26), and if we assume that X = Y ′ with Y separable, then C satisfies the same compactness properties as required in the statement of Lemma 2.11. If we further assume that (2.28) holds where X 0 is endowed with the above seminorm, we may repeat the proof below in order to obtain that (H3) holds in that situation (see also Lemma 2.19 and its proof for a slightly generalized situation).

Proof of Lemma 2.15. We start here again with (2.25). We have

1 = fn X1 ≤ C W fn X0 + v n X1 ,
and thus

fn X0 ≥ C -1 W (1 -v n X1 ) ≥ (2C W ) -1
for any n ≥ n * , with n * ≥ 1 large enough, so that fn ∈ C := C (2CW ) -1 ,1 , for n ≥ n * . By the compactness properties of C, there exist a subsequence ( fn k ) and

f 1 ∈ X + \{0} such that fn k ⇀ f 1 weakly σ(X, Y ).
We present a variant of Lemma 2.13 which is also a concrete consequence of Lemma 2.11 and Lemma 2.15.

Corollary 2.17. We assume

(H1)-(H2)-(HS1) in X = L p0 m0 , 1 ≤ p 0 < ∞, together with the facts that R B (α) is positive and bounded in B(L p0 m0 ) and B(L p1 m1 ) uniformly in α ≥ κ 0 , 0 ≤ A ∈ B(L p0 m0 ) and (R B (α)A) N is bounded in B(L p0 m0 , L p1 m1 ) uniformly in α ≥ κ 0 for some N ≥ 1, with p 1 > p 0 and m 1 such that m 0 /m 1 ∈ L ϑ , 1/ϑ := 1/p 0 -1/p 1 .
Then condition (H3) holds for both the primal and the dual problems.

Proof of Corollary 2.17. On the one hand, we have

R B (α) + • • • + (R B (α)A) N -1 R B (α) is bounded in B(X) uniformly in α ≥ κ 0 , W(α) := (R B (α)A) N is bounded in B(X, X 1 ) uniformly in α ≥ κ 0 , with X 1 := L p1 m1 ⊂ X and thus {(gm 0 ) p0 ; g ≥ 0, g X1 ≤ R 1 } is a weakly compact subset of L 1 (E) for any R 1 > 0.
Condition (H3) holds for the direct problem thanks to Lemma 2.13. On the other hand, we set Y := X ′ = L q0 ν0 , q 0 := p ′ 0 , ν 0 := m -1 0 , and we first observe that

R B * (α) + • • • + (R B * (α)A * ) N -1 R B * (α) is bounded in B(Y ) uniformly in α ≥ κ 0 .
We next observe that

(A * R B * (α)) N +1 = A * W(α) * R B * (α) is bounded in B(Y 0 , Y ) uniformly in α ≥ κ 0 , with Y 0 := L q1 ν1 , q 1 := p ′ 1 , ν 1 := m -1 1 . Because {(gν 1 ) q1 ; g ≥ 0, g Y ≤ R 1
} is a weakly compact subset of L 1 (E) for any R 1 > 0, we have from the proof of Lemma 2.13 that the set C defined by (2.26) for the norms of Y 0 and Y 1 := Y satisfies the weak compactness property required in the statement of Lemma 2.11. We may thus apply Lemma 2.15 and we deduce that condition (H3) holds for the dual problem.

Another concrete consequence of Lemma 2.11 and Lemma 2.15 is the following. Lemma 2.18. We assume X = M 1 mi (E) for a continuous weight function m i on E, i = 0 or i = 1, (H1)-(H2)-(HS1) and there exists

N ≥ 1 such that (R B (α)A) N : M 1 m0 (E) → M 1 m1 (E) uniformly in α ≥ κ 0 for another continuous weight function m 1-i on E such that m 1 (x)/m 0 (x) → ∞ as x → ∞.
We additionally assume that A ≥ 0 and R B (α) ≥ 0 for α ≥ κ 0 when i = 0. Then condition (H3) holds.

Proof of Lemma 2.18. We define X i := M 1 mi (E) and we consider the set C defined by (2.26) which is clearly compact for the weak * σ(M 1 m1 , C m1,0 ) topology. When X = M 1 m0 , the result follows from Lemma 2.11 while when X = M 1 m1 , the result is a consequence of Lemma 2.15. We may slightly improve the preceding results by considering a more abstract framework and a somehow more general boundedness condition. Lemma 2.19. We assume X = Y ′ , Y separable, (H1)-(H2)-(HS1) and there exist N ≥ 1, γ ∈ [0, 1) and ϕ ∈ Y + \ {0} such that for any α ≥ κ 0 , there holds

(2.29) W(α)f X ≤ γ f X + f, ϕ X,Y ,
for all f ∈ X + , or there holds

(2.30) W(α)f X ≤ γ f X + W(α)f, ϕ X,Y ,
for all f ∈ X + . Then condition (H3) holds true, and the limit

f 1 satisfies f 1 , ϕ X,Y ≥ 1 -γ > 0.
The case X = M 1 m1 (E) in Lemma 2.18 corresponds here to the first situation where (2.29) holds with X := M 1 m1 (E), γ := 0, Y := C m0,0 (E) and ϕ := m 0 /m 1 . Proof of Lemma 2.19. Starting with (2.25) and using (2.29), we have

fn X ≤ W(λ n ) fn X + V(λ n )ε n X ≤ γ fn X + fn , ϕ X,Y + v n X , so that fn , ϕ X,Y ≥ 1 -γ -v n X .
By compactness, there are f 1 ≥ 0 and a subsequence ( fn ′ ) such that fn ′ ⇀ f 1 weak * σ(X, Y ). Passing to the limit as n ′ → ∞ in the above estimate, we find

(2.31) f 1 , ϕ X,Y = lim n ′ →∞ fn ′ , ϕ X,Y ≥ 1 -γ,
and in particular f 1 = 0. Under the assumption (2.30), modifying slightly the previous argument, we have

fn X ≤ γ fn X + w n , ϕ X,Y + v n X , which, together with fn , ϕ X,Y = w n , ϕ X,Y + v n , ϕ X,Y , implies fn , ϕ X,Y ≥ 1 -γ -v n X + v n , ϕ X,Y .
By compactness again, there are f 1 ≥ 0 and a subsequence ( fn ′ ) such that fn ′ ⇀ f 1 weak * σ(X, Y ), and passing to the limit n ′ → ∞ in the above estimate, we conclude again to (2.31).

Let us comment on Lemma 2.19 and in particular the condition (2.30).

In the case when

X = L ∞ (E, E , µ) = L 1 (E, E , µ)
′ , we can relate condition (2.30) to the assumption that there exist

f 0 ∈ X + and ϕ ∈ Y + \ {0} such that (2.32) S L (t)f 0 X ≤ S L (t)f 0 , ϕ , ∀ t ≥ 0.
This last condition is reminiscent from conditions that appear in probabilistic inspired methods for the ergodicity of semigroups, see the condition (1b) in Theorem 1.7 but also Assumption (A2) in [START_REF] Champagnat | Exponential convergence to quasi-stationary distribution and Q-process[END_REF], both in the vein of [START_REF] Del Moral | On the stability of nonlinear Feynman-Kac semigroups[END_REF]Condition Z]. Assume indeed (2.32), let η > κ 1κ 0 > 0 and consider the trivial decomposition

L = A + B = η + (L -η). Then set κ B := κ 1 -η < κ 0 , so that for any α > κ B , B -α = L -(η + α) is invertible since η + α > η + κ B = κ 1 . We thus have for any α > κ B W(α) := η(α -B) -1 = η ∞ 0 e -(η+α)t S L (t) dt
and (2.32) then ensures that W(α)f 0 X ≤ W(α)f 0 , ϕ . We recover (2.30) with γ = 0 and the difference that f 0 is fixed here.

As a Corollary of Lemma 2.18 or Lemma 2.19 and anticipating on the material of part 3, we present now a situation very classical in stochastic processes theory.

Corollary 2.20. We consider a positive semigroup S = S L defined on a Radon space X = M 1 ψ (E) for some positive weight functions ψ on E, in particular (H1) holds. We also assume that (H2) holds for some κ 0 ∈ R. We finally assume the Lyapunov condition

(2.33) L * ψ ≤ κ B ψ + M χ, with κ B < κ 0 , M ≥ 0 and χ ∈ C ψ,0 (E), 0 ≤ χ ≤ ψ.
Then condition (H3) holds true.

Let us emphasize that we may assume some regularity on ψ by considering ψ ∈ D(L * ) so that (2.33) makes sense in X or just understand (2.33) in the weak sense:

Lf, ψ ≤ κ B f, ψ + M f, χ , ∀ f ∈ D(L) ∩ X + .
Proof of Corollary 2.20. We introduce the splitting L = A+B where A is the bounded multiplicator operator A := M χ/ψ. As a bounded perturbation of L, the operator B is the generator of a semigroup S B . Defining S t := S L (t)e -Mt ≥ 0 and A c := M (1χ/ψ) ≥ 0, we have the Duhamel formula S B = S + SA c * S B and iterating infinitely many times, we deduce the Dyson-Philips formula

S B = ∞ k=0 ( SA c ) ( * k) * S.
That implies that S B ≥ 0 as a combination of positive operators. Alternatively, from the very definition of B, we have κ -B ≤ (M + κ) -L for any κ ∈ R. Choosing κ > max(ω(S L ), ω(S B )) and using the direct implication in Lemma 2.1, we have R B (κ) ≥ R L (M + κ) ≥ 0. Using the reciprocal implication in Lemma 2.1, we obtain again that S B ≥ 0. Now, for 0 ≤ f 0 ∈ D(B) and setting f t := S B (t)f 0 , we may compute

d dt f t , ψ = Bf t , ψ ≤ κ B f t , ψ , so that S B (t)f 0 M 1 ψ ≤ e κBt f 0 M 1 ψ . Using (2.

13) we immediately and classically deduce

R B (α) B(M 1 ψ ) ≤ 1 α -κ B , ∀ α > κ B , so that R B (α) is bounded in B(M 1 ψ ) and R B (α)A is bounded in B(M 1 χ , M 1
ψ ) uniformly for α ≥ κ 0 . We apply Lemma 2.18 or Lemma 2.19 ((2.29) with N = 1, γ = 0 and ϕ = M α-κB χ) in order to conclude.

In the proof of Corollary 2.20, we may alternatively use the trivial splitting L = Ã+ B = η +(L-η) for some η > κ 1κ 0 , so that α -B is invertible for any α ≥ κ 0 , and reformulate the Lyapunov condition (α

-B * )ψ ≥ (α + η -κ B )ψ -M χ,
for any α ≥ κ 0 . Observing that W(α) := ÃR B(α) = η(α -B) -1 , we deduce

W * (α)ψ ≤ η η + α -κ B ψ + M η + α -κ B W * (α)χ.
We equivalently have

W(α)f M 1 ψ ≤ γ f M 1 ψ + W(α)f, ϕ , uniformly for any α ≥ κ 0 , with γ := η η+κ0-κB < 1 and ϕ := M r+κ0-κB χ, which is nothing but condition (2.30).
We finally come to the existence of a solution to the first eigenvalue problem and the first eigentriplet problem.

Theorem 2.21. Under conditions (H1)-(H2)-(H3), the first eigenvalue problem (2.14) has a solution (λ 1 , f 1 ) with λ 1 satisfying (2.19). When furthermore (H3) holds for the dual problem, then the first eigentriplet problem

(1.1)-(1.2) admits a solution (λ 1 , f 1 , φ 1 ) ∈ R × X × Y .
Theorem 2.21 generalizes some known versions of the existence part of the Krein-Rutman Theorem where either L is assumed additionally to be the generator of a semigroup or to have strongly power compact resolvent or even where some additional conditions are made on the positive cone X + . As mentioned in the introduction, some possible references for these previous results are Krein-Rutman [START_REF] Kreȋn | Linear operators leaving invariant a cone in a Banach space[END_REF] We now additionally assume that (H3) holds for the dual problem. As recalled during the proof of Lemma 2.3 and by definition of λ 1 , we have (λ 1 , +∞) ⊂ ρ(L) = ρ(L * ) and λ 1 ∈ Σ(L) = Σ(L * ). Taking λ n ց λ 1 , we argue as in the proof of Lemma 2.6 and we get

∃ φ n ≥ 0, λ n φ n -L * φ n → 0, φ n = 1.
Thanks to (H3) for the dual problem, we deduce that there exist a subsequence ( φ n k ) and φ 1 ∈ X ′ , φ 1 = 1 such that φ n k → φ 1 . We thus conclude that φ 1 is a solution to the dual problem (1.2) (for the same eigenvalue λ 1 ).

Let us conclude this section by some remarks.

Remark 2.22. (1) -As seen above, the condition (H1)-(H2) for the primal and the dual problems are equivalent, and thus one only has to check (H1)-(H2)-(H3) for the primal problem and (H3) for the dual problem in order to solve the first eigentriplet problem. It is worth emphasizing that condition (H3) on the dual problem is not a consequence of the condition (H3) on the primal problem. However, as presented in Lemma 2.7, Lemma 2.8 and Corollary 2.17, there exist several natural situations where both conditions (H3) for the primal and the dual problems hold together.

(2) -Alternatively, one may also assume (H1)-(H2)-(H3) for the dual problem, and then use a more classical fixed point theorem for proving the existence of a steady state for the rescaled semigroup by using for instance the Markov-Kakutani fixed point theorem [START_REF] Kakutani | Two fixed-point theorems concerning bicompact convex sets[END_REF] as in [START_REF] Kavian | The Fokker-Planck equation with subcritical confinement force[END_REF]Thm. 5.1], the Tychonov fixed point theorem as in [START_REF] Gamba | On the Boltzmann equation for diffusively excited granular media[END_REF] or [153, Thm. 1.2] or a Birkhoff-Von Neumann type Theorem as in [START_REF] Cañizo | Harris-type results on geometric and subgeometric convergence to equilibrium for stochastic semigroups[END_REF]Thm. 6.1]. For these last techniques, we also refer to Section 3, where such a dynamical approach is adapted to the present context. One may also use the usual Doblin-Harris theory, see for instance [START_REF] Hairer | Yet another look at Harris' ergodic theorem for Markov chains[END_REF][START_REF] Cañizo | Harris-type results on geometric and subgeometric convergence to equilibrium for stochastic semigroups[END_REF] and the references therein, and Sections 8.6 and 12.2 for applications of this approach.

Discussion.

We discuss now the existence results presented in the preceding section.

For further references, let us first recall that when X is a Hilbert space and L is self-adjoint, the first eigenvalue may be simply obtained thanks to the variational problem (2.35)

λ 1 = sup f ∈X+\{0} Lf, f f 2 .
We now explain how Theorem 2.21 is a generalization of the classical Krein-Rutman theorem stated in Theorem 1.2. We thus consider a Banach lattice X such that X ++ := intX + = ∅ and an operator L such that, for κ 1 ∈ R and any κ > κ 1 , R := (κ -L) -1 : X → X is compact and R : X + \{0} → X ++ , in particular (H1) holds true. As a first step, we recall the following very classical technical lemma of the KR theory.

Lemma 2.23. Assume X ++ := int X + = ∅. For g ∈ X + and f ∈ X ++ , there exists C ≥ 0 such that g ≤ Cf .

Proof of Lemma 2.23. We argue by contradiction. Otherwise, for any n ≥ 1, we would have fg/n ∈ X c + ⊂ X c ++ and that last set is closed. Passing to the limit, we get f ∈ X c ++ , which is in contradiction with the assumption f ∈ X ++ . For a given g 0 ∈ X + \{0}, we set f 0 := Rg 0 ∈ X ++ . From Lemma 2.23, there exists C 0 ≥ 0 such that (κ -L)f 0 = g 0 ≤ C 0 f 0 . That implies that Lemma 2.4-(ii) holds with κ 0 := κ -C 0 , and thus (H2) also holds. One may then define µ 1 := κλ 1 , with

λ 1 := inf{λ ∈ R; (λ ′ -L) -1 ∈ B(X), ∀ λ ′ ∈ [λ, κ]} ≥ κ 0 .
We recall that because of Lemma 2.6 (or its proof), there exist (λ n ), ( fn ) and (ε n ) such that (2.20) holds, namely

λ n ց λ 1 , fn ≥ 0, ε n := λ n fn -L fn ≥ 0, fn = 1, ε n → 0.
We may rewrite the equation as

fn = R[ε n + (κ -λ n ) fn ],
so that ( fn ) belongs to a compact set of X because of the compactness assumption made on R, so that (H3) holds true. Because of Theorem 2.21, we deduce that there exists f 1 ∈ X + such that f 1 = 1 and Lf 1 = λ 1 . That implies f 1 = µ 1 Rf 1 , and thus that the existence part of Theorem 1.2 is a consequence of Theorem 2.21 for an operator R which is the positive resolvent of an operator L.

We would like to emphasize on the fact that our definition of the first eigenvalue by (2.15)-(2.16) bears some strong similarity with the definition of the first eigenvalue for elliptic operators in non divergence form as presented in [START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF]. Indeed, if λ ∈ I, then

∃ f ∈ X + \{0}, Lf ≤ λf.
Assuming now that X is a space of functions (defined on a set E) and that f (x) > 0 for any x ∈ E, we deduce that

λ ≥ sup E Lf f ,
and thus λ 1 is characterized by

λ 1 = inf f >0 sup E Lf f ,
which is nothing but [47, (1.13)] (with a change of sign because of a different sign convention). We thus see that our formulation is a generalization at a more abstract level and for resolvent positive operators of that classical min-max approach for elliptic operators. Some more or less classical references on that subject are [START_REF] Donsker | On a variational formula for the principal eigenvalue for operators with maximum principle[END_REF][START_REF] Donsker | On the principal eigenvalue of second-order elliptic differential operators[END_REF], [START_REF] Nussbaum | On variational principles for the generalized principal eigenvalue of second order elliptic operators and some applications[END_REF], [START_REF] Pinsky | A generalized Dirichlet principle for second order nonselfadjoint elliptic operators[END_REF], [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF] and [START_REF] Berestycki | On the definition and the properties of the principal eigenvalue of some nonlocal operators[END_REF]. In particular in [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF], two generalized principal eigenvalues

λ ′ 1 := sup{κ 0 ∈ R; ∃ g 0 ∈ C 0 Lg 0 ≥ κ 0 g 0 } and λ ′′ 1 := inf{κ 1 ∈ R; ∃ g 1 ∈ C 1 Lg 1 ≤ κ 1 g 1 } are defined for appropriate cones C i ⊂ X + \{0}
for problems with lack of compactness. The links between the three quantities λ 1 , λ ′ 1 and λ ′′ 1 are discussed as well as the possible non existence of a related principal eigenfunction f 1 . The non existence of associated principal eigenfunction should not be a surprise since it is the case when one considers L = ∆ in X = L 2 (R d ) where Lg 1 = L * g 1 = λ ′′ 1 ψ with 0 < ψ = 1 / ∈ X = X ′ and λ ′′ 1 = 0, but no associated principal eigenfunction exists in X. We also refer to [START_REF] Lions | Premières valeurs et fonctions propres. Cours du Collège de France 2020-2021[END_REF] where some examples of such a situation are discussed.

For its own interest and further discussions, we finally state and prove a slightly less general variant of Theorem 1.8. Theorem 2.24. Consider a Banach lattice with positive cone X + and a linear and bounded operator R : X → X such that (i) R :

X + → X + ; (ii) ∃ g 2 ∈ X + \{0}, ∃ C 2 > 0 such that Rg 2 ≤ C 2 g 2 . We define K 2 := {g ∈ X + ; ∃a > 0, g ≤ ag 2 }, and next A(g) := inf{a > 0; g ≤ ag 2 }, if g ∈ K 2 , as well as J := {µ ≥ 0; ∃h ∈ K 2 , h ≥ µRh + g 2 }.
We further assume (iii) µ 1 := sup J < +∞.

(iv) Any upper bounded and increasing sequences (g n ) of X is convergent in the weak sense σ(X, Y ). More precisely, if g n ≤ g n+1 ≤ ḡ ∈ X for any n ≥ 1, there exists g ∈ X, g ≤ ḡ, such that g n ⇀ g. (v) Any sequence (g n ) of normalized almost first eigenvectors is relatively compact. More precisely, for any sequence (g n ) of K 2 such that A(g n ) = 1, g n = µ n Rg n + ε n , µ n ր µ 1 and ε n → 0, there exists g ∈ K 2 and a subsequence (g n k ) such that g n k → g and A(g) = 1.

Then there exists

f 1 ∈ X 2 such that f 1 = µ 1 Rf 1 and A(f 1 ) = 1.
Proof of Theorem 2.24. We split the proof into three steps.

Step 1. We first establish that for any µ ∈ J , there exists g = gµ ∈ K 2 such that (2.36) g = µRg + g 2 .

We set g0 = 0, g1 = g 2 , and we define (g n ) recursively by gn+1 = µRg n + g 2 , for any n ≥ 1. We claim that 0 ≤ gn ≤ gn+1 ≤ h, for any n ≥ 0, where h enters in the definition of µ ∈ J . That is obviously true at order n = 0. Assuming that last inequality is proved at order n -1 for n ≥ 1, we compute

gn+1 = µRg n + g 2 ≥ µRg n-1 + g 2 = gn and gn+1 = µRg n + g 2 ≤ µRh + g 2 ≤ h,
which proves the same inequality at order n, and thus for any n ≥ 0. Using the convergence property (iv) of upper bounded increasing sequences in X, we deduce that there exists g ∈ X 2 such that gn → g and thus (2.36) holds.

Step 2. We obviously have 0 ∈ J and J is an interval because if (µ, h) satisfies the condition µ ∈ J then so do (µ

(M g) -(µ + ε)R(M g) = M g 2 -M εRg ≥ M g 2 -M εARg 2 ≥ M (1 -εAC 2 )g 2 ≥ g 2 ,
so that µ + ε ∈ J .

Step 3. We first establish by contradiction that A(g µ ) ր ∞ when µ ր µ 1 . If it was not the case, there exists A ∈ (0, ∞) and a sequence (µ n ) such that A(g µ n ) ≤ A as µ n ր µ 1 . Choosing 0 < ε ≤ 1/(2AC 2 ) and M ≥ 2 as in Step 2, the same computation gives

(M gµ n ) -(µ + ε)R(M gµ n ) ≥ g 2 ,
so that µ n + ε ∈ J . That means that µ n + ε ≤ µ 1 , and a contradiction with the fact µ n ր µ 1 . We next consider µ n ր µ 1 and we define

A n := A(g µ n ), ĝn := gµ n A n , ε n := g 2
A n , ĝn = µ n T ĝn + ε n . We observe that ε n → 0 and A(ĝ n ) = 1. Because of the compactness assumption (v), we deduce that there exists f 1 ∈ K 2 and a subsequence (ĝ n k ) such that ĝn k → f 1 and A(f 1 ) = 1. We conclude by passing to the limit in the above almost first eigenvalue equations.

We may compare Theorem 2.24 with the results presented in the previous section. When L satisfies condition (H1), we may set R := R L (κ 1 ) so that R ∈ B(X) and R satisfies (i). In that case, Theorem 2.24 claims the existence of

f 1 ∈ K 2 such that Lf 1 = λ 1 f 1 , with λ 1 := κ 1 -µ 1 .
The condition (ii) on R translates as Lg 2 ≤ (κ 1 -1/C 2 )g 2 which may be seen as an equivalent of condition (H1) (when working in the space X 2 := K 2 -K 2 with norm g 2 := A(|g|) and L generates a semigroup S. The hypothesis (iii) is nothing but (H2) and the hypothesis (iv) is very natural: it holds in the space L p (E) and M 1 (E) without additional condition on R and it holds in a space of continuous functions when some additional uniform continuity assumption is made on the range of R. Assumption (v) has to be compared with condition (H3). It is worth emphasizing that when X ⊂ L p (E) and g 2 > 0 a.e., we simply have A(g) = g/g 2 L ∞ for any g ∈ X + . As a conclusion, although Theorem 2.21 and Theorem 2.24 bear some similarities none seems to be a consequence of the other. We believe that Theorem 2.21 is more flexible since it does not impose to work with the normalization associated to the seminorm g → A(|g|) of L ∞type. It is also worth emphasizing the very similarity between Step 3 in the proof of Theorem 2.24 and the proof of Lemma 2.6 and, on the other hand, that Theorem 1.2 is a particular case of Theorem 2.24 by essentially exploiting the fundamental Lemma 2.23 as shown in [START_REF] Lions | Premières valeurs et fonctions propres. Cours du Collège de France 2020-2021[END_REF]. We finally point out that when Y = X ′ , the condition (iv) is equivalent to a property of Banach lattices known as order continuous norm, see for instance [

Existence through a dynamical approach

In this part, we develop a dynamical approach for proving the existence part of the Krein-Rutman Theorem. We thus always consider a positive semigroup S = S L on a Banach lattice X. We recover Theorem 2.21 under slightly reinforced assumptions. Above all, we are able to extend the existence part of the Krein-Rutman Theorem to a more general framework, namely to the case when L only enjoys a suitable weakly dissipative condition.

About dissipativity.

Let us start by recalling some classical definitions and results. We say that an operator L defined in a Banach space X is dissipative if there is some number κ ∈ R such that

∀ f ∈ D(L), ∃ f * ∈ J f , ℜe f * , Lf ≤ κ f 2 ,
where we define the associated dual set J f ⊂ X ′ of f by (3.1)

J f := {ϕ ∈ X ′ ; ϕ, f = f = ϕ X ′ }.
In that situation and in order to be more precise, we should say that L-κ is dissipative. It is worth emphasizing that J f = ∅ thanks to the corollary (2.2) of the Hahn-Banach dominated extension theorem. We say that an operator L is hypodissipative in a Banach space X if there exist an equivalent norm ||| • ||| in X and a number κ ∈ R such that

(3.2) ∀ f ∈ D(L), ∃ f * ∈ J f,|||•||| , ℜe f * , Lf ≤ κ|||f ||| 2 ,
where

(3.3) J f,|||•||| := {ϕ ∈ X ′ ; ϕ, f = |||f ||| 2 = |||ϕ||| 2 X ′ }.
The only difference between the two definitions (3.1) and (3.3) comes from the norms in which the normalization is performed. When L is the generator of a semigroup S L , one can show that the growth bound ω = ω(S L ) defined in (2.9) also satisfies ω = inf{κ ∈ R, (3.2) holds for some equivalent norm ||| • |||}, and S L is a semigroup of contraction when L is dissipative with κ = 0. At least formally, denoting f t := S(t)f , for f ∈ D(L), we deduce from (3.2) that 1 2

d dt |||f t ||| 2 = ℜe (f t ) * , Lf t ≤ κ|||f t ||| 2 ,
and together with the Grönwall lemma, we deduce

|||S(t)f ||| ≤ e κt |||f |||, ∀ t ≥ 0,
which is nothing but (2.10). That last estimate is actually equivalent to the hypodissipativity estimate (3.2). Quite similarly, when

(3.4) ∃ ψ ∈ Y + \{0}, ∃κ ∈ R, ±L * ψ ≤ κψ,
we may compute

± d dt f t , ψ = ± Lf t , ψ = ± f t , L * ψ ≤ κ f t , ψ ,
and together with the Grönwall lemma, we get

(3.5) ± S t f, ψ ≤ ±e ±κt f, ψ , ∀ t ≥ 0.
Two important more accurate versions of the previous ones are presented now. They will be of main importance in the sequel. On the one hand, we may assume that L satisfies a Lyapunov type condition, namely there exists ψ i ∈ Y + and κ ∈ R such that

(3.6) L * ψ 2 ≤ κψ 2 + ψ 0 ,
with ψ 2 > 0 and ψ 0 /ψ 2 → 0 at infinity. For f t = S L (t)f , f ∈ D(L) ∩ X + , a similar computation as above gives

d dt f t , ψ 2 = f t , L * ψ 2 ≤ κ f t , ψ 2 + f t , ψ 0 .
Denoting [f ] i := |f |, ψ i and using the Grönwall lemma, we classically deduce

(3.7) [S(t)f ] 2 ≤ e κt [f ] 2 + t 0 e κ(t-s) [S(s)f ] 0 ds, ∀ t ≥ 0.
The Lyapunov condition (3.6) is particularly relevant and useful in a Radon measures space framework X = M 1 ψ2 (E) for some weight function ψ 2 on E. On the other hand, we may generalize the above Lyapunov condition by assuming the structure condition (HS2) there exist a splitting

L = A + B and κ B ∈ R such that A is B-bounded, that means ∃ C ≥ 0, ∀ f ∈ X, Af ≤ C( f + Bf ),
the operator B generates a semigroup S B and

(3.8) (S B A) ( * ℓ) * S B (t) B(X) = O(e αt ), ∀ t > 0,
for any ℓ ≥ 0 and α > κ B .

Here and below, for two functions U : R + → B(X 0 , X 1 ) and V : R + → B(X 1 , X 2 ), we define the convolution function

(V * U )(t) := t 0 V (t -s)U (s) ds,
when the integral is well-defined. For U : R + → B(X ), we also recursively define U ( * 0) = I and U ( * (ℓ+1)) = U ( * ℓ) * U . Using this convolution notation, the Duhamel formula writes

S L = S B + S B A * S L ,
and iterating this formula, for any N ≥ 1, we get the following iterated Duhamel formula (3.9)

S L = S B + • • • + (S B A) * (N -1) * S B + (S B A) ( * N ) * S L .
When S L is well defined in another space X 0 ⊃ X and the last iterated convolution term enjoys the regularity property (S B A) ( * N ) (t) B(X0,X) = O(e αt ) for all t > 0 and α > κ B , we deduce from the above iterated Duhamel formula, the estimate

(3.10) S(t)f ≤ C 0 e αt f + C 1 t 0 e α(t-s) S(s)f 0 ds, ∀ t ≥ 0, α > κ B , ∀ f ∈ X,
for some constants C i ≥ 1 and where • 0 stands for the norm in X 0 . We may observe that the estimate (3.7) in the case of a Lyapunov condition is a particular case of (3.10) corresponding to the norms

Existence in the dissipative case.

In this section, we give an existence result for a positive semigroup S L on a Banach lattice X satisfying a kind of regularity/compactness assumption in the spirit of the structure condition (HS2) discussed above.

Theorem 3.1. On a Banach lattice X = Y ′ , with Y separable Banach lattice, consider a positive semigroup S = S L satisfying the growth bound (2.10), and set κ 1 := ω ′ + log M for some ω ′ > ω(S L ).

We assume

(1) ∃ φ 0 ∈ Y + \{0}, ∃ κ 0 ∈ R such that [S(t)f ] 0 ≥ e κ0t
[f ] 0 for any t ≥ 0 and f ∈ X + , where we denote [f ] 0 := |f |, φ 0 ;

(2) there exist κ, C 0 , C 1 ∈ R with κ < κ 0 , C 0 ≥ 1 and C 1 ≥ 0, such that

(3.11) S(t)f ≤ C 0 e κt f + C 1 t 0 e κ(t-s) [S(s)f ] 0 ds, ∀ t ≥ 0, ∀ f ∈ X.
Then there exist λ 1 ∈ [κ 0 , κ 1 ] and

f 1 ∈ X + \{0} such that Lf 1 = λ 1 f 1 .
Let us mention that this result shares similarities with [260, Cor. 2.7] and [START_REF] Collet | Quasi-stationary distributions for structured birth and death processes with mutations[END_REF]Thm. 4.2], see also [START_REF] Lotz | Positive linear operators on L p and the Doeblin condition[END_REF][START_REF] Caselles | On the peripheral spectrum of positive operators[END_REF] for earlier works in that direction.

Remark 3.2. (1) Assumption (2) in the statement of Theorem 3.1 holds when there exist V, W such that

(3.12) S = V + W * S, W ≥ 0,
and there exist κ, C V , C W ∈ R, κ < κ 0 , C V ≥ 1, C W > 0 such that (3.13) V (t) B(X) ≤ C V e κt , W (t) B(X0,X) ≤ C W e κt .
(2) Under the structural condition (HS2) together with some regularization effect on the semigroup of the type

(S B A) ( * N ) (t) B(X0,X) = O(e κt ), ∀ t > 0, κ ∈ (κ B , κ 0 ),
we recover the above condition (3.12)-(3.13) with

(3.14) V := S B + • • • + (S B A) * (N -1) * S B , W := (S B A) ( * N ) ,
because of the iterated Duhamel formula (3.9). In that case, the representation formula (2.13) holds true for any z > λ 1 from Lemma 2.2-(ii) and we easily compute

R L (z) = V(z) + W(z)R L (z), ∀ z > λ 1 , with V(z) := ∞ 0 e -λt V (t)dt, W(z) := ∞ 0 e -λt W (t)dt, ∀ z > κ.
We observe that W satisfies (2.28) in Lemma 2.15 if W satisfies (3.13) and the set C defined by (2.26) satisfies the same compactness properties as required in the statement of Lemma 2.11. We may thus apply Lemma 2.15 (see also Remark 2.16) and deduce that (H3) holds for the primal problem. We finally obtain the same conclusion as in Theorem 3.1 thanks to Theorem 2.21.

(3) Under the same structural condition (HS2) as above, but assuming now that

W (t) B(X,X1) = O(e κt ), ∀ t > 0, κ ∈ (κ B , κ 0 ),
with W := (S B A) ( * N ) and X 1 ⊂ X with strongly compact embedding, we observe that S does not necessary satisfies the assumptions of Theorem 3.1, but it rather satisfies the assumptions of Lemma 2.7 with K T := (W * S)(T ) and T > 0 large enough. In that situation, we also obtain the same conclusion as in Theorem 3.1 thanks to Lemma 2.7 and Theorem 2.21.

Proof of Theorem 3.1. We split the proof into two steps.

Step 1. We define the set

C := {f ∈ X + , [f ] 0 = 1, f ≤ R},
for a convenient constant R > 0 to be fixed later. For any fixed t > 0, we next define the nonlinear weakly σ(X, Y ) continuous mapping

Φ t : C → X, f → S t f [S t f ] 0 .
Thanks to assumption (1), we may observe that it is well defined because

(3.15) [S t f ] 0 ≥ e κ0t [f ] 0 = e κ0t > 0.
For any f ∈ C, we thus immediately have Φ t f ≥ 0 and [Φ t f ] 0 = 1. On the other hand, from assumption (1) again and the semigroup property, we have

(3.16) [S(t)f ] 0 ≥ e κ0(t-s) [S(s)f ] 0 .
For f ∈ C and t ≥ 0, we next compute

Φ t f ≤ C 0 e -αt f + C 1 t 0 e -α(t-s) ds ≤ C 0 e -αt R + C 1 α ,
where we have set α := κ 0κ B > 0. Fixing T 0 such that C 0 e -αT0 ≤ 1/2 and next R ≥ 2C 1 /α, we have thus Φ T0 : C → C. Thanks to the Tykonov fixed point Theorem, there exists f T0 ∈ C such that Φ T0 f T0 = f T0 . In other words, we have established the existence of f T0 ∈ X such that (3.17)

f T0 ≥ 0, [f T0 ] 0 = 1, S T0 f T0 = e λ1T0 f T0 , with λ 1 := (1/T 0 ) log[S T0 f T0 ] 0 ∈ [κ 0 , κ 1 ].
Step 2. Rewriting equation (3.17) as

0 = e -λ1T0 S T0 f T0 -f T0 = (L -λ 1 )
T0 0 e -λ1t S t f T0 dt and defining

f 1 := T0 0 e -λ1t S t f T0 dt, we get that f 1 ∈ X + \{0} satisfies Lf 1 = λ 1 f 1 .
We present now a second proof based on a large times dynamical argument which is classical in the mean ergodicity theory of Von Neumann and Birkhoff introduced in [START_REF] Neumann | Proof of the quasi-ergodic hypothesis[END_REF][START_REF] Birkhoff | Proof of the ergodic theorem[END_REF] and which will be adaped in the weak dissipativity case in Section 3.5 below.

Alternative

Step 2. We define S t := S t e -λ1t , so that f T0 becomes a periodic state for S t from (3.17), namely

S t f T0 = S t-kT0 f T0 , k := [t/T 0 ], ∀ t > 0.
Using (3.15) and the above relation, we have

[ S t f T0 ] 0 = [ S t-kT0 f T0 ] 0 ≥ e (κ0-λ1)(t-kT0) [f T0 ] 0 ≥ e (κ0-λ1)T0 =: r * > 0,
for any t ≥ 0. On the other hand, thanks to the growth bound (2.10), we have

S t f T0 = S t-kT0 f T0
≤ M e (κ-λ1)(t-kT0) f T0 ≤ M e (κ-λ1)T0 R =: R * < ∞, for any t ≥ 0. We finally define

u T := 1 T T 0 S t f T0 dt.
From the previous estimates, both sequences ( S t f T0 ) and (u T ) are bounded in

K := {f ∈ X; f ≥ 0, [f ] 0 ≥ r * , f ≤ R * }.
By compactness, there exists a subsequence (u T k ) and f 1 ∈ K such that u T k ⇀ f 1 in a weak sense as k → ∞. For any fixed t > 0, we observe that

S t f 1 -f 1 = lim k→∞ 1 T k T k 0 S t S s f T0 ds - 1 T k T k 0 S s f T0 ds = lim k→∞ 1 T k T k +t T k S s f T0 ds - 1 T k t 0 S s f T0 ds = 0,
where we have used that ( S s f T0 ) is uniformly bounded in the last line. As a consequence, f 1 is a stationary state for the rescaled semigroup S t , and thus an eigenfunction associated to the eigenvalue λ 1 for the operator L.

About weak dissipativity.

In this section, we recall some definitions and results about the weak dissipativity. We say that the generator B of a semigroup S B is weakly dissipative in a Banach space X i if there exist a second Banach space X i-1 ⊃ X i and some numbers κ ∈ R and σ > 0 such that

∀ f ∈ D(B |Xi ), ∃ f * ∈ J f,Xi , f * , Bf ≤ κ f 2 Xi -σ f 2 Xi-1
, where we define the associated dual set J f,Xi ⊂ X ′ i of f (for the norm • Xi ) by (3.18)

J f,Xi := {ϕ ∈ X ′ i ; ϕ, f = f 2 Xi = ϕ 2 X ′ i }.
By translation, we may assume that κ = 0, an hypothesis we will always make in the sequel of this section. We will furthermore assume the splitting structure L = A + B with A a B-bounded operator and B weakly dissipative generator. More precisely, we assume that there exists one more Banach lattice X 0 ⊃ X 1 ⊃ X 2 := X, with norm denoted by • k := • X k , such that B generates a semigroup and is weakly dissipative in each X k : for any k = 1, 2

(3.19) ∀ f ∈ D(B |X k ), ∃ f * ∈ J f,X k , f * , Bf X ′ k ,X k ≤ -σ f 2 k-1 .
This classically implies (or we can take the next inequality as a definition of the weak dissipativity condition) that

(3.20) d dt S B (t)f k + σ S B (t)f k-1 ≤ 0, ∀ t ≥ 0, ∀ f ∈ X k , ∀ k = 1, 2.
We assume that X k is dense into X k-1 for k = 1, 2 and that X 1 is an interpolated space between X 0 and X 2 in the sense that there exists a continuous and strictly decreasing function η

: (0, 1] → [0, ∞), η(ε) → ∞ when ε → 0, η(1) = 0, such that (3.21) f 1 ≤ ε f 2 + η(ε) f 0 , ∀ ε ∈ (0, 1], ∀ f ∈ X 2 .
From (3.20) with k = 2, we deduce

(3.22) S B (t)f 2 ≤ f 2 , ∀ t ≥ 0, ∀ f ∈ X 2 .
Next, for k = 1, gathering the weak dissipativity condition (3.20), the interpolation condition (3.21) and the non expansion property (3.22) in the space X 2 , we get

d dt S B (t)f 1 + σ η(ε) S B (t)f 1 ≤ σε η(ε) S B (t)f 2 ≤ σε η(ε) f 2 ,
for any t ≥ 0, ε ∈ (0, 1) and f ∈ X 2 . We deduce

d dt S B (t)f 1 e σ η(ε) t ≤ σε η(ε) e σ η(ε) t f 2 ,
and thanks to the Grönwall lemma, we obtain

(3.23) S B (t)f 1 ≤ Θ(t) f 2 ,
for any t ≥ 0 and f ∈ X 2 , with

(3.24) Θ(t) := inf ε∈(0,1) e -σ η(ε) t + ε → 0 as t → +∞.
On the other hand, using the representation formula

R B (z)f = ∞ 0 e -zt S B (t)f dt, ∀ z ∈ ∆ 0 , ∀ f ∈ X 2 ,
together with (3.20), we get

σ R B (z)f 1 ≤ ∞ 0 σ S B (t)f 1 dt ≤ f 2 ,
for any z ∈ ∆ 0 and f ∈ X 2 . We next assume that

(3.25) Θ(t) -1 AS B (t)f 1 + ∞ 0 AS B (t)f 1 dt f 1 , that there exist α > 1, N ≥ 1, C ≥ 1 such that (3.26) sup x+iy∈∆0 AR 1+ε1 B (x + iy) . . . AR 1+εN B (x + iy)f 2 ≤ C y α f 2 , for any ε ∈ {0, 1} N , ε 1 + • • • + ε N ≤ 1, and that (3.27) sup z∈∆0 (R B (z)A) N f X1 ≤ f 1 ,
with X 1 compactly imbedded in X 1 . The necessity to add (ε i ) in (3.26) is probably purely technical and not restrictive for applications. In examples, we can take N = 2N ′ , when

(3.28) sup x+iy∈∆0 (AR B ) N ′ (x + iy)f 3 ≤ C y α f 2 ,
for some convenient space X 3 such that A : X 1 → X 3 and sup z∈∆0 R B (z) B(X3,X2) < ∞. At the level of the semigroup, (3.28) is typically a consequence of

(AS B ) ( * N ′′ ) (t) B(X2,X ζ 3 ) ∈ L 1 (R + ), with ζ > 0, where X ζ 3 := {f ∈ X 3 , L ζ f ∈ X 3
} stands for the (possibly fractional) domain for the operator defined in X 3 . However, (3.26) is a bit more general than that last estimate. We refer to [START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Mischler | Erratum: Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Mischler | Relaxation in time elapsed neuron network models in the weak connectivity regime[END_REF][START_REF] Mischler | Weak and strong connectivity regimes for a general time elapsed neuron network model[END_REF] for precise definition, examples and discussion. For further references, we observe that (3.23) and (3.25) together imply

1 T T 0 (S B * AS B )(t)f 1 dt ≤ 1 T T 0 t 0 S B (t -s)AS B (s)f 1 dsdt ≤ 1 T T 0 T 0 S B (u) B(X2,X1) AS B (s)f 1 duds 1 T T 0 Θ(u) du f 2 .
Arguing in a similar way for any ℓ ≥ 1, we establish

(3.29) 1 T T 0 (S B * (AS B ) ( * ℓ) )(t)f 1 dt 1 T T 0 Θ du f 2 → 0 as T → ∞.
For synthesizing and for further references, let us now bring out some possible general framework for semigroup enjoying weak dissipativity. We introduce the following structure condition on a semigroup S L and its generator L by assuming (HS3) there exist a splitting L = A + B, some Banach lattices X 2 ⊂ X 1 , an integer N ≥ 1 and some decaying functions Θ i : R + → R + with Θ 1 (t) → 0 as t → ∞, Θ 2 ∈ L 1 (R + ) such that A is positive, B generates a positive semigroup S B and the following estimates hold

(S B A) ( * ℓ) * S B B(X2,X1) = O(Θ 1 ), ∀ ℓ ∈ {0, . . . , N -1}, (3.30) (S B A) ( * N ) B(X1,X2) = O(Θ 2 ). (3.31)
We now particularize our discussion to a Radon measures framework. We assume that there exist some weight functions

ψ i on E, ψ 0 ψ 1 ≤ ψ 2 , with ψ 2 (x)/ψ 1 (x) → ∞ as x → ∞ so that M 1 ψ2 ⊂⊂ M 1 ψ1 (compact imbedding for the weak convergence), a function χ ∈ C c (E), 0 ≤ χ ≤ 1, and a constant M ≥ 0 such that (i) L * ψ 1 ≤ -ψ 0 + M χ; (ii) L * ψ 2 ≤ M χ; (iii) ψ 1 ≤ εψ 2 + η(ε)ψ 0 for any ε > 0, for a function η : (0, 1] → (0, ∞) such that η(1) = 0, η(ε) → ∞ when ε → 0, and (3.32) t → Θ(t) := inf ε∈(0,1) e -t η(ε) + ε ∈ L 1 (0, ∞).
It is worth emphasizing that from the very definition, we have automatically that Θ is positive and decreasing, Θ(0) = 1 and Θ(t) → 0 as t → ∞. Arguing similarly as we did during the proof of Corollary 2.20 and the end of Section 3.1, we introduce the splitting

A := M χ, B := L -A,
and we establish that S B is a positive semigroup on X = M 1 ψ2 (E). More precisely, for 0 ≤ f 0 ∈ D(B) in the domain of S B and denoting f t := S B (t)f 0 , we may compute

d dt f t ψ 2 ≤ f t B * ψ 2 ≤ 0 and similarly d dt f t ψ 1 ≤ f t B * ψ 1 ≤ -f t ψ 0 .
Integrating both differential inequalities, we deduce

S B ∈ L ∞ t (B(M 1 ψi )), i = 1, 2 and ∞ 0 S B (t)f 0 M 1 ψ 0 dt ≤ f 0 M 1 ψ 1 , ∀ f 0 ∈ M 1 ψ1 .
We may make a slight (but important) improvement of the previous estimate by proceeding similarly as we did for proving (3.23). Using the same notations as in the above computation, we indeed have

d dt f t ψ 1 + 1 η(ε) f t ψ 1 ≤ ε η(ε) f t ψ 2 ≤ ε η(ε) f 0 ψ 2 ,
where we have used (i) and (iii) in the first inequality and the previous L ∞ t (B(M 1 ψ2 )) bound in the second inequality. Integrating in time, we deduce

S B (t)f M 1 ψ 1 ≤ Θ(t) f M 1 ψ 2 , ∀ t > 0.
Taking X i := M 1 ψi and N = 1, we see that L then satisfies (HS3) with Θ i = Θ.

3.4. First existence result in the weakly dissipative case. We first come back to the proof of Theorem 2.21 and explain what goes wrong when we try to adapt it to a weak dissipativity context. More precisely, we assume that S L is a positive semigroup (so that (H1) holds) satisfying L * φ 0 ≥ 0 for some φ 0 ∈ X ′ \{0} (so that (H2) holds) and the splitting structure (HS3) for some bounded operator A and some weakly dissipative operator B, in the sense that (3.19) holds. In such a situation, we may define

λ 1 := inf{λ ∈ R; R L (κ) ∈ B(X), ∀ κ ≥ λ} ≥ 0,
and there exist sequences (λ n ) of R and ( fn ) of X + such that

λ n ց λ 1 ≥ 0, fn = 1, ε n := λ n fn -L fn → 0 in X,
thanks to Lemma 2.6. In the simplest situation, we may further assume that R B (κ) : X 1 → X 0 is uniformly bounded in κ ≥ λ 1 and A : X 0 → X 1 with X = X 1 ⊂ X 0 . The issue is that even in that case, we may write fn

= R B (λ n )A fn + R B (λ n )ε n ,
but it is not clear how to conclude that ( fn ) belongs to a compact set in

X because it is not clear that R B (λ n )ε n → 0 in X.
The next result aim precisely to establish that last convergence under suitable quite strong (although natural and true in some examples) assumptions on the operator L. The proof is adapted from [231, Sec. 6.3] and mixes some dynamical argument together with the stationary approach developed in Section 2.2.

Theorem 3.3. Consider a positive semigroup S L in a Banach lattice X = X 2 ⊂ X 1 ⊂ X 0 such that its generator L satisfies (1) there exists φ 0 ∈ D(L * ), φ 0 ≥ 0, φ 0 = 0, such that L * φ 0 ≥ 0; Then, there exist λ 1 ≥ 0 and f 1 ∈ X 1 such that

(3.33) f 1 X1 = 1, f 1 ≥ 0, Lf 1 = λ 1 f 1 .
Proof of Theorem 3.3. We split the proof into four steps.

Step 1. We know from Lemma 2.2 and Lemma 2.4-(i) that (H1) and (H2) hold. We may then define λ 1 ≥ 0 with the help of ( In the sequel, we always assume λ 1 = 0.

Step 2. Let us fix f 0 ∈ D(L) such that f 0 ≥ 0 and C 0 := f 0 , φ 0 > 0, which exists by definition of φ 0 . Denoting f (t) := S L (t)f 0 , we have

d dt f (t), φ 0 = Lf (t), φ 0 = f (t), L * φ 0 ≥ 0, which in turns implies f (t), φ 0 ≥ C 0 , ∀ t ≥ 0.
Step 3. We claim that R L (0) B(X2,X1) = +∞. That in particular implies R L (0) B(X) = +∞ and thus 0 ∈ Σ(L). We assume by contradiction that K 2,1 := R L (0) B(X2,X1) < +∞. First, because S L is positive, we have

|R L (z)f | ≤ ∞ 0 e -tℜez S L (t)|f | dt = |R L (ℜez)|f |, from which we deduce R L (z) B(X2,X1) ≤ R L (ℜez) B(X2,X1) , ∀ z ∈ ∆ 0 .
As a consequence, we have

(3.34) sup y∈R R L (iy) B(X2,X1) ≤ K 2,1 .
We write the representation formulas (taken from [278, (2.21)])

S L (t)f = T 0 (t) + lim M→∞ T 1,M (t)
with

T 0 (t) := N -1 ℓ=0 S B * (AS B ) ( * ℓ) (t)f and T 1,M (t) := i 2π a+iM a-iM e zt R L (z) (AR B (z)) N f dz,
for any f ∈ D(L), t ≥ 0 and a > 0. On the one hand, from (3.29), we have the Cesàro mean convergence

(3.35) 1 T T 0 T 0 (t) dt → 0 in X 1 , as T → ∞.
On the other hand, we estimate the contribution of T 1,M . Integrating by part, we have

T 1,M (t) = 1 t i 2π a+iM a-iM e zt d dz R L (z) (AR B (z)) N f dz, with d dz R L (z) (AR B (z)) N = ε∈N N +1 , |ε|=1 R L (z) 1+ε0 AR 1+ε1 B (z) . . . AR 1+εN B (z).
Together with condition (3.26) and estimate (3.34), we get

d dz R L (z) (AR B (z)) N f 1 ≤ (K 2,1 + K 2 2,1 )N sup ε∈N N , |ε|≤1 AR 1+ε1 B (z) . . . AR 1+εN B (z)f 2 ≤ C 1 y α f 2 ,
uniformly for any z = x + iy ∈ ∆ 0 , for some constant C 1 > 0. We deduce

(3.36) lim M→∞ T 1,M (t) 1 ≤ 1 t 1 2π R C 1 y α dy f 2 → 0,
as t → ∞. Gathering (3.35) and (3.36), we conclude in particular that

1 T T 0 S L (t)f 0 dt → 0 in X 1 , as T → ∞,
which is in contradiction with the estimate of Step 2.

Step 4. Conclusion. Taking advantage of the convenient blow up of R L (λ) as λ ց 0 established in the previous Step 3, we may now argue similarly as in the proof of Theorem 2.21. From Step 2, there exists a sequence (λ n ) such that λ n → 0 and

R L (λ n ) B(X2,X1) → ∞.
That means that there exist (f n ) and (g n ) such that

f n X1 → ∞, g n X2 = 1, f n = R L (λ n ) g n ,
or equivalently that there exist ( fn ) and (ε n ) (by defining fn :=

f n± / f n± X1 , ε n := g n± / f n± X1 ) satisfying (3.37) fn X1 = 1, fn ≥ 0, ε n X2 → 0, ε n = (λ n -L) fn .
As in the proof of Lemma 2.8, we deduce that (2.25) holds, that is

(3.38) fn = N -1 ℓ=0 (R B (λ n )A) ℓ R B (λ n )ε n + (R B (λ n )A) N fn .
Using the uniform boundedness

(R B (λ n )A) ℓ R B (λ n ) ∈ B(X 2 , X 1 ), (R B (λ n )A) N ∈ B(X 1 , X 1 ), X 1 ⊂⊂ X 1 ,
we deduce that ( fn ) belongs to a compact set of X 1 , or in other words, that there exist a subsequence of ( fn ) (not relabeled) and f 1 ∈ X 1 such that fn → f 1 in X 1 . We may pass to the limit in (3.37), and we get (3.33).

3.5.

Second existence result in the weakly dissipative case. Using a pure dynamical approach adapted from the second proof of Theorem 3.1 and from [81, Thm. 6.1], we establish a second existence result which is less demanding in terms of conditions on the semigroup S L .

Theorem 3.4. Consider a positive semigroup S = S L on a Banach lattice X = Y ′ for a separable Banach lattice Y . We assume (i) there exists

φ 0 ∈ Y + \{0} such that [S t f ] 0 ≥ [f ] 0 for any f ∈ X + and f → [f ] 0 := |f |, φ 0 is a norm on X.
We then denotes X the vector space X endowed with this norm

[•] 0 ; (ii) there exist v ∈ L ∞ (R + ; B(X)) and 0 ≤ w ∈ L 1 (R + ; B(X , X)) such that (3.39) S = v + w * S,
and we set

(3.40) M := sup t≥0 v(t) B(X) < ∞, Θ(t) := w(t) B(X ,X) ∈ L 1 (R + ).
Then there exists a pair (2) By definition of the norm [•] 0 of X , we see that X is a weighted L 1 space or a weighted Radon measures space. In many applications, when both X and X are Radon measures spaces, one can choose N = 1. On the other hand, when X is for instance a (possibly weighted) L p space with p > 1, one must take N ≥ 2 in most of the applications. In condition (ii), the first bound is not really demanding and almost automatic in view of the estimates exhibited in Section 3.3. The second bound is a kind of regularity estimate reminiscent of the enlarging and shrinkage technique developed in [START_REF] Mouhot | Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials[END_REF][START_REF] Gualdani | Factorization of non-symmetric operators and exponential Htheorem[END_REF][START_REF] Mischler | Exponential stability of slowly decaying solutions to the kinetic-Fokker-Planck equation[END_REF].

(λ 1 , f 1 ) ∈ R + × X + \{0} such that Lf 1 = λ 1 f 1 .
Proof of Theorem 3.4. We split the proof into three steps.

Step 1. We define R := max(2 Θ L 1 , g 0 ), for some g 0 ∈ X + such that [g 0 ] 0 = 1, and next the nonempty convex and compact (in the weak * σ(X, Y ) sense) set

C := {f ∈ X + ; [f ] 0 = 1, f ≤ R},
as well as the increasing function

λ(t) := inf f ∈C [S(t)f ] 0 , ∀ t ≥ 0.
We have the alternative

• (1) sup λ > 2M , • (2) sup λ ≤ 2M .
Step 2. We assume that the first term (1) of the alternative holds true, or in other words, there exists T 0 > 0 such that

(3.42) ∀ f ∈ C, [S T0 f ] 0 ≥ 2M.
We define as before

Φ T0 f := S T0 f [S T0 f ] 0 , ∀ f ∈ C.
By construction, for any f ∈ C, we have Φ T0 f ≥ 0 and [Φ T0 f ] 0 = 1. On the other hand, using the splitting structure (3.39) and the estimates (3.40), we have

S(t)f ≤ M f + t 0 Θ(t -s)[S(s)f ] 0 ds.
From hypothesis (i) and the semigroup property, we also have

[S t f ] 0 ≥ [S s f ] 0 , ∀ t ≥ s ≥ 0.
The two above estimates together imply

Φ T0 f ≤ M f [S T0 f ] 0 + T0 0 Θ(T 0 -s) [S s f ] 0 [S T0 f ] 0 ds ≤ 1 2 f + Θ L 1 ≤ R,
for any f ∈ C. We have thus proved Φ T0 : C → C. Thanks to the Tykonov fixed point Theorem, there exists f T0 ∈ C such that Φ T0 f T0 = f T0 . In other words, we have built a pair of "almost eigenvalue and eigenfunction"

f T0 ≥ 0, [f T0 ] 0 = 1, S T0 f T0 = e λ1T0 f T0 , with e λ1T0 = [S T0 f ] 0 and thus λ 1 ∈ [0, κ 1 ]
. We conclude to the existence of f 1 ∈ C such that Lf 1 = λ 1 f 1 really similarly as in Step 2 of the Second proof of Theorem 3.1.

Step 3. We assume that the second term (2) of the alternative holds true. In that case, for any n ≥ 1, there exists

f n ∈ C such that [S(n)f n ] 0 ≤ 2M . By compactness, there exists f 0 ∈ C and a subsequence (f n k ) such that f n k ⇀ f 0 ∈ C and ∀ t ≥ 0, ∀ k (n k ≥ t), [S(t)f n k ] 0 ≤ [S(n k )f n k ] 0 ≤ 2M, so that (3.43) ∀ t ≥ 0, [S(t)f 0 ] 0 ≤ 2M.
Using this particular initial datum, we argue similarly as in [81, proof of Theorem 6.1], and we conclude to the existence of a stationary state. More precisely, we come back to the splitting structure (3.39) of the semigroup S and we introduce the associated Cesàro means

(3.44) U T := 1 T T 0 S(t) dt, V T := 1 T T 0 v(t) dt, K T := 1 T T 0 (w * S)(t) dt,
for any T > 0. We obviously have

V T B(X) ≤ 1 T T 0 v(t) B(X) dt ≤ M.
On the other hand, we have thanks to the Fubini theorem and the positivity of the two operators involved in this integral formula. We deduce

K T f 0 ≤ T 0 w(τ )dτ 1 T T 0 S(s) f 0 ds ≤ ∞ 0 w(τ ) B(X ,X) dτ 1 T T 0 S(s)f 0 ds 0 = Θ L 1 [U T f 0 ] 0 ,
thanks to assumption (ii), so that K T f 0 is uniformly bounded in X thanks to (3.43) and the elementary estimate

[U T f 0 ] 0 ≤ [S T f 0 ] 0 . We then deduce that U T = V T + K T satisfies U T f 0 ≤ M f 0 + 2M Θ L 1 and 1 ≤ [S T f 0 ] 0 ≤ 2M,
for any T > 0. By compactness, there exists T k → +∞ and

f 1 ∈ X + such that U T k f ⇀ f 1 weakly * in X.
Thanks to the second inequality, we have [f 1 ] 0 ≥ 1. We then argue thanks to the usual mean ergodic theorem trick. For any fixed s > 0, we observe that

S(s)f 1 -f 1 = lim k→∞ 1 T k T k 0 S(s)S(t)f 0 dt - 1 T k T k 0 S(t)f 0 dt = lim k→∞ 1 T k T k +s T k S(t)f 0 dt - 1 T k s 0 S(t)f 0 dt
weakly * in X. By the lower semicontinuous property of the norm [•] 0 , we deduce

[S(s)f 1 -f 1 ] 0 ≤ lim inf k→∞ 1 T k T k +s T k [S(t)f 0 ] 0 dt + 1 T k s 0 [S(t)f 0 ] 0 dt = 0,
so that f 1 is a stationary solution, and thus f 1 is an eigenfunction associated to the eigenvalue λ 1 = 0.

Irreducibility and geometry of the first eigenvalue

In this section, we are concerned with the geometric part of the Krein-Rutman theorem for an unbounded operator L on a Banach lattice X. We assume that the conclusions of the existence part are achieved, namely (C1) the first primal and dual eigenvalue problem has a solution (λ 1 , f 1 , φ 1 ): there exist

λ 1 ∈ R, f 1 ∈ X + ∩ D(L)\{0}, φ 1 ∈ Y + ∩ D(L * )\{0} such that (4.1) Lf 1 = λ 1 f 1 and L * φ 1 = λ 1 φ 1 .
By construction, we also have Σ(L) ⊂ {z ∈ C, ℜe(z) ≤ λ 1 }.

Assuming that S is positive as for the existence part and an additional strong maximum principle property, we analyze the first eigenvalue problem.

4.1. More about positivity. For further references, we introduce several notions which are strongly related to the positivity property for semigroups.

The signum operator sign. In a real Banach lattice X, we say that sign f ∈ B(X, X ′′ ) is a signum operator for f ∈ X, if it satisfies the following properties

(sign f ) f = |f |, (4.2) (sign f ) g ≤ |g|, ∀ g ∈ X. (4.3)
In the sequel, we will always assume that such an operator exists. We refer to [15, Sec. C.I & C.II] for a general introduction to the topic. In practice, we will only need a weak formulation of the sign operator (see below) which may be defined only in some subspace X ⊂ X. We always additionally assume that the signum operator satisfies

(sign (-f )) (-g) = (sign f )g, ∀ g ∈ X, (sign f ) g = g, ∀ g ∈ X, if f ∈ X + ,
We also define

sign + f := 1 2 I + signf .
• When X is a space of functions, the sign operator sign f associated to f ∈ X corresponds to the multiplication by the function sign f := 1 f >0 -1 f <0 . When X := L p (E), we obviously see that sign f ∈ B(L p (E)) for any f ∈ L p (E). On the other hand, when X := C 0 (E), we only have

sign f ∈ B(C 0 (E); M ∞ (E))
, where M ∞ (E) denotes the space of uniformly bounded measurable functions, so that M ∞ (E) ⊂ (C 0 (E)) ′′ . In a space of bounded measures X = M 1 (E), we may define the sign operator by means of the Radon-Nikodym theorem. For a given f ∈ M 1 (E), using Hahn decomposition, there exists indeed a measurable function α : E → {-1, 1} such that f = α|f |, and we then define (sign f )g = αg for any g ∈ M 1 (E).

• When X is σ-order complete, in the sense that any increasing and upper bounded sequence has a supremum (a common least upper bound), the operator sign exists and is more regular, namely sign f ∈ B(X) for any f ∈ X, see [START_REF] Nagel | An abstract Kato inequality for generators of positive operators semigroups on Banach lattices[END_REF] and also [START_REF] Arendt | One-parameter semigroups of positive operators[END_REF]Sec. C.I.8]. We recover in particular that sign f ∈ B(L p (E)) for any f ∈ L p (E).

Weak principle maximum and Kato's inequality. We introduce now two definitions formulated on an operator L which are almost equivalent to the positivity property of the semigroup S when L is the generator of S.

• We say that the operator L satisfies the weak maximum principle when (4.4) κ ∈ R, f ∈ D(L) and (κ -L)f ≥ 0 imply f ≥ 0;

• We say that the operator L satisfies Kato's inequality when

(4.5) ∀ f ∈ D(L), L|f | ≥ (sign f )Lf.
Since |f | does not necessarily belong to D(L), the correct way to understand Kato's inequality is

(4.6) ∀ f ∈ D(L), ∀ ψ ∈ D(L * ) ∩ Y + , |f |, L * ψ ≥ (sign f )Lf, ψ .
We immediately see from the definitions that (4.5) is equivalent to assume

(4.7) ∀ f ∈ D(L), Lf + ≥ (sign + f )Lf.
Remark 4.1. We complement Lemma 2.1, by claiming that for a semigroup S = S L on a Banach lattice X, there is equivalence between the fact that S is positive and κ -L satisfies the weak maximum principle for any κ > ω(L), what is straightforward using that these properties are equivalent to the fact that R L (κ) ≥ 0 for any κ > ω(L). These properties also imply that Kato's inequality holds true, see [START_REF] Nagel | An abstract Kato inequality for generators of positive operators semigroups on Banach lattices[END_REF][START_REF] Arendt | Generators of positive semigroups[END_REF]

(sign f )Lf = (sign f ) lim t→0 S t f -f t ≤ lim t→0 S t |f | -|f | t = L|f |,
where we have used the very definition of the generator L and the properties (4.2)-(4.3) of sign f in the inequality.

We end this section by introducing other notions of positivity which strengthen the previously defined positivity condition.

Strict order. We may define a first stronger order > (or <) on X by writing for f ∈ X

f > 0 if f ∈ X + \{0}
and similarly a stronger order > (or <) on X ′ by writing for φ ∈ X ′ φ > 0 if φ ∈ X ′ + \{0}. We may next define the strict (and stronger) order ≫ (or ≪) on X by writing for f ∈ X

(4.8) f ≫ 0 or f ∈ X ++ iff ∀ ψ ∈ X ′ + \{0}
, ψ, f > 0, and similarly the strict order ≫ (or ≪) on X ′ by writing for φ ∈ X ′ (4.9)

φ ≫ 0 or φ ∈ X ′ ++ iff ∀ g ∈ X + \{0}, φ, g > 0.
On the two Banach lattices X and Y , we thus have three positivity notions with ≫ (associated to X ++ and Y ++ ) stronger than > (associated to X + \{0} and Y + \{0}) which itself is stronger than ≥ (associated to X + and Y + ). Let us comment on the notion of strict positivity. Remark 4.2. When X = Y ′ for instance, there are two possible strict positivity notions on X given by (4.8) for the space X (namely φ ∈ X is stricly positive when ξ, ψ > 0 for any ξ ∈ X ′ + \{0}) and by (4.9) for the space Y (namely φ ∈ X is stricly positive when ψ, g > 0 for any g ∈ Y + \{0}). They clearly coincide when X is reflexive, but in general the first one is stronger than the second one. In that situation, we will always consider that X is endowed with the weakest "dual" strict order (4.9).

Examples 4.3. In the space C 0 (E), the strict order is defined by f ≫ 0 iff f (x) > 0 for any x ∈ E. In a space L p (E, E , µ), 1 ≤ p ≤ ∞, the strict order is defined by f ≫ 0 iff f (x) > 0 for µ-a.e. x ∈ E. In the space M 1 (E) = C 0 (E) ′ , the strict order is defined by duality by f ≫ 0 iff f, ϕ > 0 for any ϕ ∈ C 0 (E), ϕ ≥ 0, ϕ ≡ 0. Remark 4.4. In a Banach lattice X such that int X + = ∅, the common definition of the strict order is X ++ := int X + . In particular, in the case when E is compact and X = C 0 (E) = C(E), we have int X + = ∅ and the definition of X ++ introduced in Examples 4.3 coincides with int X + . In all the other examples considered, we have int X + = ∅, and thus our definition of the strict order does not coincide with the one defined through the set int X + . Remark 4.5. Another notion of strict order can be defined through the notions of ideals and quasiinterior points as briefly explained now, see [START_REF] Arendt | One-parameter semigroups of positive operators[END_REF] or [START_REF] Bátkai | Positive operator semigroups[END_REF]Chap. 10] and the references therein for details. Defining the segment [g 1 , g 2 ] and the set I f for g 1 , g 2 ∈ X and f ∈ X + \{0} by

[g 1 , g 2 ] := {g ∈ X; g 1 ≤ g ≤ g 2 }, I f := k≥0 [-kf, kf ] = Span[0, f ],
one shows that I f is an ideal in the sense that g ∈ I f implies |g| ∈ I f and 0 ≤ g ≤ f implies g ∈ I f . We say that f is an order unit if I f = X. When int X + = ∅, we find that f is an order unit iff f ∈ int X + from Lemma 2.23, so that we recover the notion of strict positivity defined above. On the other hand, we say that f is a quasi-interior point if Īf = X. It can be shown that f is a quasi-interior point iff f is strictly positive in the sense of the direct strict order (4.8), see for instance [START_REF] Schaefer | Banach lattices and positive operators[END_REF]Thm. II.6.3]. These two notions of strict positivity and quasi interior point thus coincide when X is reflexive or when X = L p (E, E , µ), 1 ≤ p < ∞, see also [START_REF] Bátkai | Positive operator semigroups[END_REF]Examples 10.16] when µ is a σ-finite diffuse (or atomless) measure. On the other hand, it is important to point out again that the "dual" strict order (4.9) considered here is a weaker notion than the quasi-interior point notion. For instance, in

X = C 0 (E) ′ = M 1 (E), there is no quasi-interior point but X ++ = ∅.
We finally point out the following result. For a semigroup S = S L in a Banach lattice, under the mild assumption that there exists a strictly positive subeigenvector for the dual problem, namely

∃ φ ∈ X ′ ++ , ∃ b ∈ R, L * φ ≤ b φ, then Kato's inequality (4.5) implies that S is positive, see [13, Thm. 1.6].
4.2. Irreducibility and strong maximum principle. We present some other material involving the strict positivity. When satisfied by ξ ∈ X or Y , we will in particular make use of the property (4.10) ξ + ≫ 0 implies ξ ≫ 0 (or equivalently ξ -= 0).

For further references, we introduce some general framework for the couple of Banach lattices we will use in the sequel:

(X1) (i) X ++ = ∅, Y ++ = ∅
and the signum operator is well define in both X and Y ;

(ii) the property (4.10) holds in both spaces X and Y .

While (X1)-(i) is always satisfied in the applications, it is not the case for (X1)-(ii).

Lemma 4.6. The property (4.10) holds true in a space X endowed with the direct strict order (4.8), in particular in X = L p , p ∈ [1, ∞), and X = C 0 , and also in the space L ∞ = (L 1 ) ′ endowed with the dual strict order (4.9).

Proof of Lemma 4.6. We start recalling the proof of (4.10) in a general space X endowed with the strict order (4.8). Consider an element f of the Banach lattice X and assume that f + ≫ 0. The vectors f + and f -are disjoint, i.e. f + ∧ f -= 0, see for instance [264, Thm. 1.1.1 iv)]. On the one hand, since f + ≫ 0, we have that (nf + ) ∧ f -→ f - with respect to the norm as n → ∞, see [START_REF] Schaefer | Banach lattices and positive operators[END_REF]Thm. II.6.3]. On the other hand, we have

0 ≤ (nf + ) ∧ f -≤ (nf + ) ∧ (nf -) = n(f + ∧ f -) = 0,
for every integer n ≥ 1, where the last equality follows from the fact that f + and f -are disjoint. We deduce by passing to the limit n → ∞ that f -= 0. Thus, f = f + ≫ 0. We now establish (4.10) when X = L ∞ (notice that exactly the same arguments may be used when X = L p and X = C 0 , what provides an elementary proof in these cases too). Take f ∈ L ∞ such that f + ≫ 0. From the definition of the strict order made explicit in Examples 4.3, we have f + (x) = max(f (x), 0) > 0 a.e., so that f (x) > 0 a.e. and finally f -(x) = 0 a.e..

We give now a counter-example in the Radon measures space case.

Example 4.7. Consider M 1 ([0, 1]) = C([0, 1]) ′ endowed with the dual strict order (4.9). Let (q n ) be an enumeration of the rational numbers in [0, 1] and let r be an irrational number in [0, 1]. The functional φ given by φ, f :=

∞ n=1 1/2 n f (q n ) -f (r) satisfies φ + ≫ 0, but φ itself is not positive.
For an operator A ∈ B(X), we have yet formalized a positivity condition in section 2.1, by (P1)

A ≥ 0 if A : X + → X + .
Other possible definition of positivity may be (P2) A :

X + \{0} → X + \{0}; (P3) A : X ++ → X ++ .
We now define a stronger notion of positivity, named as strong positivity condition, by (P4) A > 0 if A : X + \{0} → X ++ . We list without proof some elementary properties about these different notions and also refer to Section 6.2 for further discussion. We have (P2) ⇒ (P1), (P3) ⇒ (P1) as well as (P4) ⇒ ((P3), (P2)). We also have A :

X + → X + iff A * : Y + → Y + ; A : X ++ → X ++ iff A * : Y ++ → Y ++ ; A : X + \{0} → X ++ iff A * : Y + \{0} → Y ++ .
We say that λ -L satisfies the strong maximum principle if

(4.11) f ∈ X + ∩ D(L), (λ -L)f ≥ 0 imply f ≫ 0 or f = 0.
It is worth emphasizing that if λ -L satisfies the strong maximum principle for some λ ∈ R then λ ′ -L satisfies the strong maximum principle for any λ ′ ≤ λ.

We say that a positive semigroup S is irreducible if

(4.12) ∀ f ∈ X + \{0}, ∀ φ ∈ Y + \{0}, ∃τ > 0 S τ f, φ > 0.
A semigroup S is classically said to be irreducible and aperiodic if the above positivity condition holds for all sufficiently large times, namely

(4.13) ∀ f ∈ X + \{0}, ∀ φ ∈ Y + \{0}, ∃ T > 0, ∀τ ≥ T S τ f, φ > 0.
Other notions of strong positivity for the semigroup S are

∃ T > 0, S T : X + \{0} → X ++ , (4.14) ∃ T > 0, T 0 S(t)dt : X + \{0} → X ++ . (4.15)
We summarize some possible implications between the previous positivity notions. (2) The integral strong positivity condition (4.15) implies the irreducibility condition (4.12), but the reverse implication is false. Similarly, the irreducibility and aperiodicity condition (4.13) implies the irreducibility condition (4.12), but the reverse implication is false;

(3) The irreducibility condition (4.12) is equivalent to the fact that R L (λ) : X + \{0} → X ++ , for any λ > λ 1 , as well as to the fact that λ -L satisfies the strong maximum principle (4.11) for any λ ∈ R.

The result is very classic, at least for strongly positive semigroup, see e.g. [START_REF] Arendt | One-parameter semigroups of positive operators[END_REF]Definition C.3.1] or [START_REF] Bátkai | Positive operator semigroups[END_REF]Prop. 14.10]. For the sake of completeness, we however present a short proof.

Proof of Lemma 4.8. We prove (1). We assume (4.14) and we fix g ∈ X + \{0}, φ ∈ Y + \{0}, so that S(T )g, φ > 0. Observing that the function t → S(t)g, φ is continuous, there exists ε > 0 such that S(t)g, φ > 0 for any

t ∈ [T -ε, T ], so that T 0 S(t)dtg, φ = T 0 S(t)g, φ dt > 0.
Because φ ∈ Y + \{0} may be chosen arbitrary, we deduce (4.15). We prove (2). We assume now (4.15) and we fix g

∈ X + \{0}, φ ∈ Y + \{0}, so that T 0 S(t)g, φ dt = T 0 S(t)dtg, φ > 0,
by assumption. We get (4.12) by observing that the function t → S(t)g, φ must be positive somewhere on [0, T ]. For the reverse implication we refer to [START_REF] Bernard | Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts[END_REF][START_REF] Gabriel | Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation[END_REF], where is studied an example of growth-fragmentation operator associated to mitosis satisfying the irreducibility condition (4.12) but not the integral strong positivity condition (4.15) nor the irreducibility and aperiodicity condition (4.13), see also Section 9.

We prove (3). We finally assume (4.12). From the above continuity argument, for any g ∈ X + \{0}, φ ∈ Y + \{0} there exist τ > ε > 0 such that S(t)g, φ > 0 for any t ∈ [τε, τ + ε]. As a consequence and thanks to the representation formula (2.13) for any fixed λ > λ 1 which holds thanks to Lemma 2.2-(ii), we have

R L (λ)g, φ = ∞ 0 e -λt S(t)dtg, φ > 0. Because φ ∈ Y + \{0} is arbitrary, we have established that R L (λ)g ∈ X ++ for any g ∈ X + \{0}. In other words, when λ > λ 1 and f ∈ X + ∩ D(L) satisfy g := (λ -L)f ≥ 0, we deduce that f = R L (λ)g ∈ X ++
, what is the strong maximum principle. This one is obviously equivalent to the strong positivity property R L (λ) : X + \{0} → X ++ . On the other way round, writing the above identity as

∞ 0 e -λt S(t)g, φ dt = R L (λ)g, φ ,
we see that the strong maximum principle implies that the RHS term is positive for any g ∈ X + \{0}, φ ∈ Y + \{0}. As a consequence, the LHS term is positive and there exists τ > 0 such that S(τ )g, φ > 0, which is nothing but the irreducibility condition (4.12).

We present two other elementary results about the strong maximum principle. Lemma 4.9. Consider L satisfying (H1) and λ ∈ R. Then the following assertions are equivalent (1) λ -L satisfies the strong maximum principle for any f ∈ D(L) ∩ X + ;

(2) λ -L satisfies the strong maximum principle for any f ∈ D(L k ) ∩ X + for some k ≥ 1;

(3) λ -L * satisfies the strong maximum principle for any φ ∈ D(L * ) ∩ Y + ; (4) λ -L satisfies the strong maximum principle for any φ ∈ D((L * ) ℓ ) ∩ Y + for some ℓ ≥ 1.

Proof of Lemma 4.9. Assume that λ -L satisfies the strong maximum principle for some λ ∈ R and k ≥ 1 and consider φ ∈ D(L * ) ∩ Y + \{0} such that (λ -L * )φ ≥ 0. For any κ > max(λ, λ 1 ) and g ∈ D(L k-1 ) ∩ X + \{0}, thanks to (H1) and the strong maximum principle, there exists

f ∈ D(L k ) ∩ X ++ such that (κ -L)f = g. As a consequence, we have φ, g = φ, (κ -L)f = (κ -L * )φ, f ≥ (κ -λ) φ, f > 0. Since g ∈ D(L k-1 ) ∩ X + is arbitrary and D(L k-1 ) ∩ X + is dense in X + , we deduce that φ ≫ 0.
We have proved that λ -L * satisfies the strong maximum principle. The other implications can be proved similarly.

Remark 4.10.

(1) In many applications, we start proving the strong maximum principle on smooth enough functions (belonging to the iterated domain) for which pointwise arguments may be used.

(2) We may replace the condition (1) by assuming that λ-L satisfies the strong maximum principle for

f ∈ C ∩ X + for a subspace C ⊂ D(L) such that (λ -L) -1 ∈ B(C) and C is dense in X.
The strong maximum principle can be seen as a consequence of the weak maximum principle together with the existence of a family of strictly positive barrier functions. We give now a typical result which can be applied (or modified in order to be applied) in many situations.

Lemma 4.11. We assume that (i) the operator λ -L satisfies the weak maximum principle and Kato's inequalities;

(ii) there exists a subset

G ⊂ X ++ ∩ {g ∈ D(L); (L -λ)g ≥ 0} such that ∀ f ∈ D(L) ∩ X + \{0}, ∃ g ∈ G such that (g -f ) + ∈ D(L).
Then λ -L satisfies the strong maximum principle.

Proof of Lemma 4.11. We consider f ∈ D(L) ∩ X + \{0} such that (λ -L)f ≥ 0 and choose g ∈ G such that h := (gf ) + ∈ D(L). We remark that from Kato's inequality (4.7), we have

(L -λ)h ≥ sign + (g -f )(L -λ)(g -f ) ≥ 0.
As a consequence of the weak maximum principle, we have h ≤ 0. That implies h = 0, so that gf ≤ 0 and finally f ≫ 0.

The above barrier functions technique is also useful for obtaining the condition (H2) (possibly in a constructive way). Lemma 4.12. For an operator L, we assume that (i) the condition (H1) holds with a constant κ 1 ∈ R;

(ii) the hypothesis (ii) in Lemma 4.11 holds with λ = κ 1 ;

(iii) there exists h 0 ∈ X + \{0} such that for any g ∈ G there exists ε > 0 such that g ≥ εh 0 .

Then the property (H2) holds true.

Proof of Lemma 4.12. Thanks to assumption (i), we may define f 0 ∈ D(L) ∩ X + \{0} as the solution to the equation (κ 1 -L)f 0 = h 0 . From the proof of Lemma 4.11 and condition (iii), there exists g ∈ G and next ε > 0 such that f 0 ≥ g ≥ εh 0 . Coming back to the equation, we have

Lf 0 = κ 1 f 0 -h 0 ≥ (κ 1 -ε -1 )f 0 ,
so that condition (H2) holds true with κ 0 := κ 1ε -1 thanks to Lemma 2.4-(ii).

4.3.

The geometry of the first eigenvalue problem. We come back on and state a result about the geometry of the first eigenvalue.

We consider an operator L on X which satisfies the conclusion (C1) about the existence of a solution (λ 1 , f 1 , φ 1 ) to the first eigentriplet problem. We next assume (H1 ′ ) the weak maximum principle

(4.16) λ > λ 1 , f ∈ D(L), (λ -L)f ≥ 0 imply f ≥ 0 and its Kato's inequalities counterpart (4.17) (signf )Lf ≤ L|f |, (sign + f )Lf ≤ Lf + ,
as well as (H4) the strong maximum principle

(4.18) λ ∈ R, f ∈ X + ∩ D(L), (λ -L)f ≥ 0 imply f ≫ 0 or f = 0.
We also assume the same properties for the adjoint operator L * . We may then state our main result in this section, where we recall that N (A) denotes the null space associated to the operator A.

Theorem 4.13. We assume that X and Y are Banach lattices satisfying (X1). We consider an unbounded operator L on X which satisfies the conclusion (C1) about the existence of a solution (λ 1 , f 1 , φ 1 ) to the first eigentriplet eigenvalue problem. We also assume that L and L * both satisfy the weak maximum principle and Kato's inequalities (H1 ′ ) as well as the strong maximum principle (H4).

Then the following hold i) f 1 ≫ 0, φ 1 ≫ 0 and λ 1 is the unique eigenvalue associated to a positive eigenvector. We next make the normalization choice

(4.19) φ 1 = 1, φ 1 , f 1 = 1.
ii) λ 1 is algebraically simple:

N ((L -λ 1 ) k ) = Span(f 1 ), ∀ k ≥ 1, (4.20) N ((L * -λ 1 ) k ) = Span(φ 1 ), ∀ k ≥ 1. (4.21)
In particular f 1 (resp. φ 1 ) is the unique positive and normalized eigenvector of L (resp. of L * ) associated to λ 1 . Finally, the projection on the first eigenspace (associated to λ 1 ) is given by

Π f := f, φ 1 f 1 .
Remark 4.14. (1) It is worth emphasizing again that (4.16) and (4.17) for both L and L * are true when L is the generator of a positive semigroup (see Lemma 2.1 and Remark 4.1) and that (4.18) is true when S L enjoys additional strong positivity (or irreducibility) condition as formulated in (4.12), (4.13), (4.14) or (4.15). As a consequence, the conclusion of Theorem 4.13 holds true when L is the generator of a positive semigroup which satisfies the hypotheses of the existence part of the Krein-Rutman Theorem 2.21 and one of the additional above strict positivity conditions. (4) From ii), we deduce that L decomposes according to X = X 0 ⊕ X 1 with X 1 := Span f 1 and

X 0 := (Span φ 1 ) ⊥ = {f ∈ X; f, φ 1 = 0} in the sense of [229, § III.5.6]. More precisely, X = X 0 ⊕ X 1 is a topological direct sum, L : X 0 ∩ D(L) → X 0 and L : X 1 → X 1 .
(5) One can observe from the proof below that the conclusion (i) in Theorem 4.13 holds under the sole assumptions (X1)-(i) for X and Y , (C1) and (H4) for L and L * . The conclusion (4.20) holds when assuming furthermore that (4.10) holds in X and L satisfies (H1 ′ ). The similar conclusion (4.21) holds when assuming furthermore that (4.10) holds in Y and L * satisfies (H1 ′ ). [START_REF] Alfaro | Confining integro-differential equations originating from evolutionary biology: ground states and long time dynamics[END_REF] We finally recall that under condition (H1), the strong maximum principle (H4) for L is equivalent to the strong maximum principle (H4) for L * (see Lemma 4.9). When furthermore condition (H2) holds and λ 1 in (C1) is defined by (2.16), the weak maximum principle (4.16) for L is equivalent to the weak maximum principle (4.16) for L * (see the proof of Lemma 2.3).

The proof of Theorem 4.13 is split into the following Lemma 4.15, Lemma 4.17, Lemma 4.18 and Lemma 4.20.

Lemma 4.15. Under assumptions (X1)-(i), (C1) and (H4) for both L and L * , the solution (λ 1 , f 1 , φ 1 ) to the first eigentriplet problem satisfies

(4.22) f 1 ≫ 0 and φ 1 ≫ 0.
Proof of Lemma 4.15. We just apply the strong maximum principle to the two eigenfunctions

f 1 ∈ X\{0} and φ 1 ∈ Y \{0}.
Remark 4.16. It is worth emphasizing that the same conclusion clearly holds when instead of (C1) we only assume that

f 1 ∈ X + \{0} and φ 1 ∈ Y + \{0} satisfy (4.23) Lf 1 = λ 1 f 1 , L * φ 1 = λ * 1 φ 1 . In that case, we deduce that λ * 1 = λ 1 by writing λ 1 f 1 , φ 1 = Lf 1 , φ 1 = f 1 , L * φ 1 = λ * 1 f 1 , φ 1 , and observing that f 1 , φ 1 = 0.
Lemma 4.17. Under assumptions (X1)-(i), (C1) and (H4) for L * (resp. L), λ 1 is the unique eigenvalue associated to a positive eigenvector for L (resp. for L * ).

Proof of Lemma 4.17. Consider λ ∈ C and f ∈ X + \{0} such that Lf = λf and observe that from the proof of Lemma 4.15, we have φ 1 ≫ 0. We compute

0 = (λ -L)f, φ 1 = f, (λ -L * )φ 1 = (λ -λ 1 ) f, φ 1 ,
and thus λ = λ 1 since f, φ 1 > 0. The same proof applies to the dual problem.

Lemma 4.18. We assume again (X1)-(i), (C1) and (H4) for both L and L * . Under the additional condition (H1 ′ ) for L and (4.10) in X (resp. (H1 ′ ) for L * and (4.10) in Y ), we have N (Lλ 1 ) = Span(f 1 ) (resp. N (L *λ 1 ) = Span(φ 1 )). In particular, f 1 (resp. φ 1 ) is unique (because of the positivity and the normalization conditions).

Proof of Lemma 4.18. Consider a eigenfunction f ∈ X\{0} associated to the eigenvalue λ 1 . First, we observe from Kato's inequality that

λ 1 |f | = λ 1 sign(f )f = sign(f )Lf ≤ L|f |.
That inequality is in fact an equality, otherwise we would have

λ 1 |f |, φ 1 = L|f |, φ 1 = |f |, L * φ 1 = λ 1 |f |, φ 1 ,
and a contradiction. As a consequence, |f | is a solution to the eigenvalue problem

λ 1 |f | = L|f |, so that λ 1 f ± = Lf ± , by writing f ± = (|f | ± f )/2.
The strong maximum principle assumption implies f ± ≫ 0 or f ± = 0, and thus f + ≫ 0 or f -≫ 0 since f = 0. Without loss of generality we may assume f + ≫ 0. From (4.10), we then deduce f ≫ 0. We introduce the normalized eigenfunctions f := rf and f1 = r 1 f 1 with

(4.24) f , φ 1 = f1 , φ 1 = 1.
Now, thanks to Kato's inequality again, we write

λ 1 ( f -f1 ) + = sign + ( f -f1 )L( f -f1 ) ≤ L( f -f1 ) + ,
and for the same reason as above that last inequality is in fact an inequality. The strong maximum principle implies that either ( f -f1 ) + = 0, which also reads f ≤ f1 , or ( f -f1 ) + ≫ 0, which implies that f ≫ f1 by using again (4.10). Because of the identity (4.24) and the fact that φ 1 ∈ X ′ + \{0}, the second case in the above alternative is not possible. Repeating the same argument with ( f1 -f ) + we get that f1 ≤ f and we conclude with f = f1 . The same proof applies to the dual problem.

∈ span(φ 1 ) if ψ ∈ Y + satisfies L * ψ ≥ λ 1 ψ and g ∈ span(f 1 ) if g ∈ X + satisfies Lg ≥ λ 1 g. In the second case, we indeed cannot have L * g -λ 1 g ∈ X + \{0}, since this would implies Lg -λ 1 g, φ 1 > 0,
and this would be in contradiction with the fact that

Lg -λ 1 g, φ 1 = g, L * φ 1 -λ 1 φ 1 = 0.
We thus must have Lgλ 1 g = 0 and we conclude thanks to Lemma 4.18. The same proof applies to the dual problem. Lemma 4.20. Under the same assumptions as in Lemma 4.18, λ 1 is algebraically simple for L (resp. for L * ).

Proof of Lemma 4.20. We use an induction argument. We have already proved that N ((Lλ 1 ) k ) = Span(f 1 ) for k = 1. Assume then the result proved for any ℓ, 1 ≤ ℓ ≤ k, and consider

f ∈ N ((L -λ 1 ) k+1 ). That means that (L -λ 1 )f ∈ N ((L -λ 1 ) k ), and thus (L -λ 1 )f = rf 1 , with r ∈ R, thanks to the induction hypothesis. If r = 0, then f ∈ N (L -λ 1 ) = Span(f 1 ). Otherwise, r = 0, and then λ 1 f, φ 1 = f, L * φ 1 = Lf, φ 1 = λ 1 f + rf 1 , φ 1
, which in turn implies r f 1 , φ 1 = 0 and a contradiction. That concludes the proof. 4.4. Mean ergodicity. We deduce from the above analysis a first classical and general but rough information about the long-time behaviour of the trajectories associated to a semigroup. More precisely, assuming the existence and uniqueness of the first eigentriplet (λ 1 , f 1 , φ 1 ) for the generator L of a semigroup S and introducing the rescaled semigroup S t := e -λ1t S(t), we wish to establish the following mean ergodic property (E1) for any f ∈ X, there holds

(4.25) 1 T T 0 S t f dt → f, φ 1 f 1 , as T → ∞,
in a sense to be specified.

We start with a general result, taken from [152, Thm. V.4.5], which states that, under the conclusions of Theorem 4.13, (E1) holds for the strong topology if the semigroup ( S t ) is bounded.

Theorem 4.21. Consider a positive semigroup S on a Banach lattice X and assume that its generator L satisfies the conclusions of Theorem 4.13 about the existence and uniqueness of the first eigentriplet (λ 1 , f 1 , φ 1 ). Assume furthermore that ( S t ) t≥0 is bounded. Then, the above mean ergodic property (E1) holds for the strong topology.

Proof of Theorem 4.21. Following the proof of [152, Thm. V.4.5], we consider the subspace

X 0 := Span f 1 ⊕ Span{f -S(t)f : f ∈ X, t ≥ 0}
of X and we take φ ∈ Y which vanishes on X 0 . Since φ vanishes on each element of the form f -S(t)f , this implies that S * (t)φ = φ for all t ≥ 0. We deduce that L * φ = λ 1 φ, and consequently φ ∈ Span φ 1 due to the point ii) in Theorem 4.13. Since we also have φ, f 1 = 0, we deduce that φ = 0 and therefore X 0 = X. We observe now

T 0 S(s)ds I -S(t) = I -S(T ) t 0

S(s)ds

for all t, T ≥ 0, which is an immediate consequence of the semigroup property. The above relation and the boundedness assumption on ( S T ) T ≥0 imply that the convergence (4.25) holds for f = g -S(t)g with g ∈ X, t ≥ 0, and thus for any f ∈ X 0 . Finally, since X 0 is dense in X and using again the fact that ( S t ) t≥0 is bounded, we can readily extend the validity of (4.25) to any f ∈ X.

We will now give weaker versions of Theorem 4.21 with proofs which are based on compactness arguments. The motivation for providing such alternative proofs that require stronger assumptions is that, unlike the proof of Theorem 4.21, the methods can be adapted to derive stronger ergodicity results, namely without averaging in time, see Section 5.5.

Theorem 4.22. Consider a positive semigroup S on a Banach lattice X and assume that its generator L satisfies the conclusions of Theorem 4.13 about the existence and uniqueness of the first eigentriplet (λ 1 , f 1 , φ 1 ). With the above notations, we assume furthermore that (1) ( S t ) is bounded;

(2) B X is weakly compact for a topology which makes f → f, φ 1 continuous. Then, the above mean ergodic property (E1) holds for the topology introduced in (2).

Proof of Theorem 4.22. Fix f ∈ X and define

u T := 1 T T 0 S t f dt.
From (1), we have

u T ≤ 1 T T 0 S t f dt ≤ M f , ∀ T > 0.
We also compute

u T , φ 1 = 1 T T 0 S t f, φ 1 dt = f, φ 1 , ∀ T > 0.
Thanks to assumption (2), we deduce that there exists f * ∈ X and a sequence (T k ) such that

u T k → f * and f * , φ 1 = f, φ 1 .
Because ( S t f ) is bounded, we may use the usual ergodicity trick as in the second proof of Theorem 3.1 and for any t > 0, we have

S t f * -f * = lim k→∞ 1 T k T k +t T k S s f ds - t 0 S s f ds = 0.
We have established (Lλ 1 )f * = 0, so that f * ∈ Span(f 1 ) and more precisely f * = f, φ 1 f 1 . By uniqueness of the limit, it is the whole family (u T ) which converges to f * .

We present a variant of the previous result in which we see that in a very general framework (including all the applications we present in the second part of this work) the above hypotheses (1) and (2) are not needed (or more precisely are automatically satisfied).

Theorem 4.23. (1) Consider a Banach lattice X ⊂ L 1 loc (E, E , µ) and Y ⊂ L 1 loc (E, E , µ) (so that in particular φ 1 ∈ L 1
loc and L 1 φ1 is well-defined) and a positive semigroup S on X such that its generator L satisfies the conclusions of Theorem 4.13 about the existence, positivity and uniqueness of the first eigentriplet (λ 1 , f 1 , φ 1 ). Then the mean ergodic convergence (E1) holds for the weak topology of L 1 φ1 . (2) Assuming additionally that S is strongly continuous and that (4.26)

X k := (D(L k ), • X k ) ⊂ L 1 loc with strong compact embedding for some k ≥ 1, where f X k := f L 1 φ 1 + • • • + L k f L 1 φ 1 , ∀ f ∈ D(L k ),
then the mean ergodic convergence (E1) holds for the strong topology of L 1 φ1 . Proof of Theorem 4.23. Step 1. We first recall a very classical result about conservative semigroups. Denoting S t := e -λ1t S(t), we observe that this rescaled semigroup satisfies

(i) S t ≥ 0; (ii) S t f 1 = f 1 for any t ≥ 0;
(iii) S t g, φ 1 = g, φ 1 for any g ∈ X and t ≥ 0.

We denote [f ] 1 := |f |, φ 1 which is a norm on X (we use here that φ 1 ≫ 0) and S t is obviously a contraction for this one. Indeed, for any f ∈ X, there holds

| S t f | = | S t f + -S t f -| ≤ S t f + + S t f -= S t |f |,
using (i) in the inequality, and next (4.27) [

S t f ] 1 = | S t f |, φ 1 ≤ S t |f |, φ 1 = [f ] 1 ,
using (iii) in the last equality. Abusing notations, we also denote by X the completion of X for the L 1 φ1 norm (so that we may identify X to a closed subspace of L 1 φ1 ). We may then extend S t to X by uniform continuity and this extension still satisfies the properties (i)-(ii)-(iii) on X . Consider now f ∈ X such that H(f /f 1 )f 1 ∈ X for some convex function H : R → R, where we use here that X ⊂ L 1 loc , and thus in particular f 1 > 0 a.e. on E, in order to give a sense to the term

H(f /f 1 )f 1 . From (ii), we have ℓ[( S t f )/f 1 ]f 1 = S t [ℓ(f /f 1 )f 1 ]
, for any real affine function ℓ. Next from (i) and (2.7), we have

H[( S t f )/f 1 ]f 1 ≤ S t [H(f /f 1 )f 1 ],
because of H = sup ℓ≤H ℓ and the supremum can be taken on a numerable set of affine functions. Thanks to (iii), we conclude that

(4.28) H[( S t f )/f 1 ]f 1 , φ 1 ≤ H[f /f 1 ]f 1 , φ 1 , ∀ t ≥ 0.
Step 2. We normalize

f 1 , φ 1 = 1. For f ∈ X ⊂ L 1 φ1 so that f φ 1 = (f /f 1 )f 1 φ 1 ∈ L 1
, the de la Vallée Poussin theorem tells us that there exists an even and convex function

H : R → R + such that H(s)/s → +∞ as s → ∞ and H(f /f 1 )f 1 φ 1 ∈ L 1 .
Using the notations of the proof of Theorem 4.22, the Jensen inequality and the above estimate (4.28), we deduce

E H(u T /f 1 )f 1 φ 1 dµ ≤ 1 T T 0 E H[( S t f )/f 1 ]f 1 φ 1 dµdt ≤ E H(f /f 1 )f 1 φ 1 dµ,
for any T > 0. Now, for any A ∈ E and T, K > 0, we have

E u T 1 A φ 1 dµ = E u T f 1 1 |u T | f 1 >K 1 A f 1 φ 1 dµ + E u T f 1 1 |u T | f 1 ≤K 1 A f 1 φ 1 dµ ≤ K H(K) E H(u T /f 1 )f 1 φ 1 dµ + K E 1 A f 1 φ 1 dµ ≤ K H(K) E H(f /f 1 )f 1 φ 1 dµ + K E 1 A f 1 φ 1 dµ,
from what we immediately deduce that (u T ) belongs to a weak compact set of L 1 φ1 . We conclude that (4.25) holds for the weak convergence in L 1 φ1 as in the proof of Theorem 4.22.

Step 3. We now additionally assume that (4.26) holds with strong compact embedding for some

k ≥ 1. Taking f ∈ D(L k ), we compute |L j ( S t f )|, φ 1 = | S t (L j f )|, φ 1 ≤ |L j f |, φ 1 ,
for any j ≤ k and any t ≥ 0, and thus the same bound holds for (u T ). From (4.26), we deduce that up to the extraction of a subsequence, (u T ) converges a.e. on E. Together with the weak convergence in L 1 φ1 yet established, we classically deduce that the whole family (u T ) converges for the strong topology in L 1 φ1 . We conclude that the same holds for any f ∈ X by taking advantage of the fact that D(L k ) is dense in X for the strong topology of X, and thus for the strong topology of X , and of the estimate of contraction (4.27).

Remark 4.24. (1) A similar conclusion holds as in Theorem 4.23 when we assume

X ⊂ M 1 loc , D(L k ) ⊂ L 1 loc and D(L * k ) ⊂ L 1 loc for some k ≥ 1 instead of X, Y ⊂ L 1 loc . For f ∈ D(L k ) ⊂ L 1 loc
, we may indeed repeat the proof of Theorem 4.23 and we obtain the same conclusion. We next define X as the closure of D(L k ) for the norm [•] 1 . We conclude that (4.25) holds weakly in L 1 φ1 for any f ∈ X by a density argument.

(2) The proof of Theorem 4.23 is based on so-called General Relative Entropy (GRE) techniques as developed for instance in [START_REF] Loskot | Relative entropy and stability of stochastic semigroups[END_REF], [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF] and [START_REF] Bernard | Loskot-Rudnicki's Inequality and General Relative Entropy Inequality for Cauchy Problems Preserving Positivity[END_REF]. These ones are known to be useful for some classes of PDEs and for stochastic semigroups in order to establish uniform in time estimates and longtime convergence results.

The main interest of the two previous results is that they do not ask any new information on the semigroup but they are just based on the eigentriplet stationary problem. The shortcoming is that they are formulated in terms of the norm [•] 1 instead of the norm of X. We present a second variant of Theorem 4.22 which is well adapted to the splitting framework developed in Sections 2 and 3 and is precisely formulated in a weak or strong topology of a space X 0 ⊃ X. Theorem 4.25. Consider a positive semigroup S = S L such that L satisfies the conclusions of Theorem 4.13 about the existence and uniqueness of the first eigentriplet (λ 1 , f 1 , φ 1 ). Assume furthermore that S satisfies the splitting structure introduced in (HS2) in section 3.2 or (HS3) in Section 3.2, or more precisely, there exist two families of operators (V (t)) and (W (t)) such that

S = V + W * S,
a real number κ ≤ λ 1 and some decaying functions

Θ i : R + → R + with Θ 1 (t) → 0 as t → ∞, Θ 2 ∈ L 1 (R + ) such that the following estimates hold V (t)e -κt B(X) = O(1), V (t)e -κt B(X,X0) = O(Θ 1 ), (4.29) W (t)e -κt B(X0,X1) = O(Θ 2 ), (4.30) with X 1 ⊂ X 0 ⊂ X 0 , where X 0 is the space X endowed with the norm [g] 1 := |g|, φ 1 .
(1) Assume furthermore that X 1 ⊂ X 0 with compact embedding for the weak or the strong topology in X 0 and this topology makes f → f, φ 1 continuous. Then the mean ergodic convergence (E1) holds true for the above strong or weak topology.

(2) Assume furthermore that X ⊂ L 1 loc , S is strongly continuous, and that the space X k defined by (4.26) is strongly compact embedded in L 1 loc for some k ≥ 1. Then the mean ergodic convergence (E1) holds true for the strong topology of X 0 .

Proof of Theorem 4.25. We define

V (t) := V (t)e -λ1t , W (t) := W (t)e -λ1t , so that S = V + W * S, and
M := sup t≥0 V (t) B(X) < ∞, V B(X,X0) Θ 1 ∈ C 0 (R + ), (4.31) Θ 2 (t) := W (t) B(X0,X1) ∈ L 1 (R + ).
Step 1. We furthermore assume [START_REF] Abdallah | Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system[END_REF] and that the weak topology of X 0 makes f → f, φ 1 continuous. We denote by T the weak or the strong topology X 0 (depending of the assumption made on the embedding X 1 ⊂ X 0 ). For f 0 ∈ X, we split

f (t) := S t f 0 = v(t) + k(t), v(t) := V (t)f 0 , k(t) := ( W * S)(t)f 0 ,
and we observe that v(t) X0 → 0 as t → ∞ from the second estimate in (4.31). On the other hand, we have

sup t≥0 k(t) X1 ≤ W L 1 sup t≥0 S t f 0 X0 ≤ W L 1 f 0 X0 , from (4.27).
In particular, k(t) belongs to a compact set of T , so that (f (t)) t≥0 also belongs to a compact set for the same topology T . The same argument used on the Cesàro function (u T ) defined during the proof of Theorem 4.22 implies that there exist f * ∈ X and a sequence (T k ) such that u T k → f * in the sense of T and f * , φ 1 = f, φ 1 , the last identity following from the assumption that f → f, φ 1 continuous for T . We may then conclude as in the proof of Theorem 4.22.

Step 2. We furthermore assume [START_REF] Agoshkov | Problema sledov funktsiȋ iz prostranstv H 1 p (Ω × D) i issledovanie nekotorykh variatsionnykh kraevykh zadach perenosa[END_REF], and by linearity we may assume f 0 ∈ X, f 0 , φ 1 = 0. We recall that (4.25) holds for the strong topology of L 1 φ1 from Theorem 4.23 and that v(t) X0 → 0 as t → ∞ from Step 1. Arguing as in Step 3 of the proof of Theorem 3.4, we have

K(T ) := 1 T T 0 (W * S)(t) dt = 1 T T 0 W (s) T -s 0 S(u) duds = T 0 W (s) T -s T U (T -s) ds,
where U T := U T e -λ1T , U t is defined by (3.44), so that u T = U T f 0 and [u T ] 1 → 0 as T → ∞ from Theorem 4.23. As a consequence, we have

K(T )f 0 X1 ≤ T /2 0 Θ 2 (s)[ U (T -s)] 0 ds + T T /2 Θ(s)[ U (T -s)] 0 ds ≤ Θ 2 L 1 sup t≥T /2 [ U (t)] 0 + ∞ T /2 Θ 2 (s)ds sup t≥0 [ U (t)] 0 → 0,
as T → ∞. All together, we have established that u T X0 → 0 as T → ∞.

The geometry of the boundary point spectrum

We summarize the results established up to now by assuming that the main conclusions in the previous sections are achieved, namely (C2) the first eigentriplet problem (4.1) has a unique solution (λ 1 , f 1 , φ 1 ), and furthermore, f 1 ≫ 0 and φ 1 ≫ 0. In that situation, we make the usual normalization (4.19).

In this section, we aim to describe one step further the geometry of the spectrum and more precisely to get some some information on the boundary point spectrum

Σ + P (L) := Σ P (L) ∩ ∆ λ1 = Σ P (L) ∩ Σ + (L).
That will be possible by introducing first a suitable and usual complexification framework and next by assuming a stronger positivity property on L or on the associated semigroup. Here and for further references below, we recall that we define the sets

Σ d (L) ⊂ Σ P (L) ⊂ Σ(L),
where the point spectrum set Σ P (L) is the set of eigenvalues, namely λ ∈ Σ P (L) if N (L-λ) = {0}, and the discret spectrum set Σ d (L) is the set of eigenvalues which are isolated and have finite algebraic multiplicity.

Complexification and the sign operator.

We present some materials, most of them being very classical, about the sign operator in a complex Banach lattice and we refer to [START_REF] Schaefer | Banach lattices and positive operators[END_REF][START_REF] Arendt | One-parameter semigroups of positive operators[END_REF] for more details.

Complexification. The complexification space X C associated to a real Banach lattice X is defined by

X C := X + iX so that f ∈ X C if f = g + ih, g, h ∈ X.
In general, we just write X without mentioning the field, although when we need to specify it, we write X C or X R . We extend on X C the order defined on X R by writing

f = g + ih ≥ 0 if g ≥ 0 and h = 0.
The conjugate f of a complex vector f = g + ih is classically defined by f = gih. We then define the modulus

(5.1) |f | := sup θ∈[0,2π] (g cos θ + h sin θ),
which indeed exists for such a family of vectors. One checks the usual modulus properties:

|f | ≥ 0, |f | = 0 iff f = 0, |λf | = |λ| |f |, |f + g| ≤ |f | + |g|,
for any f, g ∈ X and λ ∈ C. We finally define the norm on X C by

f := |g + ih| X R ,
and we observe that X C has a complex Banach lattice structure. We extend the definition of A ∈ B(X R ) to X C by setting

A(g + ih) = Ag + iAh, ∀ g + ih ∈ X C .
The operator sign. We classically extend the sign operator defined in Section 4.1 to the present complex Banach lattice framework. Instead of dealing with the most general case, we will use some regularity assumption on the Banach lattice X which is suitable for our purpose and that we present below. Similarly as in Remark 4.5, for f ∈ X, we define

X f := n {g ∈ X; |g| ≤ n|f |},
and next, similarly as in Theorem 2.24, we define

A f [g] := inf{C > 0; |g| ≤ C|f |}, ∀ g ∈ X f .
We summarize the regularity conditions we need on the Banach lattice X by assuming : (X2) For any f ∈ X such that |f | ∈ X ++ , there exists a sign operator sign f ∈ B(X), with the following properties

sign f • sign f = I, (sign f )f = |f |, (5.2) (sign f ) g = (sign (uf )) (ug), |(sign f ) g| ≤ |g|, ∀ g ∈ X, ∀ u ∈ S 1 . (5.3)
and furthermore (X3) for any f ∈ X such that |f | ∈ X ++ , the inclusion X f ⊂ X is dense for the strong, the weak, or the weak- * topology, and for all f ∈ X and g ∈ X f

(5.4) g ∈ X R and |g| ≤ C|f | ⇔ A f [g -ir|f |] ≤ C 2 + r 2 , ∀ r ∈ R.
For a space of functions, the sign operator is defined as the multiplication by (abusing notations)

(5.5) sign f := f /|f |, ∀ f ∈ X, |f | ∈ X ++ .
Lemma 5.1. With (5.5), the properties (X2) and (X3) hold when

X = L p (E, E , µ) or X = C 0 (E).
Proof of Lemma 5.1. For f ∈ X, |f | ∈ X ++ , we just indicate the proof of X f = X, the other algebraic properties being clear from the definition (5.5). When f ∈ L p such that |f | > 0 µ-a.e. and 0 ≤ g ∈ L p , we set g n := g ∧ (n|f |). We have 0 ≤ g n ≤ g and g n → g strongly L p if p < ∞ and weakly- * L ∞ if p = ∞. The general case g ∈ L p is dealt in the usual way by introducing positive and negative parts and next real and imaginary part. That concludes the proof of

X f = L p . The proof of X f = C 0 (E) is similar.
A sign operator satisfying (X2) and (X3) can actually be built by using Kakutani's theorem in general Banach lattices whenever |f | is a quasi-interior point, see for instance [START_REF] Bátkai | Positive operator semigroups[END_REF]Chapter 14.3].

In X = L ∞ (E, E , µ), being a quasi-interior point is more demanding than belonging to X ++ , and our framework is thus more general in that case. In X = M 1 (E), the situation is even worst since there is no quasi-interior point, so the approach via Kakutani's theorem does not provide any sign operator. However, we can associate to f ∈ M 1 (E) such that |f | ≫ 0 a sign operator by means of the Radon-Nikodym theorem. Denoting α : E → S 1 the measurable function such that f = α|f |, the multiplication by ᾱ/|α| defines a sign operator sign f ∈ B(X), or in other words (abusing notations)

(5.6) sign f := ᾱ/|α|, ∀ f = α|f | ∈ M 1 , |f | ∈ M 1 ++ . Lemma 5.2.
With the definition (5.6), X = M 1 (E) enjoys the properties (X2) and (X3).

Proof of Lemma 5.2. As for Lemma 5.1, we only sketch the proof of the density property X f = X, which holds here for the weak- * topology, the other algebraic properties being clear from the definition (5.6). Without loss of generality, we may take f ∈ X ++ , meaning that f (O) > 0 for any open set O ⊂ E. For ε > 0 and ϕ ∈ C 0 (E), we can find a partition E 1 , . . . , E n of E and some elements x 1 , . . . , x n of E such that for any i ∈ {1, • • • , n}:

f (E i ) > 0, x i ∈ E i and sup x∈Ei |ϕ(x) -ϕ(x i )| < ε.
For g ∈ X and ε > 0, defining g ε by

g ε := n i=1 g(E i ) f (E i ) f |Ei ∈ X f , we have g ε , ϕ - n i=1 ϕ(x i )g(E i ) ≤ n i=1 |g(E i )| Ei |ϕ(x) -ϕ(x i )| f (dx) f (E i ) ≤ ε g X ,
as well as

g, ϕ - n i=1 ϕ(x i )g(E i ) ≤ n i=1 Ei |ϕ(x) -ϕ(x i )| |g|(dx) ≤ ε g X .
We have established that | g εg, ϕ | ≤ 2ε g for any ε > 0, from what we deduce that g belongs to the weak- * closure of X f .

Lemma 5.3. Assume (X2)-(X3), and f ∈ X ++ . Consider a linear operator Q : X f → X f such that Qf = f and A f (Qg) ≤ A f (g) for any g ∈ X f . Then Q ≥ 0.
Proof of Lemma 5.3. Take 0 ≤ g ∈ X f such that g ≤ 2Cf , C > 0, and observe that

-Cf ≤ g -Cf ≤ Cf.
For any r ∈ R, we compute

A f [(Qg) -Cf -irf ] = A f [Q(g -Cf -irf )] ≤ A f [g -Cf -irf ] ≤ C 2 + r 2 ,
by using the non expansion property of Q and the claim (5.4). Using again (5.4), we deduce -Cf ≤ (Qg) -Cf ≤ Cf and the conclusion.

We generalize Kato's inequality (4.5) to the present complex framework by saying that an operator L on X satisfies (the complex) Kato's inequality if

(5.7) ∀ f ∈ D(L), ℜe(signf )Lf ≤ L|f |,
possibly in a dual sense as in (4.6). As for the real Kato's inequality, when L is the generator of a semigroup, Kato's inequality (5.7) is a consequence of the positivity of the semigroup, and we refer to Remark 4.1 for references about this claim.

5.2.

On the subgroup and discrete structure of the boundary point spectrum.

In this section, we establish that the boundary point spectrum enjoys a subgroup structure under the same kind of hypotheses as considered in the previous sections.

Lemma 5.4. Under assumptions (C2), (X2) and the complex Kato's inequality (5.7), for any Lemma 5.4. By definition Lf = λf and f ∈ D(L). By linearity of the operator sign and thanks to (5.2) and Kato's inequality (5.7), we have

λ ∈ Σ + P (L)\{0} the associated normalized eigenfunction f satisfies |f | = f 1 . Proof of
λ 1 |f | = ℜe[λ(signf )f ] = ℜe(signf )(λf ) = ℜe(signf )Lf ≤ L|f |.
By the duality argument introduced during the proof of Lemma 4.18, we must have λ 1 |f | = L|f | and the conclusion.

Theorem 5.5. Assume (C2), (X2), (X3) and that the complex Kato's inequality (5.7) holds true. Denoting L = Lλ 1 , the set S := Σ P ( L) ∩ iR is an additive subgroup and dimN ( Liα) k = 1 for any iα ∈ S and k ≥ 1.

Theorem 5.5 is similar but more general than [15, C-III, Cor. 2.12] and [START_REF] Bátkai | Positive operator semigroups[END_REF]Prop. 14.15]. Our proof is also very similar to the proof of [START_REF] Bátkai | Positive operator semigroups[END_REF]Prop. 14.15]. However, it is more direct and avoid the use of the C(K) algebra and Kakutani's Theorem [START_REF] Kakutani | Concrete representation of abstract (M )-spaces. (A characterization of the space of continuous functions[END_REF] (see also [START_REF] Meyer-Nieberg | Banach lattices. Universitext[END_REF]Thm. 2.1.3]).

Proof of Theorem 5.5. We split the proof into three steps.

Step 1. We consider f associated to an eigenvalue iα ∈ Σ P ( L)\{0}, and we define

T (t) := (sign f )e -iαt S(t)(sign f ).
Observing that S(t)f = e iαt f , we have

T (t)|f | = (sign f )e -iαt S(t)f = (sign f )f = |f | = S(t)|f |.
On the other hand, we have

|T (t)g| ≤ | S(t)(sign f )g| ≤ S(t)|g|, ∀ g ∈ X,
which, by positivity of S(t), yields

|T (t)g| ≤ A f (g) S(t)|f | = A f (g)|f |, ∀ g ∈ X f .
Because |f | = f 1 ≫ 0 from Lemma 5.4, we can apply Lemma 5.3 to |f | and Q := T (t). We deduce that T (t) ≥ 0 on X |f | = X f , and then on X = X f . As a consequence, 0 ≤ T (t)g = |T (t)g| ≤ S(t)g for any g ≥ 0. In other words, we have 0 ≤ S(t) -T (t) and then 0 ≤ S(t) * -T (t) * . We must have S(t) * -T (t) * = 0. Otherwise, there would exist ψ ∈ Y + \{0} such that ( S(t) * -T (t) * )ψ ∈ Y + \{0}, and we find a contradiction by computing

0 < ( S(t) * -T (t) * )ψ, f 1 = ψ, ( S(t) -T (t))f 1 = 0.
We have thus established that S(t) = T (t).

Step 2. Consider α, β ∈ R and f, g ∈ X\{0} such that Lf = iαf and Lg = iβg, and suppose first that (sign f ) : D(L) → D(L). From Step 1 and the fact that (sign f ) • sign f = I, for any h ∈ D(L), we may compute

Lh = lim t→0 1 t ( S(t)h -h) = (sign f ) lim t→0 1 t (e -iαt S(t)(sign f )h -(sign f )h) = (sign f )( L -iα)(sign f )h,
or in other words Liα = (sign f ) L(sign f ). We have similarly Liβ = (sign ḡ) L(sign g). Both equations together imply

L -i(α + β) = (sign f )(sign ḡ) L(sign g)(sign f ).
Defining h := (sign f )(sign ḡ)f 1 , so that (sign g)(sign f )h = f 1 , we get Lh = i(α + β)h, and finally i(α + β) ∈ S, so that the additive subgroup structure is established. When the condition (sign f ) : D(L) → D(L) is not granted, we modify the above argument by using a resolvent approach. For some λ > 0, we compute thanks to (2.13)

(λ -L) -1 = ∞ 0 e -λt S(t) dt = (sign f ) ∞ 0 e -(λ+iα)t S(t) dt (sign f ) = (sign f )(λ + iα -L) -1 (sign f ).
Repeating the argument, we obtain

(λ + i(α + β) -L) -1 = (sign f )(sign ḡ)(λ -L) -1 (sign g)(sign f ).
Applying that last identity to the vector h = (sign f )(sign ḡ)f 1 and using that (λ-L)

-1 f 1 = λ -1 f 1 , we deduce (λ + i(α + β) -L) -1 h = λ -1 h.
In other words, we have again Lh = i(α + β)h, and we conclude as above.

Step 3. From the fact that (sign f ) is an invertible operator and the equation

( L -iα) k = (sign f ) -1 ( L) k (sign f ), we see from Theorem 4.13-(ii) that N ( L -iα) k = (sign f ) -1 N ( L) k = (sign f ) -1 Spanf 1 for any k ≥ 1, so that its dimension is one.
Making an additional splitting structure hypothesis as yet introduced in Section 2.2, we may significantly improve the conclusion. We first recall a classical result on the spectrum of an operator which holds under some power compactness assumption on the resolvent.

Theorem 5.6. We assume that L satisfies the splitting structure (HS1) introduced in Section 2.2 with W(z) ∈ K (X) for some N ≥ 1 and any z ∈ ∆ κ0 . Then Σ(L) ∩ ∆ κ0 ⊂ Σ d (L).

Theorem 5.6 is a consequence of [START_REF] Voigt | A perturbation theorem for the essential spectral radius of strongly continuous semigroups[END_REF]Cor. 1.1]. We also refer to [278, proof of Thm. 3.1] for a possible elementary proof.

A sketch of the proof of Theorem 5.6. Iterating the formula

R L = R B + R B AR L , we deduce J (z)R L (z) = V(z) with J := I -(AR B ) N and V := R B + • • • + R B (AR B ) N -1 . Because J is holomorphic on ∆ κ0 , it
is a compact perturbation of the identity and J (z) → I when ℜez → ∞, one may use the theory of degenerate-meromorphic functions of Ribarič and Vidav [START_REF] Ribarič | Analytic properties of the inverse A(z) -1 of an analytic linear operator valued function A(z)[END_REF] (also established independently by Steinberg, see in particular [341, Cor. 1]), and conclude that J (z) is invertible outside of a discrete set D of ∆ κ0 . That implies that Σ(L) ∩ ∆ κ0 = D is a discrete set of ∆ κ0 . On the other hand, thanks to the Fredholm alternative [START_REF] Fredholm | Sur une classe d'équations fonctionnelles[END_REF], one deduces that the eigenspace associated to each spectral value λ ∈ D is non zero and finite dimensional, so that λ ∈ Σ d (L). See also [START_REF] Tamarkin | On Fredholm's integral equations, whose kernels are analytic in a parameter[END_REF][START_REF] Šmul'yan | Completely continuous perturbations of operators[END_REF] for pioneering works in the subject.

We end this section by a result which gives a more accurate description of the geometry of the boundary spectrum, and is a variant of the classical results [15, C-III, Thm. 3.12], [152, Thm. VI.1.12], [41, Thm. 14.17].

Theorem 5.7. Assume (C2), (X2), (X3), that the complex Kato's inequality (5.7) holds true and additionally that the splitting structure (HS1) holds with W(z) ∈ K (X) for some N ≥ 1 and any z ∈ ∆ κ0 . Then the set Σ P ( L) ∩ iR is a discrete additive subgroup of iR and any of its elements is an algebraically simple eigenvalue. More precisely, -either Σ P ( L) ∩ iR = {0} and the projection on the first eigenspace (associated to λ 1 ) writes

Π f := f, φ 1 f 1 ;
-or Σ P ( L) ∩ iR = iαZ for some α > 0 and there exists a sequence

(g k , ψ k ) k∈Z such that Lg k = (λ 1 + ikα)g k , L * ψ k = (λ 1 + ikα)ψ k , and g k , ψ ℓ = δ kℓ .
Proof of Theorem 5.7. Combining Theorem 5.5 and Theorem 5.6, we immediately get that the subgroup S := Σ P ( L) ∩ iR satisfies S ⊂ Σ d (L), it is thus discrete and made of algebraically simple eigenvalues. The first case Σ P ( L) ∩ iR = {0} falls yet in the conclusions of Theorem 4.13.

In the second case, where the boundary spectrum is not trivial, the existence of a projection on the boundary eigenspace Span(g k ) k∈Z is ensured by the Jacobs-de Leeuw-Glicksberg theorem provided that L is the generator of a relatively compact semigroup, see for instance [41, Thm. A.39 and Prop. A.40] and the references therein. We also refer to [229, paragraphs III.6.4 and III.6.5] for very classical results on the projector on the direct sum of eigenspaces associated to eigenvalues belonging to a subset of the spectrum. We can even give an explicit expression of this projection in terms of (g k ) and (ψ k ) under the form of a Fejér type sum, see Theorem 5.25.

Stronger positivity.

In order to go one step further and establish the triviality of the boundary point spectrum, we need to reinforce the positivity of the semigroup or its generator. One possible condition is based on the following notion.

The reverse strong positivity condition

For A ≥ 0, we recall that from (2.6), we have

(5.8) |Af | ≤ A|f |, ∀ f ∈ X,
and we observe that the above inequality is an equality when Af = uA|f | for some u ∈ S 1 . We focus now on the case of equality in (5.8).

Definition 5.8. We say that A satisfies the "reverse strong positivity condition" for a subclass of vectors C ⊂ X if for any f ∈ C

(5.9)

|Af | = A|f | implies ∃ u ∈ S 1 , Af = uA|f |.
We start observing that A > 0 (as defined in Section 4.2) implies the strict positivity for non-signed vectors in X R .

Lemma 5.9. Consider an operator A > 0 and assume X is reflexive.

For f ∈ X R such that ±f / ∈ X + , there holds |Af | ≪ A|f |. Proof of Lemma 5.9. Let us consider f ∈ X R such that f ± = 0. We claim that |Af | ≪ A|f |. Observing that Af + = Af + Af -≥ Af,
we deduce Af + ≥ (Af ) + , and similarly Af -≥ (Af ) -. We first consider the case (Af ) + > 0. For φ ≫ 0, we have

0 < (Af ) + , φ = sup 0≤ψ≤φ Af, ψ = Af, ψ * = f, A * ψ * ,
for some 0 ≤ ψ * ≤ φ, where we have used the very definition of X ++ , the definition of (Af ) + as an element of X ′′ and that B X ′ is compact for the weakly * topology σ(X ′ , X). We deduce in particular that ψ * = 0, so that ψ * > 0 and finally A * ψ * ≫ 0 because A * > 0 (as an elementary consequence of the fact that A > 0 listed in Section 4.2). We deduce

(Af ) + , φ = f, A * ψ * < f + , A * ψ * = Af + , ψ * ≤ Af + , φ
where for the strict inequality we have first used the assumption f -= 0 and next elementary arguments. We thus have (Af ) + < Af + . Similarly, we establish (Af ) -< Af -when (Af ) -> 0.

As a conclusion, in the three cases Af = 0, (Af ) + = 0 and (Af ) -= 0, we have

|Af | = (Af ) + + (Af ) -≪ Af + + Af -= A|f |,
which is the desired strict inequality.

We believe that a similar result also holds true for complex vectors in a general Banach lattice framework. We do not try to prove such a statement but we rather establish the corresponding complex version for our examples of concrete Banach spaces in which the definition of the absolute value |f | of a vector f ∈ X is more tractable.

Lemma 5.10. Consider an operator A > 0 on X ⊂ L 1 loc (E) for some locally and σ-compact metric space E. For f ∈ X such that |f | ≫ 0, we have

|Af | = A|f | implies ∃ u ∈ S 1 , f = u|f |,
and thus (5.9) holds.

Proof of Lemma 5.10. We assume by contradiction that ∀v ∈ S 1 , |f | > ℜe(vf ), in particular writing f = g + ih, g, h ∈ X R , we have g, h ∈ X\{0}. On the one hand, because of A > 0 and A is linear, for any v = e iα ∈ S 1 , we have

A|f | ≫ Aℜe(e iα f )) = cos α (Ag) -sin α (Ah).
On the other hand, in the Banach lattice we consider here, there exists β : E → R measurable such that |Af | = e iβ Af and thus

A|f | = |Af | = ℜe|Af | = cos β (Ag) -sin β (Ah),
and a contradiction. We have established that there exists

v ∈ S 1 such that |f | ≡ ℜe(f v). Now, we have (ℜe(f v)) 2 + (ℑm(f v)) 2 = |f v| = |f | = ℜe(f v), which in turn implies ℑm(f v) = 0, since ℜe(f v) ≫ 0.
That is here that we use the assumption |f | ≫ 0 and not only f ∈ X + \{0}. We conclude that |f | = f v and thus that f = u|f |, with

u := v -1 ∈ S 1 .
A similar result also holds in the Radon space of measures. For a measurable space (E, E ), we call transition kernel, a mapping

Q : E × E → [0, ∞] such that (i) ∀ B ∈ E , x → Q(x, B) is measurable; (ii) ∀ x ∈ E, B → Q(x, B) is a measure.
We recall the classical Markov-Riesz representation theorem which claims that for any linear and positive operator B : C 0 (E) → C 0 (E) there holds

(Bφ)(x) = E φ(x)Q(x, dy), ∀ φ ∈ C 0 (E),
for a transition kernel Q such that in the condition (i) above the mapping is furthermore continuous.

Lemma 5.11. Consider an operator A > 0 in

X = M 1 = M 1 (E, E ), for some Borel space (E, E )
where E is a locally and σ-compact metric set. For f ∈ X such that |f | ≫ 0, we have (5.9).

Proof of Lemma 5.11. By definition, the operator A is the dual of a positive operator on C 0 (E).

Using the representation formula recalled above for that adjoint operator, we get

(Af )(dy) = E Q(x, dy)f (dx), ∀ f ∈ M 1 ,
for a transition kernel Q. We deduce that

Af, φ = E×E φ(y)Q(x, dy)f (dx),
for any bounded Borel function φ : E → C. In particular, the strict positivity A > 0 translates as

Q(x, •) ≫ 0 in M 1 for any x ∈ E. We fix now φ ∈ C 0 (E) such that φ ≫ 0 and f ∈ M 1 such that |f | ≫ 0,

The reverse Kato's inequality condition

We recall that it has been stated in section 4.1 that the generator L of a positive semigroup S(t) satisfies Kato's inequality (4.5) which in a complex framework writes

(5.10) ∀ f ∈ X, ℜe(signf )Lf ≤ L|f |.
We also observe that if f = u|f | for some u ∈ S 1 , we have

ℜe(sign f )Lf = sign(u -1 f )L(u -1 f ) = L|f |,
which is the case of equality in Kato's inequality.

Definition 5.12. We say that L satisfies a "reverse Kato's inequality condition" for a class of vectors C ⊂ D(L) if for any f ∈ C the case of equality in Kato's inequality

L|f | = ℜe(signf )Lf implies ∃ u ∈ C, f = u|f |.
In some situation, we may prove directly that the "reverse Kato's inequality condition" holds by reasoning at the level of the operator L, see for instance [231, Proof of Theorem 5.1]. That is also a consequence of the strong positivity of the semigroup as we see below.

Lemma 5.13. Consider a semigroup S and its generator L. On the set C of vectors f ∈ X\{0} such that

(5.11) ∃ λ ∈ C, Lf = λf, L|f | = (ℜeλ)|f |,
there is equivalence between: (i) S(t) satisfies the "reverse strong positivity condition" for some (and thus any) t > 0;

(ii) L satisfies the "reverse Kato's inequality condition".

Remark 5.14. When X ⊂ L 1 loc , the "reverse Kato's inequality condition" (ii) implies the "reverse strong positivity condition" (i) on the class C of vectors such that f ∈ D(L), 0 ≪ |f | ∈ D(L). Assume indeed that L satisfies (ii) and consider f ∈ C such that |S t f | = S t |f | for any t ≥ 0. By differentiating, we get

(5.12) (sign f )Lf = L|f |.
From the "reverse Kato's inequality condition", we deduce that f = u|f | for some u ∈ S 1 , so that (i) holds.

Proof of Lemma 5. [START_REF] Arendt | Kato's inequality: a characterisation of generators of positive semigroups[END_REF]. In what follows, we fix f ∈ X\{0} such that (5.11) holds, and we compute

(5.13) ℜe(sign f )Lf = ℜe(sign f )(λf ) = (ℜeλ)|f | = L|f |.
For any t > 0, we also have S t f = e λt f , S t |f | = e ℜeλt |f |, and thus

(5.14) |S t f | = S t |f |.
Assuming the "reverse Kato's inequality condition", we deduce from (5.13) that f = u|f | for some u ∈ S 1 , thus S t f = uS t |f | for some u ∈ S 1 , which is the conclusion of the "reverse strong positivity condition" when (5.14) holds.

On the other way round, assuming the "reverse strong positivity condition" for some T > 0, we deduce from (5.14) for T > 0 that there exists v ∈ S 1 such that

e λT f = S T f = vS T |f | = ve ℜeλT |f |.
That implies that f = u|f | with u = ve -i(ℑmλ)T , which is nothing but the conclusion of the "reverse Kato's inequality condition" when (5.13) holds.

We summarize the material developed above in the following main result of the section.

Theorem 5.15. Assume that S is a positive semigroup on X with X ⊂ L 1 loc (E) or X = M 1 (E) for some locally and σ-compact metric space E and denote by (E k ) a sequence of increasing compact sets such that E = lim E k . We furthermore assume that for any k ≥ 1 there exists T > 0 such that S T is strictly positive on E k , in the sense that

(5.15) ∀ f ∈ X + \{0}, f |E k ≡ 0, ∀ φ ∈ X ′ + \{0}, supp φ ⊂ E k , S T f, φ > 0.
Then L satisfies the "reverse Kato's inequality condition" on the set C of eigenvectors introduced in Lemma 5.13.

Proof of Theorem 5.15. Let us consider f ∈ X\{0} such that (5.11) holds, so that S t |f | = e (ℜeλ)t |f | for any t ≥ 0. On the one hand, we may fix k ≥ 1 such that |f | ≡ 0 on E k . Then for any ℓ ≥ k, there exists T ℓ > 0 such that (5.15) holds, so that

e (ℜeλ)T ℓ |f |, φ = S T ℓ |f |, φ > 0,
for any φ ∈ Y + \{0}, supp φ ⊂ E ℓ . That implies |f |, φ > 0 on for any φ ∈ Y + \{0}, and thus |f | ≫ 0. Next, as in the proof of Lemma 5.13, we observe that

|S T ℓ f | = S T ℓ |f |, ∀ ℓ ≥ k.
Repeating the proof of Lemma 5.10 and Lemma 5.11, we deduce that there exists

u ℓ ∈ S 1 such that S T ℓ f = u ℓ S T ℓ |f | on E ℓ , or equivalently there exists v ℓ ∈ S 1 such that f = v ℓ |f | on E ℓ , with v ℓ := u ℓ e -i(ℑmλ)T ℓ . Because E ℓ ⊃ E 1 , we have established that f = v 1 |f
| on E which is the conclusion of the "reverse Kato's inequality condition" when (5.11) holds.

5.4. On the triviality of the boundary spectrum. As in section 4.3, we still assume the existence (C1) of a solution (λ 1 , f 1 , φ 1 ) ∈ R × X + × Y + to the first eigenvalue problem (4.1) and that L enjoys the weak maximum principle (4.16) and Kato's inequalities (4.17) as formulated in condition (H1 ′ ) as well as the strong maximum principle (H4). Because we deal with complex eigenvalue, we also assume that the complex Kato's inequality variant (5.10) holds.

We introduce a first additional assumption:

(H5) the"reverse Kato's inequality condition" (as defined in Definition 5.12) holds true for the class C defined in Lemma 5.13: for f ∈ X\{0} such that (5. [START_REF] Arkeryd | A global existence theorem for the initial-boundary value problem for the Boltzmann equation when the boundaries are not isothermal[END_REF])

∃ λ ∈ C, Lf = λf, L|f | = (ℜeλ)|f | = ℜe(signf )Lf, we have ∃ u ∈ C, f = u|f |.
On the other hand, we do not need the structure assumption (X3).

We are then able to make a more accurate analyse of the geometry of the spectrum.

Theorem 5.16. Consider an unbounded operator L on a Banach lattice X which satisfy (C2), (H4), (4.17) and (H5). Then the conclusion (S3 2 ) about the uniqueness of λ 1 as the eigenvalue with largest real part holds: Σ + P (L) = {λ 1 }. Remark 5.17. (1) It is worth emphasizing again that (4.17) is true when L is the generator of a positive semigroup and that (H5) is true when S L (T ) satisfies the "reverse strong positivity condition" for some T > 0 as a consequence of Lemma 5.13, see also Theorem

We consider two cases:

When the above inequality is not an equality, we have

(ℜeλ) |f |, φ 1 < L|f |, φ 1 = |f |, L * φ 1 = λ 1 |f |, φ 1 ,
and thus ℜeλ < λ 1 .

When on the contrary the above inequality is an equality, then |f | is a positive eigenvector associated to the eigenvalue ℜeλ. Because of (H4), we have |f | ∈ X ++ and repeating the proof of Lemma 4.17, we get ℜeλ = λ 1 . The condition (C2) implies |f | = f 1 . On the other hand, f satisfies (5.16) and thus f ∈ Span(f 1 ) from assumption (H5), in particular λ = λ 1 .

When L is the generator of a positive and irreducible semigroup S, we may introduce the alternative assumption:

(H5 ′ ) the semigroup S is aperiodic as defined in (4.13), namely

∀ f ∈ X + \{0}, ∀ φ ∈ Y + \{0}, ∃ T > 0, ∀τ ≥ T S τ f, φ > 0.
Theorem 5.18. Let X be Banach lattice in which the property (4.10) holds true. Consider a positive and irreducible semigroup S on X which satisfies the aperiodicity condition (H5 ′ ) and such that its generator L satisfies (C2). Then the conclusion (S3 2 ) holds: 19. It is worth pointing out that since (H5 ′ ) is stronger than (H4), see the points (2) and (3) in Lemma 4.8, we can use Theorem 4.13 and replace in Theorem 5.18 the assumption that (C2) is satisfied by the assumption that (C1) and (H1 ′ ) for both L and L * are satisfied, together with the structure assumption (X1) on X and Y . Proof of of Theorem 5.18. We introduce the notations S t := S t e -λ1t and L := Lλ 1 . Assume that f = g + ih ∈ X, g, h ∈ X R , is an eigenfunction associated to the eigenvalue λ = λ 1 + iα ∈ C, α > 0, so that

Σ + P (L) = {λ 1 }. Remark 5.
L(g + ih) = iα(g + ih) = 2πi t 0 (g + ih),
for some t 0 > 0. On the one hand, because α = 0, we must have g = 0 and h = 0, and because of

α g, φ 1 = Lh, φ 1 = h, L * φ 1 = 0,
and φ 1 ≫ 0, we have g + = 0 and g -= 0. As a consequence, and because of (4.10), there exists ψ ∈ Y + \{0} such that g + , ψ = 0. On the other hand, we compute

S t0 (g + ih) = e iαt0 (g + ih) = g + ih,
from what we deduce S t0 g = g, because S t is real. On the other hand, because S t is positive, we have g + = ( S t0 g) + ≤ S t0 g + , and next φ 1 , g + ≤ φ 1 , S t0 g + = S * t0 φ 1 , g + = φ 1 , g + , so that the inequalities are equalities (remind again that φ 1 ≫ 0), and thus

S t0 g + = g + . We conclude that S kt0 g + , ψ = g + , ψ = 0, ∀ k ≥ 0,
what is in contradiction with (H5 ′ ). We have established that Σ + P (L) = {λ 1 }.

We end this section with a third situation where the triviality of the boundary spectrum is an immediate consequence of Theorem 5.5 and Theorem 5.6.

Theorem 5.20. (1) We make the same assumptions as in Theorem 5.5 and also that there exists M > 0 large enough such that λ -L is invertible in B(X) for any λ ∈ C, ℜeλ = λ 1 , |λ| ≥ M . Then λ 1 is the unique eigenvalue with largest real part as formulated in (S3 2 ).

(2) We furthermore assume that the hypothesis of Theorem 5.6 are met and that λ -L is invertible in B(X) for any for any λ ∈ C, ℜeλ ≥ λ 1ε, |λ| ≥ M . Then a (non constructive) spectral gap (S3 3 ) holds.

We summarize the main results established in this section as follows.

(C3) the first eigentriplet problem (4.1) has a solution (λ 1 , f 1 , φ 1 ), furthermore this one is unique, f 1 ≫ 0, φ 1 ≫ 0, λ 1 is algebraically simples (for both L and L * ) and Σ + P (L) = {λ 1 }.

5.5. Ergodicity. Thanks to the above analyze, we are able to formulate some convergence results on the trajectories associated to a semigroup. More precisely, assuming the existence and uniqueness of the first eigentriplet (λ 1 , f 1 , φ 1 ) for the generator L of a semigroup S and still denoting the rescaled semigroup S t := e -λ1t S(t), we wish in particular to establish the following ergodic property (E2) for any f ∈ X, there holds (5.17)

S t f → f, φ 1 f 1 , as t → ∞,
in a sense to be specified.

We start with a simple result which take advantage of some dissipativity property of the semigroup formulated by a "reverse positivity condition". We next present some more involved results which use directly the spectral information. It is worth emphasizing that our results in this section do not use any spectral gap property what contrasts with the results we will present in the next section.

Theorem 5.21. Consider a positive semigroup S on a Banach lattice X such that its generator L enjoys the conclusions (C2) of existence, uniqueness and strict positivity of the first eigentriplet (λ 1 , f 1 , φ 1 ) and let us set S t := e -λ1t S t . We denote X the space X endowed with the norm [•], with [f ] := |f |, φ 1 . Assume furthermore that (1) for any f ∈ X, the trajectory ( S t f ) t≥0 is continuous in X and belongs to a compact set of a normed space X 1 , with X 1 ⊂ X ;

(2) (S t ) satisfies the reverse positivity condition for semigroups

(5.18) |S t f | = S t |f |, ∀ t > 0, implies ∃ T > 0, ∃ u T ∈ S 1 , S T f = u T S T |f |.
Then, the ergodicity property (E2) holds in the sense of the norm of X 1 .

Let us comment on hypotheses made in the statement of Theorem 5.21. Hypothesis (1) can be obtained as a consequence of a Lyapunov (or growth) condition reminiscent of the structure condition (HS3) introduced in Section 3.3 and an irreducibility condition. Typically, we assume first

S(t)f ≤ M f + K sup 0≤τ ≤t [ S(τ )f ] 0 , with [g] 0 := |g|, ψ 0 , ψ 0 ∈ Y + \{0}
, what can be established under the very general condition (ii) of Theorem 3.4. Next we need to be able to prove that ψ 0 ≤ rφ 1 for some r > 0. In concrete situations, we may take ψ 0 with compact support and then the above inequality is a consequence of the standard strong maximum principle. We deduce

S(t)f ≤ M f + Kr sup 0≤τ ≤t | S(τ )f |, φ 1 ≤ M f + Kr sup 0≤τ ≤t S(τ )|f |, φ 1 = M f + Kr |f |, φ 1 ,
so that ( S t ) is bounded. The hypothesis (1) is in fact a bit more demanding, but also quite natural. Assume that S L enjoys the splitting structure introduced in section 3.1 and section 3.3, so that (5. 19)

S = V + K, with V := S B + • • • + ( S B A) * (N -1) * SB , K := ( S B A) ( * N ) * S, S B (t) := e -λ1t S B (t).
In some applications, we typically have

V (t)f 0 ≤ Θ(t) f 0 , ( S B A) ( * N ) B(X ,X1) ≤ Θ with Θ ∈ L 1 (R + ) ∩ C 0 (R + ), X 1 ⊂ X compact.
In that situation, we deduce (1).

Proof of Theorem 5.21. We fix f ∈ X and without loss of generality, we may assume that f, φ 1 = 0. We observe that (5.20)

| S t f |, φ 1 = | S t-s S s f |, φ 1 ≤ S t-s | S s f |, φ 1 = | S s f |, φ 1 ,
for any t ≥ s. We deduce that ( S t ) is a dynamical system with compact trajectoires in X 1 and H(f

) := |f |, φ 1 is a Lyapunov functional.
As a consequence, from the La Salle invariance principle, we have

(5.21) inf g∈ωH | S t f -g|, φ 1 → 0 as t → ∞, with (5.22) ω H := {g ∈ X; g, φ 1 = 0, ∀ t ∈ R, H( S t g) = inf s>0 H( S s f )}.
We next characterize ω H . Picking up g ∈ ω H , we observe that

| S t g|, φ 1 = |g|, φ 1 = |g|, S * t φ 1 = S t |g|, φ 1 , ∀ t ≥ 0, so that S t |g| -| S t g|, φ 1 = 0, ∀ t ≥ 0.
In particular, using that | S t g| ≤ S t |g|, we have

(5.23) S t |g| = | S t g|, ∀ t ≥ 0.
Because of the reverse positivity condition for semigroups (5.18), there exist T > 0 and u T ∈ S 1 such that S T g = u T S T |g|. As a consequence, by definition of the set ω H , we have

0 = g, φ 1 = S T g, φ 1 = u T S T |g|, φ 1 = u T |g|, φ 1 .
Because u T = 0, we conclude that g = 0. In other words, we have established that ω H = {0} and together with (5.21), we obtain (5.17).

We present a more concrete situation where the previous result can be invoked. Although the hypotheses are somehow restrictive, it is yet useful in many applications and its proof is very simple.

Corollary 5.22. Consider a strongly continuous and positive semigroup S on a Banach lattice X such that its generator L enjoys the conclusions (C2) of existence, uniqueness and strict positivity of the first eigentriplet (λ 1 , f 1 , φ 1 ). Assume further that the reverse Kato's inequality condition (as defined in Definition 5.12) holds true for the (large) class

C := {f ∈ D(L); L|f | = ℜe(signf )Lf }, that X ⊂ L 1
loc (E, E , µ) and that the space X k defined in (4.26) satisfies X k ⊂ L 1 loc with strongly compact embedding for some k ≥ 1. Then the ergodicity property (E2) holds in the sense of strong topology of L 1 φ1 . Proof of Corollary 5.22. Because of Step 3 in the proof Theorem 4.23, we see that condition (1) in Theorem 5.21 holds with X 1 := X k . On the other hand, because of Remark 5.14 and the reverse Kato's inequality condition in C, we see that condition (2) also holds, so that we may apply Theorem 5.21 and conclude.

We present now a variant of the previous result which provides a convergence for various topologies, and relies on the (very general) assumption that the boundary spectrum is trivial rather than on the reverse positivity condition.

Theorem 5.23. Consider a positive semigroup S on a Banach lattice X such that its generator L enjoys the conclusions (C3) on the existence, uniqueness and strict positivity of the first eigentriplet problem (λ 1 , f 1 , φ 1 ) and triviality of the boundary point spectrum. Setting S t := e -λ1t S t , we assume that we are in one of the following situations:

(1) S is strongly continuous and the trajectories ( S t f ) t≥0 are relatively compact for all f ∈ X, and we denote by T the strong topology of X; (2) X = Y ′ , Y separable, and the trajectories ( S t f ) t≥0 are bounded for all f ∈ X, and we denote by T the weak * σ(Y ′ , Y ) topology; (3) X ⊂ L 1 loc (E, E , µ), and we denote by T the weak topology of L 1 φ1 ; (4) X ⊂ L 1 loc , S is strongly continuous, and for some k ≥ 1 the space X k defined in (4.26) satisfies X k ⊂ L 1 loc with strongly compact embedding, and we denote by T the strong topology of L 1 φ1 . Then the ergodicity property (E2) holds in the sense of the topology T .

Remark 5.24. The case (4) of Theorem 5.23 enjoys some strong similarities with the main consequences of the General Relative Entropy technique developed in [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF], see in particular [269, Thm. 3.2], [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF]Thm. 4.3] and [269, Thm. 5.2]. In particular, the aperiodicity condition that the boundary point spectrum is trivial may be compared with the fact that the first eigenvector f 1 is the unique (normalized and nonnegative) vector f ∈ X with vanishing dissipation of entropy D(f ) = 0 as defined in [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF] or more generally that Span(f 1 ) is the unique invariant space on which the functional D vanishes. The present formulation is more abstract and probably more general. The drawback is the condition X k ⊂ L 1 loc with strongly compact embedding which can be avoided in [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF], by using some weak compactness argument and the lower semicontinuity property of D. That is explained by the fact that our proof uses rather the La Salle invariance principle (similarly as in the proof of [153, Thm. 3.2]) instead of a entropy dissipation argument.

In the case when the boundary point spectrum is not trivial but a discrete set, the same method of proof as for Theorem 5.22 allows us to accurately describe the periodic long time behaviour of the semigroup. Theorem 5.25. Consider a positive semigroup S on a Banach lattice X such that its generator L enjoys the conclusions (C2) on the existence and uniqueness of the first eigentriplet problem (λ 1 , f 1 , φ 1 ), and satisfies the complex Kato's inequality (5.7). Suppose furthermore that X and Y both enjoy the structure conditions (X2) and (X3), that λ 1 is an isolated eigenvalue and that the boundary spectrum is not trivial, i.e. Σ + P = {λ 1 }. Setting S t := e -λ1t S t , we assume that we are in one of the situations (1), ( 2), ( 3) or (4) listed in statement of Theorem 5.23. Then Σ + P = {λ 1 +ikα, k ∈ Z} for some α > 0, there exists a sequence (g k , ψ k ) k∈Z such that Lg k = (λ 1 +ikα)g k , L * ψ k = (λ 1 + ikα)ψ k and g k , ψ k = 1, and for all f ∈ X, in the sense of the topology T , the projection

Πf = lim n→∞ 1 n n ℓ=0 ℓ k=-ℓ f, ψ k g k is well defined and S t f -S t Πf → 0 as t → +∞.
Remark 5.26. In Theorem 5.25, the assumptions that λ 1 is isolated and Σ + P = {λ 1 } might seem difficult to check in practice. We indicate here some ways to verify them. (i) The condition that λ 1 is an isolated eigenvalue is for instance guaranteed under the assumptions of Theorem 5.6 or Theorem 6.5. (ii) The condition that Σ + P is not restricted to {λ 1 } can be guaranteed by verifying that (E2) does not hold. Indeed, if Σ + P = {λ 1 }, then Theorem 5.23 imposes (E2) to hold. The result in Theorem 5.25 can be compared for instance to [START_REF] Bátkai | Positive operator semigroups[END_REF]Thm. 14.19], although our hypotheses are slightly more general. Our proof is also more direct than in [START_REF] Bátkai | Positive operator semigroups[END_REF] and it additionally provides an explicit expression of the projection on the boundary eigenspace Span(g k ) k∈Z . The proof of Theorems 5.23 and 5.25 relies on the theory of almost periodic functions which dates back to H. Bohr. There is a large literature on the subject and we refer for instance to the book of Corduneanu [START_REF] Corduneanu | Almost periodic oscillations and waves[END_REF] for a comprehensive introduction. There are several equivalent definitions of almost periodic functions and we will use the following one. A function f ∈ C b (R, X), i.e. a bounded continuous function from R to X, is said to be almost periodic if the set {f (• + τ ), τ ∈ R} is relatively compact in C b (R, X). The set of almost periodic functions is a sub-algebra of C b (R, X), and also the closure of the space of periodic functions in C b (R, X). We start with the proof of Theorem 5.23 and Theorem 5.25 in the case when S satisfies the condition (1). Then we deduce the cases (2), ( 3) and (4) from the case (1).

Proof of Theorems 5.23 and 5.25 in the case (1). Step 1. Let f ∈ X. Since the trajectory ( S t f ) t≥0 is relatively compact, we infer from [START_REF] Haraux | A simple almost-periodicity criterion and applications[END_REF]Thm. 8] (with U (τ, t) = S t and thus no periodicity condition on U ) the existence of an almost periodic eternal solution g of the rescaled semigroup S, i.e. a function g : R → X such that g(t + τ ) = S τ g(t) for all t ∈ R and τ ≥ 0, such that

lim t→+∞ S t f -g(t) = 0.
The end of the proof consists in characterizing the function g in the situations of Theorems 5.23 and 5.25. For λ ∈ R, we define the Bohr transformation of the almost-periodic function g by

c λ (g) = lim T →+∞ 1 T T 0 e -iλt g(t) dt,
which is known to exists, see [START_REF] Corduneanu | Almost periodic oscillations and waves[END_REF]Thm. 3.4], since e -iλt g(t) is also almost periodic. Since e -iλt g(t) is besides an eternal solution of the semigroup e -iλt S t with infinitesimal generator L λ = L-λ 1 -iλ, we have that

L λ T 0 e -iλt g(t) dt = g(T ) -g(0).
Dividing by T the above expression, passing to the limit T → +∞ and using that L λ is a closed operator, we get L λ c λ (g) = 0. In other words, we have established

Lc λ (g) = (λ 1 + iλ)c λ (g) and λ 1 + iλ is an eigenvalue of L if c λ (g) = 0.
Step 2. We deduce that if the boundary spectrum is trivial, as in Theorem 5.23, then necessarily c λ (g) = 0 for all λ = 0. By the uniqueness theorem, see for instance [START_REF] Corduneanu | Almost periodic oscillations and waves[END_REF]Thm. 4.7], we get that g is constant. Due to the conservation law S t f, φ 1 = f, φ 1 and the simplicity of the eigenvalue 0, we get that g = f, φ 1 f 1 and the result of the case (1) in Theorem 5.23 is proved.

Step 3. In the case of Theorem 5.25, the boundary spectrum is not trivial and we know from Theorem 5.5 that Σ + P ( L) is an additive subgroup of iR, made of algebraically simple eigenvalues. Due to the assumption that λ 1 is isolated, this subgroup must be discrete and Σ + P (L) is thus given by {λ 1 + iαk, k ∈ Z} for some α > 0. As a consequence, any λ such that c λ (g) = 0 is necessarily of the form λ = αk for some k ∈ Z. By the uniqueness theorem, g is then a α-periodic function which is given, due to Fejér's theorem, by

g(t) = lim n→∞ 1 n n ℓ=0 ℓ k=-ℓ c αk (g)e iαkt .
Consider (g k , ψ k ) two positive direct and dual eigenvectors of L associated to the eigenvalue iαk such that g k , ψ k = 1. From the conservation laws S t f, ψ k = f, ψ k e iαkt and the algebraic simplicity of the eigenvalues iαk, we get that c αk (g) = f, φ k g k , and the result is proved.

Proof of Theorems 5.23 and 5.25 in the case [START_REF] Agoshkov | Problema sledov funktsiȋ iz prostranstv H 1 p (Ω × D) i issledovanie nekotorykh variatsionnykh kraevykh zadach perenosa[END_REF]. Since Y is separable, we can find a sequence (ϕ n ) n≥1 ⊂ Y which satisfies ϕ n = 1 and span(ϕ n ) is dense in Y . We can then define on X the norm • * by setting

(5.24) f * = ∞ n=1 2 -n | f, ϕ n |.
On bounded subsets of X, the topology of this norm is the same as the weak-* topology, or more explicitly it is worth emphasizing

f n ⇀ f * σ(Y ′ , Y ) ⇔ (sup f n < ∞ and f n -f * → 0).
Since by assumption the trajectory ( S t f ) is bounded, it is weakly-* relatively compact, and so relatively compact in (X, • * ). It is also clear that the semigroup S is continuous for the weak norm • * . The normed space (X, • * ) is not a Banach space, but the proof of Theorem 5.25 actually only requires, for applying [START_REF] Haraux | A simple almost-periodicity criterion and applications[END_REF]Thm. 8], that the closed balls of X are complete metric spaces, which is the case for the distance induced by • * . Applying the case (1) of Theorems 5.23 and 5.25 then yields the result.

Proof of Theorems 5.23 and 5.25 in the case (3). We consider f ∈ X and, repeating the proof of Step 2 in Theorem 4.23, we get that (S t f ) t≥0 belongs to a weak compact set G of L 1 φ1 . We define the norm • * by (5.24) for a sequence (ϕ n ) n≥1 ⊂ C c (E) which satisfies ϕ n L ∞ = 1 and span(ϕ n ) is dense in C 0 (E). This norm induces a metric on G which is topologically equivalent to the weak convergence on L 1 φ1 . The trajectory ( S t f ) is then relatively compact in (G, • * ) and the semigroup S is continuous for the weak norm • * . We conclude as in the proof of the case (2).

Proof of Theorems 5.23 and 5.25 in the case (4). From the step 3 of the proof of Theorem 4.23, we know that for any f ∈ X k the trajectory ( S t f ) is compact for the strong topology of L 1 φ1 . We may then conclude similarly as in the case (1), using that X k is dense in X for the norm of L 1 φ1 .

5.6.

A word about spectral analysis argument. The aim of this section is to recall some more or less classical results which makes possible to slightly improve the conclusions of the results presented in the previous section by additionally assume some spectral gap at the level of the operator or the semigroup. More precisely, we are interested by some accurate versions of a partial, but principal spectral mapping theorem which asserts that (5.25) Σ(e tL ) ∩ B c (0, e κt ) = e tΣ(L)∩∆κ , ∀ t ≥ 0, for some κ < λ 1 , and even more precisely, we wish to establish the following geometric (or exponential) asymptotic stability (E3 1 ) there exist some constants κ < λ 1 and C ≥ 1 such that for any f ∈ X, there holds

(5.26) S(t)f -f, φ 1 f 1 ≤ Θ(t) f -f, φ 1 f 1 , ∀ t ≥ 0, ∀ f ∈ X,
with the decay rate function Θ(t) := C e (κ-λ1)t .

In order to discuss the several results we present, we recall the splitting framework (5.27)

S = V + W * S, V (t) B(X) + W (t) B(X) e κt ,
for the same κ ∈ R as above. We start by recalling the spectral mapping theorem for the point spectrum, and its proof, which is instructive.

Lemma 5.27 (Spectral mapping theorem for point spectrum). For a semigroup (S t ) t≥0 with infinitesimal generator L we have

Σ P (S t ) \ {0} = e tΣP (L) , ∀ t ≥ 0.
Proof of Lemma 5.27. The inclusion e tΣP (L)

⊂ Σ P (S t ) \ {0} is clear. Now let ξ ∈ Σ P (S t ) \ {0}, that is ξ ∈ C \ {0} such that S t f = ξf for some f ∈ X C \ {0}
, and let λ ∈ C such that ξ = e λt and φ ∈ X ′ such that φ, f = 0. For any k ∈ Z we have ξ = e λt+2ikπ and so

0 = e -(λ+ 2ikπ t )t S t f -f = L -λ - 2ikπ t t 0 e -(λ+ 2ikπ t )s S s f ds.
If the last integral is non-zero for some k ∈ Z, we deduce that λ + 2ikπ t is an eigenvalue of L and the result is proved. Assume by contradiction that t 0 e -(λ+ 2ikπ t )s S s f ds = 0 for all k ∈ Z. This means that the continuous and periodic complex-valued function s → e -λs φ, S s f has all its Fourier coefficients equal to zero, which is not possible since this function is not equally zero (its value at s = 0 is not zero).

We next present a very classical result about the exponential stability of f 1 which is based on the quasi-compact semigroup framework of Voigt [START_REF] Voigt | A perturbation theorem for the essential spectral radius of strongly continuous semigroups[END_REF] (see also [15, B-IV-2] and [152, Sec. V.3]) and which is a more accurate version of Lemma 2.7 and Theorem 5.7.

Theorem 5.28. Let (S t ) t≥0 be a positive irreducible semigroup on a Banach lattice X satisfying the hypotheses of Lemma 2.7 and Theorem 5.7, in particular (H2) holds for a constant κ 0 ∈ R and there exists T > 0 such that the splitting (5.28)

S T = V T + K T ,
holds with V T B(X) ≤ e κT , κ < κ 0 , and K T ∈ K (X). Then there exists a unique solution (λ 1 , f 1 , φ 1 ) to the eigentriplet and the exponential stability (E3 1 ) holds (without constructive estimate).

Remark 5.29. In the splitting framework (5.27) the critical hypothesis K T ∈ K (X) may be obtained by assuming that

W (t) B(X,X1) e κt , ∀ t ≥ 0, X 1 ⊂ X compact.
In fact, in many applications, we are also able to establish X 1 ⊂ D(L β ), for some β > 0, without too much more work. ). We give however a short proof of Theorem 5.28 since it is simpler and more direct than the ones we usual find in the literature and in particular does not refer to subtil results about the spectrum and its essential part.

Proof of Theorem 5.28. First step. From Lemma 2.7, we already know that (H1), (H2) and (H3) hold. Together with the irreducibility which is nothing but (H4) from Lemma 4.8, we may apply Theorem 4.13 and conclude to the existence, uniqueness and strict positivity result about the eigentriplet solution (λ 1 , f 1 , φ 1 ).

Second step. We claim that Σ(L) ∩ {z ∈ C, ℜe(z) ≥ κ 0 } is also made of a finite number of isolated eigenvalues with finite geometric multiplicity. We indeed set β 0 := e κ0T . Since for any

λ ∈ B c β0 := {z ∈ C, |z| ≥ β 0 } the operator λ -V T is invertible, we see that λ ∈ B c β0 is in the spectrum of S T if and only if 0 is in the spectrum of I -(λ -V T ) -1 K T , or in the spectrum of I -K T (λ -V T ) -1 . Indeed, solving (λ -S T )f = g is equivalent to, on the one hand, (I -(λ -V T ) -1 K T )f = (λ -V T ) -1 g,
and in the other hand,

(I -K T (λ -V T ) -1 )(λ -V T )f = g. So if λ ∈ Σ(S T ) ∩ B c β0 then 1 ∈ Σ((λ -V T ) -1 K T ). Since (λ -V T ) -1 K
T is a compact operator, the classical Fredholm alternative (see for instance [START_REF] Brezis | Functional analysis, Sobolev spaces and partial differential equations[END_REF]Thm. 6.6]) asserts that its spectrum is made of eigenvalues with finite geometric multiplicity, and then so does for Σ(S T ) ∩ B c β0 . We can also prove, by adapting the proof of [START_REF] Brezis | Functional analysis, Sobolev spaces and partial differential equations[END_REF]Lem. 6.2], that these eigenvalues are isolated, and thus Σ(S T ) ∩ B c β0 is made of a finite number of isolated eigenvalues with finite geometric multiplicity. Since e T Σ(L) ⊂ Σ(S T ), we deduce that Σ(L) ∩ {z ∈ C, ℜe(z) ≥ κ 0 } is also made of a finite number of isolated eigenvalues with finite geometric multiplicity.

Third step. We prove the existence of a spectral gap and we conclude. Since Σ(L)∩{z ∈ C, ℜe(z) ≥ κ 0 } is finite, λ 1 is simple, and the boundary spectrum of L is a group, we deduce the existence of ε > 0 such that Σ(L) ∩ {z ∈ C, ℜe(z) ≥ λ 1 -ε} = {λ 1 }. The spectral mapping theorem in Lemma 5.27 then ensures that Σ(S T ) ∩ {z ∈ C, |z| ≥ e (λ1-ε)T } = {e λ1T } and that e λ1T is simple with eigenspace spanned by f 1 . The restriction S ⊥ T of S T to the invariant subspace X ⊥ := {f ∈ X, φ 1 , f = 0} thus has a spectral radius smaller than e (λ1-ε)T . The spectral radius formula (see [START_REF] Rudin | Functional analysis[END_REF]Thm. 10.13] for instance) then ensures that

lim n→∞ S ⊥ nT 1/n = r(S ⊥ T ) ≤ e (λ1-ε)T .
This guarantees, for any η ∈ (0, ε), the existence of a constant C η > 0 such that for all f ∈ X ⊥ and all t ≥ 0 e -λ1t S t f ≤ C η e -ηt f , and the proof is complete.

Let us now present a variant of another classical result known as the Gearhart-Prüss Theorem in [START_REF] Gearhart | Spectral theory for contraction semigroups on Hilbert space[END_REF][START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF], see also the contributions of Herbst [START_REF] Herbst | The spectrum of Hilbert space semigroups[END_REF] and Greiner [15, A-III.7] as well as the more constructive proof [152, Thm. V. 1.11] and recently [START_REF] Helffer | Improving semigroup bounds with resolvent estimates[END_REF] based on techniques developed in or related to [START_REF] Yao | On the inversion of the Laplace transform of C 0 semigroups and its applications[END_REF][START_REF] Blake | A spectral bound for asymptotically norm-continuous semigroups[END_REF].

Theorem 5.30. Consider a positive semigroup S on a Banach lattice X such that its generator L satisfies the conclusions (C2) about the existence, positivity and uniqueness of the first eigentriplet (λ 1 , f 1 , φ 1 ). We assume furthermore that X is an Hilbert space and that there exist κ < λ 1 and

R > 0 such that (i) sup z∈∆κ\BR R L (z) B(X) < ∞; (ii) Σ(L) ∩ ∆ κ ⊂ Σ d (L) ∩ B R .
Then the exponential stability (E3 1 ) holds (without constructive estimate).

Proof of Theorem 5.30. The spectral information (C2) and (ii) together imply (C3) (because of Theorem 5.5) and that there exists

κ * ∈ (κ, λ 1 ), such that Σ(L) ∩ ∆ κ * = {λ 1 }. The operator L on X 0 := (vect{f 1 }) ⊥ thus satisfies sup z∈∆ κ * R L (z) B(X0)
< ∞, and we conclude thanks to [152, Thm. V. 1.11]. The lack of constructively here only comes from the fact that our assumptions do not provide any information on the spectral gap λ 1κ > 0.

Remark 5.31. Except of the Hilbert space framework, the assumptions made in Theorem 5.30 are slightly weaker than those of Theorem 5.28, and are indeed established during the proof of Theorem 5.28: such an information at the level of the resolvent is a bit easier to establish than a similar estimate at the level of the semigroup. In the splitting framework (5.27) and its resolvent counterpart (2.22), we typically only have to show

(5.29) sup κ≤ℜez≤κ1 V(z) B(X) < ∞, lim r→∞ sup κ≤ℜez≤κ1, |ℑmz|≥r W(z) B(X) = 0,
for some κ < λ 1 , and W(z) ∈ K (X) for any z ∈ ∆ κ . That last claim is classical (see for instance [START_REF] Gualdani | Factorization of non-symmetric operators and exponential Htheorem[END_REF]) and we only briefly sketch the proof. On the one hand, from the first and the last estimates, we deduce that Σ(L) ∩ ∆ κ ⊂ Σ d (L) thanks to Theorem 5.6. As in the proof of Theorem 5.6 and with the usual notations, we also have

(I -W(z))R L (z) = V(z), ∀ z ∈ ∆ κ ,
where I -W(z) is invertible and (I -W(z)) -1 B(X) ≤ 2 for any z ∈ C such that κ ≤ ℜez ≤ κ 1 , |ℑmz| ≥ R and R is large enough. We immediately deduce that the condition (i) in Theorem 5.30 holds.

We end this section by a more recent result which is similar to the Gearhart-Prüss Theorem but is not restricted to an Hilbert space.

Theorem 5.32. Consider a positive semigroup S on a Banach lattice X such that its generator L satisfies the conclusions (C2) about the existence, positivity and uniqueness of the first eigentriplet (λ 1 , f 1 , φ 1 ). We further assume that L = A + B with 0 ≤ A ∈ B(X), S B ≥ 0 and the associated operators V and W defined by (3.14) satisfy (5.27) for some κ < λ 1 and that the resolvent counterpart W defined by (2.22) satisfies (5.29) and more precisely

sup κ≤ℜez≤κ1 z α W(z) B(X) < ∞, with α > 1.
Then the exponential stability (E3 1 ) holds (without constructive estimate).

The proof of Theorem 5.32 is a mere adaptation of [START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF]Thm. 3.1] (see also [START_REF] Mischler | Erratum: Spectral analysis of semigroups and growth-fragmentation equations[END_REF]) and it is thus skipped. The needed estimates are a bit stronger than those of Remark 5.31, but in the applications, they are not really more demanding. They also hold at the level of the resolvent instead of what is assumed in the statement of Theorem 5.28. We conclude by emphasizing again on the fact that all the above results are not constructive. We propose in the next part an alternative approach which is constructive.

Quantitative stability

In this section we establish some quantitative stability results in the spirit of the Doblin, Harris, Meyn-Tweedie theory for Markov semigroup.

6.1. About quantified positivity conditions. We briefly discuss some positivity conditions related to the strong maximum principle and barriers techniques. The issue is about how quantify the strong maximum principle

f ∈ X + \{0}, (κ 1 -L)f ≥ 0 imply f > 0 or f ≫ 0
or the related strong positivity of the associated semigroup. A possible way can be achieved with the help of a barrier functions family G ⊂ X + and a second weaker (semi)norm [•] used for normalization. Let us then introduce the two conditions

(6.1) ∀ R > 0, ∃g i ∈ G, ∀ f ∈ X + , [f ] = 1, f ≤ R, we have (i) S T f ≥ g 1 (for some T > 0) or (ii) f ≥ g 2 if (κ 1 -L)f ≥ 0. Point (ii)
is a quantified version of the strong maximum principle when G ⊂ X ++ and it is always a consequence of the positivity condition (i). Assume indeed that (i) holds (for some T > 0) and that f satisfies the requirements (6.1) and (κ 1 -L)f ≥ 0. We then write d dt (e (L-κ1)t f ) = e (L-κ1)t (Lκ 1 )f ≤ 0, so that f ≥ e (L-κ1)T f = e -κ1T S T f ≥ e -κ1T g 2 =: g 1 , with g 2 given by condition (i). The reciprocal implication is not clear, see however Lemma 4.8-(3).

Let us now make a list of possible quantified positivity conditions of Doblin-Harris type for a linear (and continuous) operator A : X → X:

(P1 ′ ) ∃ g 0 ∈ X + \{0}, ∃ ψ 0 ∈ X + \{0}, ∀ f ∈ X + , Af ≥ g 0 f, ψ 0 ; (P2 ′ ) ∃ g 0 ∈ X + \{0}, ∃ ψ 0 ∈ X ′ ++ , ∀ f ∈ X + , Af ≥ g 0 f, ψ 0 ; (P3 ′ ) ∃ g 0 ∈ X ++ , ∃ ψ 0 ∈ X ′ + \{0}, ∀ f ∈ X + , Af ≥ g 0 f, ψ 0 ; (P4 ′ ) ∃ g 0 ∈ X ++ , ∃ ψ 0 ∈ X ′ ++ , ∀ f ∈ X + , Af ≥ g 0 f, ψ 0 .
We summarize some elementary relations between these conditions and those listed in Section 4.2. Lemma 6.1. We have (P2 ′ ) ⇒ (P2) ⇒ (P1), (P3 ′ ) ⇒ (P3) ⇒ (P1), (P4 ′ ) ⇒ ((P4), (P3 ′ ), (P2 ′ )) as well as (P4) ⇒ ((P3), (P2)).

We also have:

A satisfies (P2 ′ ) iff A * satisfies (P3 ′ ); A satisfies (P3 ′ ) iff A * satisfies (P2 ′ ); A satisfies (P4 ′ ) iff A * satisfies (P4 ′ ).
We finally have:

A satisfies (P2 ′ ) implies ∃ g 0 ∈ X + \{0}, ∃ κ > 0, Ag 0 ≥ κg 0
Proof of Lemma 6.1. We assume Af ≥ g 0 f, ψ 0 for any f ∈ X + and some g 0 ∈ X + , ψ 0 ∈ X ′ + . For any φ ∈ X ′ \{0} and f ∈ X + , we have

A * φ, f = φ, Af ≥ φ, g 0 f, ψ 0 ,
which implies A * φ ≥ ψ 0 φ, g 0 . We thus deduce that A satisfies (P2 ′ ) (resp. (P3 ′ ), (P4)) implies that A * satisfies (P3 ′ ) (resp. (P2 ′ ), (P4)). The other implications can be established in a similar or even simpler way.

We conclude this introductory section by emphasizing on the fact (as already mentioned above) that S L satisfies (Pi ′ ) implies R L (λ) satisfies (Pi ′ ) for any λ ≥ λ 1 and i = 1, . . . , 4. 6.2. Asymptotic stability under Doblin condition. We start with a simple situation. We assume the Doblin condition, namely

(6.2) ∃ T > 0, ∃ ψ 0 ≫ 0, ∃ g 0 > 0, ∀ f ≥ 0, S T f ≥ g 0 ψ 0 , f ,
together with the companion positivity condition

(6.3) ∃ r 0 > 0, φ 1 , g 0 ≥ r 0 ,
as well as the strong additional boundedness assumption

(6.4) ∃ R 0 > 0, φ 1 ≤ R 0 ψ 0 . When ψ 0 := 1 ∈ X ′ ⊂ L ∞ , the condition in (6.4) is automatically satisfied with R 0 := φ 1 = 1.
Let us first emphasize that (6.3) is a natural condition when S * L enjoys a splitting structure similar to (5.19). More precisely, when

S * (t)φ ≤ Θ(t) φ + t 0 Θ(t -s)[ S * (s)φ] g0 ds, with Θ ∈ L 1 (R + ) ∩ C 0 (R + ), we deduce that 1 = φ 1 = S * (t)φ 1 ≤ Θ(t) + t 0 Θ(t -s)[φ 1 ] g0 ds, ∀ t > 0.
Passing to the limit t → ∞, we get (6.3) with r 0 := Θ -1 L 1 . Also (6.4) can be deduced from a splitting structure condition on the dual problem. More precisely, we assume that D(L ∞ ) ⊂ L 1 loc and the splitting property

L = A + B with A ∈ B(X), R B (λ) ∈ B(X) ∩ B(X + ) for any λ ≥ κ, with κ < κ 0 ≤ λ 1 ,
and the additional regularity condition

(6.5) (R B * (λ)A * ) N : L 1 g0 → L ∞ ψ -1 0 , ∀ λ > κ.
Since the dual eigenvector φ 1 satisfies

(λ 1 -B * )φ 1 = A * φ 1 , λ 1 > κ,
and then φ 1 = (R B * (λ 1 )A * ) N φ 1 , we may use estimate (6.5) and we get that (6.3)-(6.4) holds with the normalization condition r 0 := 1 and

R 0 := (R B * (λ)A * ) N B(L 1 g 0 ,L ∞ ψ - 1 0
) .

We are then able to formulate a first quantified stability result.

Theorem 6.2. Consider a semigroup S on a Banach lattice X such that its generator L enjoys the conclusion (C1) on the existence of the first eigentriplet (λ 1 , f 1 , φ 1 ). We assume furthermore the Doblin condition (6.2)-(6.4)-(6.3). Then the exponential stability (E3 1 ) in the norm [•] ψ0 holds true, with constructive constants.

The proof closely follows the usual contraction argument in the Doblin result, see for instance [START_REF] Meyn | Markov chains and stochastic stability[END_REF], [START_REF] Gabriel | Measure solutions to the conservative renewal equation[END_REF]Thm. 11] or [81, Thm. 2.1]. We do not explicitly assume the irreducibility of the semigroup, but the Doblin condition (6.2)-(6.4)-( 6.3) is in many aspects a strong positivity condition. In particular, our result implies the uniqueness of the first eigentriplet (λ 1 , f 1 , φ 1 ) and the triviality of the boundary spectrum.

Proof of Theorem 6.2. The two conditions (6.2) and (6.4) together imply the modified Doblin condition

∃ T > 0, ∃ g 1 > 0, ∀ f ≥ 0, S T f ≥ g 1 φ 1 , f , with g 1 := g 0 /R 0 . Take f such that φ 1 , f = 0, so that φ 1 , f ± = r := φ 1 , |f | /2 ≥ 0 and thus S T f ± ≥ g 1 φ 1 , f ± = rg 1 .
We write

|S T f | ≤ |S T f + -rg 1 | + |S T f --rg 1 | = S T |f | -2rg 1 .
We deduce

φ 1 , |S T f | ≤ S * T φ 1 , |f | -2r φ 1 , g 1 = e λ1T -φ 1 , g 1 φ 1 , |f | .
In other words, setting S t := e -λ1t S t , we have

[ S T f ] φ1 ≤ γ[f ] φ1 ,
with γ < 1 which depends explicitly of r 0 , R 0 , T and the estimates on λ 1 . We then classically deduce the exponential convergence in the [•] φ1 norm. Now, the dual condition associated to the Doblin hypothesis (6.2) is ∀ ψ ∈ X ′ + , S * T ψ ≥ ψ 0 ψ, g 0 . In particular, the first dual eigenvector φ 1 satisfies (6.6) φ 1 = e -λ1T S * T φ 1 ≥ e -λ1T ψ 0 φ 1 , g 0 = e -λ1T r 0 ψ 0 . Together with condition in (6.4), we see that [•] φ1 and [•] ψ0 are equivalent norm, and we immediately obtain the exponential convergence in the [•] ψ0 norm (with constructive constants). 6.3. Asymptotic stability under Harris condition. The Doblin condition (6.2)-(6.4)-(6.3) is too much demanding for many applications. In this section, we make the following somehow more general Harris type condition complemented with a Lyapunov condition. More precisely, we assume that there exists T > 0 such that S T := S T e -λ1T first satisfies the Lyapunov condition (6.7)

S T f ≤ γ L f + K[f ] φ1 ,
with γ L ∈ (0, 1), K ≥ 0. We next assume that S T satisfies the Harris condition (6.8)

∃ A > K/(1 -γ L ), ∃ g A > 0 such that ∀ f ≥ 0, f ≤ A[f ] φ1 there holds S T f ≥ g A [f ] φ1 .
We finally replace the positivity condition (6.3) by (6.9)

∃ r A > 0, φ 1 , g A ≥ r A .
As we have seen several times, condition (6.7) is some kind of regularity hypothesis which is natural under a splitting structure on the semigroup S L . We emphasize that conditions (6.7)-(6.8)-(6.9) slightly generalize the usual set of hypothesizes for the Harris theorem, see for instance [START_REF] Cañizo | Harris-type results on geometric and subgeometric convergence to equilibrium for stochastic semigroups[END_REF]Sect. 3].

We also point out that there is a connection between the condition (6.8) and the notion of partial integral or partial kernel operators, see for instance [START_REF] Gerlach | On characteristics of the range of integral operators[END_REF]Cor. 5.3]. The long term convergence of semigroups that contain a partially integral operator was studied in particular in [START_REF] Pichór | Continuous Markov semigroups and stability of transport equations[END_REF][START_REF] Gerlach | On the peripheral point spectrum and the asymptotic behavior of irreducible semigroups of Harris operators[END_REF][START_REF] Glück | Uniform convergence of stochastic semigroups[END_REF].

Theorem 6.3. Consider a semigroup S on a Banach lattice X such that its generator L enjoys the conclusions (C1) on the existence of the first eigentriplet (λ 1 , f 1 , φ 1 ). We assume furthermore the Harris condition (6.8) together with the Lyapunov condition (6.7) and the positivity condition (6.9).

Then the exponential stability (E3 1 ) in the norm of X holds true, with constructive constants.

Of course, in order that Theorem 6.3 really gives a constructive convergence result, we have to establish (6.8), (6.7) and (6.9) in a constructive way.

Proof of Theorem 6.3. On the one hand, we have

(6.10) [ S T f ] φ1 ≤ S T |f |, φ 1 = |f |, S * T φ 1 = [f ] φ1 .
On the other hand, we wish to establish the coupling property (6.11) [

S T f ] φ1 ≤ γ H [f ] φ1 if f ≤ A ′ [f ] φ1 and f, φ 1 = 0,
for some γ H ∈ (0, 1) and with A ′ := A/2. We thus consider f ∈ X, such that f, φ 1 = 0 and

f ≤ A ′ [f ] φ1 , so that f ± ≤ f ≤ A ′ [f ] φ1 = A[f ± ] φ1 .
Using the Harris condition (6.8), we deduce

S T f ± ≥ ϑg A , ϑ := 1 2 e -λ1T [f ] φ1 .
Similarly as in the proof of Theorem 6.2, we next compute

| S T f | ≤ | S T f + -ϑg A | + | S T f --ϑg A | ≤ S T |f | -2ϑg A and then [ S T f ] φ1 ≤ S T |f | -2ϑg A , φ 1 = |f |, S * T φ 1 -2ϑ g A , φ 1 = 1 -e -λ1T g A , φ 1 [f ] φ1 ,
which in turn implies (6.11) with γ H := 1e -κ0T r A . Now, the two estimates (6.10) and (6.11) together give (6.12)

[

S T f ] φ1 ≤ γ H [f ] φ1 + 1 -γ H A ′ f .
From (6.12) and the Lyapunov condition (6.7), we deduce that

U n+1 = M U n with U n := S n T f [ S n T f ] φ1 = S nT f [ S nT f ] φ1 and M := γ L K 1-γH A γ H .
The eigenvalues of M are

µ ± := 1 2 T ± T 2 -4D , with
T := trM = γ L + γ H , D := detM = γ L γ H -(1 -γ H ) K A ′ . We observe that γ L γ H > D > γ L γ H -(1 -γ H )(1 -γ L ) = T -1, so that (γ H -γ L ) 2 = T 2 -4γ L γ H < T 2 -4D < T 2 -4(T -1) = (T -2) 2
and finally

α := max(|µ + |, |µ -|) < max(γ H , γ L , |T -1|, 1) = 1.
We conclude that M n α n , from what we immediately conclude.

Remark 6.4. It is useful to emphasize that the existence of f 1 is not required in the proof of Theorem 6.3 for proving that M n α n , and this estimate can actually be used to derive the existence of f 1 . In ordet to prove that last claim, we first observe that Theorem 6.3 ensures that ( S nT f 0 ) n is a Cauchy sequence for any f 0 ∈ X. Indeed, for any p ∈ N, f = f 0 -S pT f 0 verifies

f, φ 1 = f 0 , φ 1 -f 0 , S *
pT φ 1 = f 0 , φ 1f 0 , φ 1 = 0, and we then have

S nT f 0 -S (n+p)T f 0 + [ S nT f 0 -S (n+p)T f 0 ] φ1 α n f 0 -S pT f 0 + [f 0 -S pT f 0 ] φ1 . Choosing f 0 ∈ X + such that [f 0 ] φ1 = 1, we deduce that ( S nT f ) converges to a fixed point f 1 of S T , which is not zero because [f 1 ] φ1 = lim[ S nT f 0 ] φ1 = [f 0 ] φ1 = 1
, and f 1 is the unique fixed point with normalization [f 1 ] φ1 = 1. Besides, f 1 ∈ X + because of the positivity of S and f 0 . This ensures that

[ S t f 1 ] φ1 = S t f 1 , φ 1 = f 1 , S * t φ 1 = f 1 , φ 1 = 1
, for any t > 0. Since on the other hand

S T S t f 1 = S t+T f 1 = S t S T f 1 = S t f 1 ,
we deduce from the uniqueness of the fixed point that S t f 1 = f 1 , which yields that f 1 ∈ D(L) and Lf 1 = λ 1 f 1 .

6.4.

Quantified isolation of the first eigenvalue. In terms of the geometry of the spectrum, an immediate consequence of Theorem 6.3 is that the conditions (6.8), (6.7) and (6.9) ensure the existence of a spectral gap, namely the existence of ε > 0 such that

Σ(L) ∩ ∆ λ1-ε = {λ 1 }.
We still assume that the Lyapunov condition (6.7) holds for some T > 0, γ L ∈ (0, 1) and K ≥ 0, but we relax (6.8) into the time-averaged condition (6.13)

     ∃ A > K/(1 -γ L ), ∃ g A > 0 such that ∀ f ≥ 0, f ≤ A[f ] φ1 there holds T 0 S t f dt ≥ g A [f ] φ1 .
It is worth emphasizing that (6.13) does not imply anymore the existence of a spectral gap, and there can be a non-trivial boundary spectrum, see Section 9.2 for an example. However, it is strong enough for guaranteeing that λ 1 is isolated from the rest of the spectrum, in the sense that

(6.14) Σ(L) ∩ B(λ 1 , ε) = {λ 1 },
for some ε > 0. In particular, if not trivial, the boundary spectrum must be discrete from Theorem 5.5 (under the additional assumptions listed in the statement of this last result).

Theorem 6.5. Consider a semigroup S on a Banach lattice X such that its generator L enjoys the conclusions (C1) on the existence of the first eigentriplet (λ 1 , f 1 , φ 1 ). We assume furthermore the time-averaged Harris condition (6.13) together with the Lyapunov condition (6.7) and the positivity condition (6.9). Then (6.14) holds true for some constructive constant ε > 0.

Proof. First, we readily deduce from (6.13) and the inversion formula (2.13) that (6.15)

∃ A > K/(1 -γ L ), ∃ g A > 0 such that ∀ f ≥ 0, f ≤ A[f ] φ1 there holds R(λ)f ≥ g A [f ] φ1 , ∀ λ > λ 1 ,
where R(λ) := (λ-λ 1 )R L (λ) and g A := (λ-λ 1 )e -λT g A . It is worth emphasizing that R(λ)f 1 = f 1 and 1 ∈ Σ( R(λ) ⊂ B(0, 1). Next, we claim that the Lyapunov condition (6.7) ensures the existence of λ > λ 1 such that (6. [START_REF] Arkeryd | A global existence theorem for the initial-boundary value problem for the Boltzmann equation when the boundaries are not isothermal[END_REF])

R(λ)f ≤ γ ′ L f + K ′ [f ]
φ1 for all f ∈ X and some γ ′ L < 1 and K ′ > 0. Indeed, by iteration of (6.7), we have

S nT f ≤ γ n L f + K 1 -γ L [f ] φ1 ,
for all integer n, from which we deduce

S t f ≤ Cγ ⌊t/T ⌋ L f + CK 1 -γ L [f ] φ1 ,
for all t ≥ 0 and where C = sup 0≤t≤T S t . We finally infer from the inversion formula (2.13) that

R(λ)f ≤ C 1 (λ -λ 1 ) λ -λ 1 + log 1 γL f + C 2 1 -γ L [f ] φ1 ,
for all λ > λ 1 and some C 1 , C 2 > 0. Then we only need to choose λ close enough to λ 1 so that

C1(λ-λ1) λ-λ1+log 1 γ L
< 1 and we obtain (6.16).

We have proved that R(λ) satisfies (6. [START_REF] Arkeryd | A global existence theorem for the initial-boundary value problem for the Boltzmann equation when the boundaries are not isothermal[END_REF]) and (6.15). Together with the positivity condition (6.9), we can thus repeat the proof of Theorem 6.3 for the operator R instead of S and we obtain the existence of constructive constants α ∈ (0, 1) and C ≥ 1 such that

R(λ) n f ≤ Cα n f , ∀ n ≥ 1,
for any f ∈ X, f, φ 1 = 0. By the spectral radius formula, we deduce

Σ R(λ) ∩ {z ∈ C, |z| > α} = {1}.
The spectral mapping theorem for the resolvent, which ensures that

Σ R(λ) \ {0} = λ -λ 1 λ -Σ(L) ,
then yields (6.14) with ε = (α -1 -1)(λλ 1 ).

6.5. The weak dissipativity case. In this section, we consider a weak dissipative semigroup (S t ) as considered in Section 3.3 and in a sense we make precise now. We consider four Banach lattices X 3 ⊂ X 2 ⊂ X 1 ⊂ X 0 = X. We first make the same kind of Harris type condition as in the previous section, namely Hypothesis (H) (Doblin-Harris) condition (6.8) holds for the same time T > 0 and for both norms • = • X0 and • = • X2 as well as the companion positivity condition (6.9) holds. Instead of the strong Lyapunov condition (6.7), we assume Hypothesis (L) (weak Lyapunov) there exist a constant K ≥ 0 such that

Sf 1 + Sf 0 ≤ f 1 + K[f ] φ1 , ∀ f ∈ X 1 , Sf 3 + Sf 2 ≤ f 3 + K[f ] φ1 , ∀ f ∈ X 3 ,
with S = S T e -λ1T . Hypothesis (I) (interpolation) there exists an increasing function ξ : R

+ → R + , λ → ξ λ , such that λ f 1 ≤ f 0 + ξ λ f 3 , ∀ λ > 0, ξ λ /λ → 0 as λ → 0.
Theorem 6.6. Consider a semigroup S on a Banach lattice X such that its generator L enjoys the conclusions (C1) on the existence of the first eigentriplet (λ 1 , f 1 , φ 1 ). We assume furthermore the three above conditions of weak confinement (L), Doblin-Harris strong irreducibility (H) and interpolation (I). Then, there exist some constructive decay rate functions Θ and Θ such that (6.17)

S n f X1 Θ(n) f X3 , ∀ n ≥ 1, and
(6.18) S n f Θ(n) f X3 , ∀ n ≥ 1, for any f ∈ X 3 , f, φ 1 = 0.
||| Sf ||| 1 + α Sf 0 ≤ |||f ||| 1 , ∀ f ∈ X 1 , f, φ 1 = 0, (6.21) ||| Sf ||| 3 + α Sf 2 ≤ |||f ||| 3 , ∀ f ∈ X 3 , f, φ 1 = 0. (6.22)
Proof of Proposition 6.7. We define (6.23) |||f

||| 1 := [f ] φ1 + δ f 0 + β f 1 ,
with β > δ > 0 conveniently chosen. We take

β := (1 -γ H )/K, δ := (1 -γ H )/A. We define ||| • ||| 3 in the same way.
In what follows, we then only establish (6.21), the proof of (6.22) being exactly the same.

We fix f ∈ X 1 , f, φ 1 = 0, and we recall

(6.24) [ Sf ] φ1 ≤ [f ] φ1 .
We also recall that from (6.11), for any A > 0, there exists γ H = γ H (A) ∈ (0, 1) such that the following coupling property holds

(6.25) [ Sf ] φ1 ≤ γ H [f ] φ1 if f 0 ≤ A[f ] φ1 .
We fix A > K and we observe that the following alternative holds (6.26)

f 0 ≤ A[f ] φ1 or (6.27) f 0 > A[f ] φ1 .
Case 1. Under condition (6.26), we use (6.25) and the first estimate in (L), and we deduce

||| Sf ||| 1 = [ Sf ] φ1 + δ Sf 0 + β Sf 1 ≤ γ H [f ] φ1 + β f 1 + βK[f ] φ1 -(β -δ) Sf 0 .
From our choice of β > 0 we have γ H + βK = 1, and we conclude that (6.21) holds with α := βδ > 0.

Case 2. Under condition (6.27), the first Lyapunov condition in (L) implies

Sf 1 + Sf 0 ≤ f 1 + K A f 0 .
Together with the non expansivity estimate (6.24), we get

[ Sf ] φ1 + β Sf 1 + β Sf 0 ≤ [f ] φ1 + β f 1 + δ f 0 ,
and we conclude to (6.21) again.

The subgeometric convergence result is a straightforward consequence of Proposition 6.7 and an interpolation argument.

Proposition 6.8. Assume that S satisfies the hypotheses of Theorem 6.6. Then (6.17) and (6.18) hold true with the same decay rate functions Θ and Θ given by (6.19) (up to a modification of the constant ζ).

Proof of Proposition 6.8. We recall that we have already proven (6.21) and (6.22). From (6.21) and the interpolation condition (I), we deduce

||| Sf ||| 1 + λα Sf 1 ≤ |||f ||| 1 + ξ λ α Sf 3 .
We observe next that from the very definition of the ||| • ||| 1 norm

||| Sf ||| 1 + α λ Sf 1 ≥ Z λ ||| Sf ||| 1 , Z λ = 1 + κλ ∈ (1, 2],
for some κ > 0 and any λ ∈ (0, λ 0 ), λ 0 > 0, and that from the very definition of the ||| • ||| 3 norm

αξ λ Sf 3 ≤ Bξ λ ||| Sf ||| 3 ,
for some B > 0. The three above estimates together imply

Z λ ||| Sf ||| 1 ≤ |||f ||| 1 + Bξ λ ||| Sf ||| 3 .
Using the second estimate (6.22) and repeating the same proof, we have

Z λn+1 ||| S n+1 f ||| 1 ≤ ||| S n f ||| 1 + Bξ λn+1 |||f ||| 3 ,
for any n ≥ 0 and for any λ n+1 > 0. The discrete Grönwall lemma implies (6.28)

||| S n f ||| 1 ≤ A n |||f ||| 1 + n k=1 A k,n ξ λ k B|||f ||| 3 , ∀ n ≥ 0,
where we have defined

A n := n k=1 a k , A k,n = A n /A k = n i=k+1 a i , a i := Z -1 λi .
Observing that

A k,n e -κ n i=k λi e κ(Λ(k)-Λ(n)) , with Λ(t) := t 0 λ s ds,
and

λ s := λ i if s ∈ (i -1, i],
we immediately conclude that the first estimate (6.17) holds true. We come back to the first inequality in (6.21) that we iterate and sum up in order to obtain

||| S n f ||| 1 + α n k=[n/2]+1 S k f 0 ≤ ||| S [n/2] f ||| 1 ,
for any n ≥ 1. Together with the non expansion inequality

[ S n f ] φ1 ≤ [ S k f ] φ1 S k f 0 , ∀ n ≥ k,
and the first estimate (6.17), we deduce

n -[n/2] -1 α[ S n f ] φ1 Θ([n/2])|||f ||| 3 ,
which is nothing but (6.18).

Parabolic equations

In this part, we consider a general elliptic operator in divergence form (7.1)

Lf := ∂ i (a ij ∂ j f ) + b i ∂ i f + ∂ i (β i f ) + cf, f ∈ H 1 0 (Ω),
where Ω ⊂ R d is a bounded domain (i.e. an open and connected set) or Ω = R d , and we always assume d ≥ 3 (in order to simplify the discussions when using the Sobolev inequality). We also always assume at least a boundedness and ellipticity condition on the (a ij ) matrix, namely

(7.2) a ij ∈ L ∞ (Ω), ∃ν > 0, ∀ξ ∈ R d , a ij ξ i ξ j ≥ ν|ξ| 2 ,
and some conditions on the coefficients b i , β j and c which will be described below. We aim to establish the existence of (λ 1 , f 1 , φ 1 ) solution to the first eigentriplet problem

(7.3) λ 1 ∈ R, 0 < f 1 ∈ H 1 0 , Lf 1 = λ 1 f 1 , 0 < φ 1 ∈ H 1 0 , L * φ 1 = λ 1 φ 1 ,
and the existence of some (constructive) rate function Θ such that the rescaled semigroup S associated to the generator L = Lλ 1 satisfies (7.4)

S(t)f -f, φ 1 f 1 H0 ≤ Θ(t) f -f, φ 1 f 1 H ,
for any t ≥ 0 and any f ∈ H, with H ⊂ H 0 ⊂ L 2 .

7.1. Diffusion with rough coefficients in a bounded domain. In this section, we consider the general elliptic operator in divergence form (7.1) in the case of a bounded and smooth enough domain Ω ⊂ R d with general elliptic condition on a ij as formulated above. We further assume that

(7.5) b i , β j ∈ L r (Ω), c ∈ L r/2 (Ω), r > d.
In that situation, the first eigentriplet problem (7.3) has been considered by Chicco in [START_REF] Chicco | Principio di massimo generalizzato e valutazione del primo autovalore per problemi ellittici del secondo ordine di tipo variazionale[END_REF][START_REF] Chicco | Some properties of the first eigenvalue and the first eigenfunction of linear second order elliptic partial differential equations in divergence form[END_REF] and revisited in a slightly less general framework (all the coefficients belong to L ∞ ) in [START_REF] Lions | Premières valeurs et fonctions propres. Cours du Collège de France 2020-2021[END_REF], where the conclusions (C2) are established. We explain with all details the existence proof by following more or less the arguments presented in [START_REF] Lions | Premières valeurs et fonctions propres. Cours du Collège de France 2020-2021[END_REF] stressing on the constructive way for obtaining the estimates, and next we present a proof of the geometric part and the stability part by taking advantage of the abstract material developed in the previous sections. It is worth emphasizing that our proof of the uniqueness of the first eigenfunction significantly differs from the one presented in [START_REF] Lions | Premières valeurs et fonctions propres. Cours du Collège de France 2020-2021[END_REF] which is based on a dissipativity argument, probably related to the reverse Kato's inequality condition. The framework considered here is the usual generalized solutions or weak solutions framework which goes back at least to Stampacchia [START_REF] Stampacchia | Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus[END_REF][START_REF] Stampacchia | Èquations elliptiques du second ordre à coefficients discontinus[END_REF], but it is reminiscent of previous contributions by Friedrichs [START_REF] Friedrichs | The identity of weak and strong extensions of differential operators[END_REF][START_REF] Friedrichs | On the differentiability of the solutions of linear elliptic differential equations[END_REF], Gårding [START_REF] Gårding | Dirichlet's problem for linear elliptic partial differential equations[END_REF], De Giorgi [127], Nash [START_REF] Nash | Continuity of solutions of parabolic and elliptic equations[END_REF], Morrey [START_REF] Morrey | Second order elliptic equations in several variables and Hölder continuity[END_REF],

Moser [START_REF] Moser | A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations[END_REF][START_REF] Moser | On Harnack's theorem for elliptic differential equations[END_REF][START_REF] Moser | A Harnack inequality for parabolic differential equations[END_REF], Ladyzhenskaya, Solonnikov, Ural'ceva [START_REF] Ladyzhenskaya | tseva. Linear and quasilinear elliptic equations[END_REF][START_REF] Ladyženskaja | Linear and quasilinear equations of parabolic type[END_REF], Oleinik, Kruzhkov [303] and many others. Lot of the functional arguments are picked up from the book of Gilbarg and Trudinger, and more specifically from [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Chapter 8], and also in recent notes by Kavian [START_REF] Kavian | Remarks on regularity theorems for solutions to elliptic equations via the ultracontractivity of the heat semigroup[END_REF] and Vasseur [START_REF] Vasseur | The De Giorgi method for elliptic and parabolic equations and some applications[END_REF]. It is worth emphasizing that the present analysis does not apply directly to elliptic operators in non divergence form, although this framework is considered in [START_REF] Lions | Premières valeurs et fonctions propres. Cours du Collège de France 2020-2021[END_REF]. We expect that all the results developed below can be generalized to a non divergence form framework, for example the one developed in [START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF], but we do not follow this line of research in the present work.

The proof of (7.3) and (7.4) are straightforward consequences of the abstract results developed in the previous sections once we have been able to check that the corresponding hypotheses are fulfilled. In the sequel, we will then show how these hypotheses are met in the present context.

Condition (H1). We recall that a weak (or variational) solution to the elliptic equation

Lf = g ∈ H -1 (Ω), f ∈ H 1 0 (Ω), is a function f ∈ H 1 0 (Ω) such that (7.6) D L (f, w) = g, w , ∀ w ∈ H 1 0 (Ω)
, where the (negative) Dirichlet form D L is defined by

D L (f, w) := - Ω (a ij ∂ j f + β i f )∂ i w + Ω (b i ∂ i f w + cf w),
for any f, w ∈ H 1 0 (Ω). Most of the time, we will simply write (7.7) Lf, w = g, w , ∀ w ∈ H 1 0 (Ω), instead of (7.6). For the reader convenience, we repeat here some estimates picked up in [START_REF] Stampacchia | Èquations elliptiques du second ordre à coefficients discontinus[END_REF]. For λ ∈ R and f ∈ H 1 0 (Ω), we start with

(λ -L)f, f = Ω a ij ∂ i f ∂ j f + Ω (β i -b i )∂ i f f + Ω (λ -c)f 2 ≥ f √ c - 2 L 2 + ν ∇f 2 L 2 -|β -b|f L 2 ∇f L 2 - √ c + f 2 L 2 + λ f 2 L 2 ≥ f √ c - 2 L 2 + ν 2 ∇f 2 L 2 - 1 2ν |β -b|f 2 L 2 - √ c + f 2 L 2 + λ f 2 L 2 ,
using the Cauchy-Schwarz inequality and the Young inequality, and next

(λ -L)f, f ≥ f √ c - 2 L 2 + ν 4 ∇f 2 L 2 + (λ - M 2ν -M 1/2 ) f 2 L 2 + ν 4 C Ω f 2 L 2 * - 1 2ν |β -b|1 |β-b|≥M f 2 L 2 - √ c + 1 c+≥M f 2 L 2 ≥ f √ c - 2 L 2 + ν 4 ∇f 2 L 2 + (λ - M 2ν -M 1/2 ) f 2 L 2 + ν 4 C Ω - 1 2ν |β -b|1 |β-b|≥M 2
L d -c + 1 c+≥M L d/2 f 2 L 2 * ,
using the Sobolev inequality (with associated constant C Ω ) and the Holder inequality. Choosing M > 0 large enough in such a way that the last term is positive, and next κ 1 > 0 large enough, we deduce for instance that

(7.8) (λ -L)f, f ≥ f √ c - 2 L 2 + ν 4 ∇f 2 L 2 + f 2 L 2 , ∀ λ ≥ κ 1 .
Thanks to the Lax-Milgram theorem and the above coercivity estimate, we deduce that λ -L is invertible, and more precisely the mapping (λ -L) -1 : H -1 → H 1 0 (Ω) is well defined. We also claim that λ -L enjoys a weak principle maximum, and more precisely (7.9) f ∈ H 1 0 (Ω), (λ -L)f ≥ 0 imply f ≥ 0. Indeed, for such a function f ∈ H 1 0 (Ω), we take w = f -∈ H 1 0 (Ω), as a test function, and elementary Sobolev space calculus together with the previous estimate yields

0 ≤ (λ -L)f, f -= -(λ -L)f -, f - ≤ -f - √ c - 2 L 2 - ν 4 ∇f - 2 L 2 -f - 2 L 2 ≤ 0,
so that f -= 0 and f ≥ 0. We thus deduce (λ -L) -1 : L 2 + → L 2 + , and from J.-L. Lions theory on parabolic equation (see for instance [251, Chapter 3]), we next deduce that L is the generator in L 2 of a positive semigroup S L , so that (H1) holds. It is worth emphasizing at this point that the semigroup S built thanks to Lions's theory is defined by S(t)f 0 = f for any f 0 ∈ L 2 , where

f ∈ E := C([0, ∞); L 2 ) ∩ L 2 loc ([0, ∞); H 1 0 ) ∩ H 1 loc ([0, ∞); H -1
) is the unique (variational) solution to the equation

(7.10) (f (T ), g(T )) L 2 -(f 0 , g(0)) L 2 = T 0 { ∂ t g, f H -1 ,H 1 0 + D L (f, g
)}ds, for any T > 0 and g ∈ E. Choosing g = f in the above equation, we classically compute

1 2 f (t) 2 L 2 - 1 2 f 0 2 L 2 - t 0 D L (f, f )ds = 0, ∀ t > 0,
which together with (7.8) implies

1 t t 0 ν 4 ∇f 2 L 2 ds ≤ - f (t) -f 0 t , f (t) + f 0 2 L 2 + κ 1 t t 0 f 2 L 2 ds, ∀ t > 0.
When f 0 ∈ D(L), the RHS is bounded and there thus exists a sequence t n → 0 such that ∇f (t n ) L 2 is bounded. That implies f 0 ∈ H 1 0 (Ω) and thus D(L) ⊂ H 1 0 (Ω). Similarly, we may consider the dual Dirichlet form D * (f, g) := D L (g, f ) and build an associated positive semigroup S * through Lions's theory described above. More precisely S * (t)g 0 = g for any t ≥ 0 and g 0 ∈ L 2 , where g ∈ E is the unique (variational) solution to the equation

(g(t), f (t)) L 2 -(g 0 , f (0)) L 2 = t 0 { ∂ t f, g H -1 ,H 1 0 + D * (g, f )}ds,
for any t > 0 and f ∈ E. Now, we fix T > 0, g T ∈ L 2 and we set g(t) := S * (Tt)g T , so that g is a solution to the backward evolution equation

-∂ t g = L * g, g(T ) = g T , with L * g := ∂ j (a ij ∂ i g) -∂ i (b i g) -β i ∂ i g + cg.
The variational formulation of this last problem is

(7.11) (g T , f (T )) L 2 -(g(0), f (0)) L 2 = T 0 { ∂ t f, g H -1
,H 1 0 -D * (g, f )}ds, for any f ∈ E. Summing up (7.10) and (7.11) with f (t) := S(t)f 0 for f 0 ∈ L 2 and g(t) := S * (Tt)g T for g T ∈ L 2 , we deduce

(S(T )f 0 , g T ) L 2 = (S * (T )g T , f 0 ) L 2 .
In other words, we have established that S * = (S L ) * and thus that L * is the generator of the semigroup S * .

Condition (H2).

Let us consider a ball B R , R > 0, such that B 4R ⊂ Ω and next the solution (7.12)

f 0 ∈ H 1 0 (Ω), (κ 1 -L)f 0 = 1 BR ,
f + L ∞ (Ω) f + L 2 (Ω) + g L r/2 (Ω)
holds for any subsolution

f ∈ H 1 0 (Ω), (λ -L)f ≤ g ∈ L r/2
(Ω). The local estimate variant [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Thm. 8.18] (or weak Harnack inequality)

(7.14) f L p (B2R) inf BR f + g L r/2 (Ω) , ∀ p ∈ [1, 2 * /2),
also holds for a nonnegative supersolution

f ∈ H 1 (Ω), f ≥ 0 on B 4R ⊂ Ω, (λ -L)f ≥ g ∈ L r/2 (Ω),
from what one deduces that a strong maximum principle [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Thm. 8.19] holds. More precisely, under the additional one side pointwise bound

(7.15) c + divβ ≤ c 0 or c -divb ≤ c 0 ,
for some c 0 ∈ R, we have that, for any f ∈ H 1 0 (Ω), (7.16) Lf ≤ 0 in Ω, f ≥ 0 in Ω imply f ≡ 0 or f > 0 a.e. in Ω.

When indeed f ≡ 0, we may choose B 4R ⊂ Ω such that f L 1 (B2R) > 0 and thus inf BR f > 0 from (7.14) (with g = 0) and because constants are supersolutions thanks to the first condition in (7.15). In the case only the second condition holds in (7.15), the same argument implies that L * satisfies the strong strong maximum principle and thus also L thanks to Lemma 4.9. We conclude that f is positive by a connexity argument.

f C α (Ω) ≤ C (λ -L)f L ∞ (Ω)
holds true for some α = α(a ij ) ∈ (0, 1) and C > 0. These last two pieces of information together and the fact that f 0 ≡ 0 imply that there exists a constant θ > 0 such that f 0 ≥ θ1 BR , and thus

Lf 0 ≥ (κ 1 -θ -1 )f 0 .
That is condition (i) in Lemma 2.4, so that condition (H2) holds thanks to Lemma 2.4. Presented in that way, the above estimate is not really constructive, but the constant θ := inf BR (κ 1 -L) -1 1 BR can also be considered as a geometric quantity associated to geometric properties of the operator and the domain.

First constructive argument for (H2). In the case when L is self-adjoint, that corresponds to the case a ij = a ji and b i + β i = 0, we classically know (that has been recalled in Section 2.3, see (2.35)) that

λ 1 = inf f ∈X+\{0} Lf, f f 2 = inf f ∈H 1 0 , f L 2 =1 O a∇f • ∇f + cf 2 ,
from what and the Sobolev imbedding, we get

λ 1 ≥ inf f ∈H 1 0 , f L 2 =1 (νC Ω -c -1 c-≥M L d/2 ) f 2 L 2 * -M ≥ -M,
by choosing M large enough. That gives an explicit lower bound on λ 1 .

Second constructive argument for (H2). We give another constructive argument without assuming any self-adjointness property. We rather assume

(7.18) (∂ i b i -c) + ∈ M 1 (Ω), b i + β i -∂ j a ij ∈ M 1 (Ω).
We fix

h 0 ∈ C 2 c (Ω) such that c 0 1 Bρ ≤ h 0 ≤ c 0 1 B 3ρ/2 with B 8ρ ⊂ Ω and h 0 L 2 = 1.
We next define f 0 as the (positive) solution to (7.19) (7.13) and (7.17), and similarly (7.20)

f 0 ∈ H 1 0 (Ω), (κ 1 -L)f 0 = h 0 , so that f 0 ∈ C α (Ω) from
f 0 ∈ H 1 0 (B 2ρ ), (κ 1 -L) f 0 = h 0 , so that f 0 ∈ C α (B 2ρ
) from (7.13) and (7.17). We observe that 0 ≤ f 0 ≤ f 0 thanks to the weak maximum principle. We then compute

1 = h 0 2 L 2 = B2ρ h 0 (κ 1 -L) f 0 = B2ρ f 0 (κ 1 -L * )h 0 ≤ f 0 L ∞ (κ 1 -L * )h 0 M 1 ,
where the last term is finite because of the additional hypothesis (7.18). We conclude to a first constructive lower bound f 0 L ∞ (B2ρ) ≥ c 1 > 0. Because of the Holder continuity, we also have

f 0 L 1 (B2ρ) ≥ c 2 with constructive constant c 2 = c 2 (c 1 , α, d) > 0.
Thanks to (7.14) (with g = 0), we obtain

f 0 ≥ 1 B 3ρ/2 inf B 3ρ/2 f 0 ≥ 1 B 3ρ/2 C wH f 0 L 1 (B 3ρ/2 ) ≥ 1 B 3ρ/2 C wH f 0 L 1 (B 3ρ/2 ) ≥ C wH c 2 c -1 0 h 0 .
Because all the inequalities are constructive and proceeding as above, we deduce that condition (ii) in Lemma 2.4 holds and thus also (H2) with constructive constant κ 0 := κ 1 -C -1 wH c -1 2 c 0 . Finally, because of (κ 1 -L)f 0 = 0 on Ω\B 3ρ/2 , we may apply the Harnack inequality [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Cor. 8.21], and we classically deduce there exist constructive constants C > 0 and C ̺ > 0 for any ̺ > 0 such that (7.21)

C ̺ 1 ω̺ ≤ f 0 ≤ C,
with ω ̺ := {x ∈ Ω; δ(x) > ̺} and δ(x) := d(x, ∂Ω) is the distance to the boundary function.

We can also get a constructive argument for (H2) by asking that condition (i) in Lemma 2.4 holds. We may for instance verify that the dual counterpart of the above constructive argument holds when (c

+ ∂ i β i ) -∈ M 1 and b i + β i + ∂ j a ji ∈ M 1 .
More precisely, we establish in a similar way as above that the solution to the problem

(7.22) φ 0 ∈ H 1 0 (Ω), (κ 1 -L * )φ 0 = h 0 , satisfies (7.23) κ 0 φ 0 ≤ L * φ 0 ≤ κ 1 φ 0 ,
for some constructive constants κ 0 ≤ κ 1 . Similarly as above again, there exist constructive constants C > 0 and C ̺ > 0 for any ̺ > 0 such that (7.24)

C ̺ 1 ω̺ ≤ φ 0 ≤ C.
Third constructive argument for (H2). We write

(7.25) Lf = a ij ∂ 2 ij f + bi ∂ i f + cf, with bi := b i + ∂ j a ji + β i and c := c+ ∂ i β i .
We further assume bi , c ∈ L ∞ . In that case, we may also obtain an explicit lower bound on λ 1 by proceeding in the following way. We define f 0 (x) := χ(|x|)

with χ ∈ C 1 c (R + ) ∩ W 2,∞ (R + ), 1 [0,1/3] ≤ χ ≤ 1 [0,1] , χ ′ ≤ 0 on [0, 1], χ(s) := n 2 (1 -s) 2 /2 on [ι n , 1],
ι n := 1 -1/(2n), for some n ≥ 1 to be chosen. As a consequence,

χ ′′ = n 2 on [ι n , 1], |χ ′ | ≤ n on [ι n , 1] and χ ≥ 1/2 on [0, ι n ]. Denoting s := |x|, we compute Lf 0 = a ij χ ′′ (s)x i xj + χ ′ (s) δ ij -xi xj s + b(x) • xχ ′ (s) + c(x)χ(s).
For n large enough, we get

Lf 0 ≥ n 2 ν -n2A -nB -C ≥ 0 on B 1 \B ιn , Lf 0 ≥ -A χ ′′ L ∞ + χ ′ (s)/s L ∞ -B χ ′ L ∞ -C ≥ κ 0 χ on B ιn , with A := a L ∞ (B1) , B := b L ∞ (B1) , C := c L ∞ (B1) and κ 0 ∈ R -.
As a conclusion, we have again established condition (ii) in Lemma 2.4, so that condition (H2) holds.

Fourth constructive argument for (H2). We present a last situation when we are able to prove a quantitative version of condition (H2). We assume that a ∈ C 0 ( Ω), divβ ∈ L r/2 , as well as bi ∈ L r and c ∈ L r/2 in the definition of (7.25). We define h 0 and f 0 as in the second constructive argument for (H2), so that (7.18) holds. Choosing p ∈ (1, 2) defined by 1/p := 1/r + 1/2 > 2/r + 1/2 * , we observe that

κ 1 f 0 -bi ∂ i f 0 -cf 0 -h 0 L p κ 1 f 0 L 2 + bi L r ∂ i f 0 L 2 + c L r/2 f 0 L 2 * + h 0 L 2 h 0 L 2 ,
from equation (7.19) and the coercivity estimate (7.8). From the Calderon-Zygmond regularity theory [START_REF] Calderon | On the existence of certain singular integrals[END_REF] or [179, Thm. 9.14], we also know that

(7.26) f 0 W 2,p (Ω) a ij ∂ 2 ij f 0 L p (Ω) . Writing a ij ∂ 2 ij f 0 = κ 1 f 0 -bi ∂ i f 0 -cf 0 -
h 0 and using the two above estimates, we deduce (7.27)

f 0 W 2,1 (Ω) h 0 L 2 (Ω) .
On the other hand, from (7.7) and the Poincaré inequality, we have

1 = h 0 2 L 2 = (κ 1 -L)f 0 , h 0 ∇f 0 L 2 ∇h 0 L 2 .
Together with the estimate (7.27) and the Gagliardo-Niremberg inequality

∇f L 2 D 2 f 1/2 L 1 f 1/2
L ∞ , we obtain a lower bound f 0 L ∞ ≥ C 0 > 0. We then conclude as in the second constructive argument for (H2).

Condition (H3). Because of Rellich-Kondrachov theorem on the compact embedding

H 1 0 ⊂ L 2 , the mapping (λ -L) -1 : L 2 → L 2 is compact for any λ ≥ κ 1 . As a consequence, introducing the splitting L = A+ B with A := κ 1 -κ B , κ B ∈ R arbitrary, the operator R B (λ) = (λ+ κ 1 -κ B -L) -1
is bounded uniformly on λ ≥ κ B and it is compact for any λ ≥ κ B . We deduce from Lemma 2.8-( 2) that (H3) holds for both the primal and the dual problems.

We may thus apply Theorem 2.21 and deduce the existence of a solution (λ 1 , f 1 , φ 1 ) to the first eigentriplet problem (7.28)

λ 1 ∈ R, 0 ≤ f 1 ∈ H 1 0 , Lf 1 = λ 1 f 1 , 0 ≤ φ 1 ∈ H 1 0 , L * φ 1 = λ 1 φ 1 ,
where both equations must be understood in the variational sense as a consequence of the discussion at the end of the proof of condition (H1).

Condition (H4). The strong maximum principle holds as already mentioned in the paragraph dedicated to condition (H2). As a consequence and thanks to Theorem 4.13, we know that the first eigentriplet problem (7.3) has a unique solution (λ 1 , f 1 , φ 1 ) which satisfies

f 1 > 0, φ 1 > 0, N (L -λ 1 ) k = Span(f 1 ) and N (L * -λ 1 ) k = Span(φ 1 ) for any k ≥ 1. Condition (H5). Consider f ∈ D(L ∞ ) such that 0 < |f | ∈ D(L ∞

We next compute

ℜe Lf, f = - Ω a kj ℜe(∂ j f ∂ k f ) + Ω (b k -β k )ℜe( f ∂ k f ) + Ω c|f | 2 , and L|f |, |f | = - Ω a kj ∂ j |f |∂ k |f | + Ω (b k -β k )ℜe( f ∂ k f ) + Ω c|f | 2 ,
where in the last equality, we have used that

∂ k |f | = 1 |f | ℜe( f ∂ k f ).
From the three above equations, we deduce

Ω a kj [∂ j |f |∂ k |f | -ℜe(∂ j f ∂ k f )] = 0.
Introducing the real and complex part decomposition f = u + iv, and similarly as in [231, Proof of Theorem 5.1], we next compute

∂ j |f |∂ k |f | -ℜe(∂ j f ∂ k f ) = 1 |f | 2 uv(∂ k u∂ j v + ∂ k v∂ j u) -u 2 ∂ j v∂ k v -v 2 ∂ j u∂ k u = 1 |f | 2 (u∂ j v -v∂ j u)(u∂ k v -v∂ k u),
so that from the ellipticity condition on a, we have u∂ k v -v∂ k u = 0 a.e. on Ω. On the other hand, from De Girogi-Nash-Moser regularity estimates (7.13) and (7.17), f has Hölder regularity. In particular both functions u and v are continuous. Because |f | ≡ 0, one of the two function is not identically vanishing, say for instance v ≡ 0. There exists some points x 0 ∈ Ω such that v(x 0 ) = 0, say for instance v(x 0 ) > 0. Denoting by ω the connected component of the set {x ∈ Ω; v(x) > 0} containing x 0 , we have ∇(u/v) = 0 on ω. Hence u = α v on ω for some α ∈ R, which implies that there exists σ ∈ S 1 such that f = σ|f | on ω. If ω = Ω, we would have |f | = 0 on ∂ω ∩ Ω = ∅, which would be a contradiction with the fact that |f | > 0. We conclude that ω = Ω and thus that f = σ|f |, which is nothing but the reverse Kato's inequality condition (H5).

At this stage, we may use Theorem 5.16, in order to get the conclusion (C3) on the triviality of the boundary punctual spectrum.

In order to go one step further and establish the asymptotic stability of f 1 , we may use the two following approaches which are consequences respectively of Lemma 7.1 and Lemma 7.2.

Lemma 7.1. For any R > 0, the set

K := {f ∈ D(L); [f ] ≤ R, [Lf ] ≤ R} is strongly compact in L 1 loc (Ω), where [g] := g L 1 φ 1 . Proof of Lemma 7.1. Consider f ∈ K so that f ∈ H 1 0 (Ω) and ∂ i (a ij ∂ j f ) + b i ∂ i f + ∂ i (β i f ) + cf = g ∈ L 2 (Ω).
From the renormalization theory of elliptic equations and the GRE trick (see for instance [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF] and the references therein) for any renormalizing function H ∈ C 2 (R), there holds

H ′′ (u)f 1 φ 1 a∇u • ∇u = div(aφ 1 ∇(H(u)f 1 )) -div(f 1 H(u)a∇φ 1 ) + div((b + β)H(u)f 1 φ 1 ) + gH ′ (u)f 1 φ 1 , with u := f /f 1 . Considering H ∈ W 2,∞
the even (and convex) function such that H(0) = 0 and

H ′′ := 1 [n,n+1]
, so that in particular |H ′ (s)| ≤ 1, and integrating the previous equation, we deduce

ν |u|∈[n,n+1] |∇u| 2 f 1 φ 1 ≤ |g|f 1 φ 1 ≤ f 1 L ∞ R.
We proceed along the line of the proof of [START_REF] Boccardo | Nonlinear elliptic and parabolic equations involving measure data[END_REF]Thm. 1]. For a fixed ω ⊂⊂ Ω, we define

B n := {x ∈ ω; |u(x)| ∈ [n, n + 1]}. Using that f 1 > 0 and φ 1 > 0, there exists a constructive constant C ω,R > 0 such that Bn |∇u| 2 ≤ C 2 ω , ∀ n ≥ 0.
From the Cauchy-Schwarz inequality, we have (7.29)

Bn |∇u| ≤ C ω meas(B n ) 1/2 , ∀ n ≥ 0.
On the other hand, denoting by 1 * := d/(d -1) the Sobolev exponent, we have

Bn |∇u| ≤ C ω,R n -1 * Bn |u| 1 * 1/2 .
Summing up and using the Cauchy-Schwarz inequality again, we have

n≥1 Bn |∇u| ≤ C ω,R n≥1 n -1 * 1/2 n≥1 Bn |u| 1 * 1/2 ≤ C ω,R n≥1 n -1 * 1/2 u 1 * /2 L 1 * .
Together with (7.29) for n = 0, we deduce

∇u L 1 (ω) ≤ C ′ ω,R (1 + ∇u 1 * /2 L 1 (ω)
). Because 1 * /2 ≤ 3/4 < 1 (recall that d ≥ 3), we can kill the last term, and we obtain the estimate

∇(f /f 1 ) L 1 (ω) ≤ C ′′ , ∀ f ∈ K, for some constant C ′′ := C ′′ ω,R > 0.
We classically conclude thanks to the Rellich-Kondrachov theorem.

From the above lemma and Theorem 5.23, we deduce that S(t)f → f, φ 1 f 1 in the L 1 φ1 norm sense as t → ∞ for any f ∈ L 2 (Ω). The alternative approach is based on the following result.

Lemma 7.2. Setting κ := κ 0 -1, there exist A, α, R > 0 such that (i) sup z∈∆κ y α R B (z) B(L 2 ;H 1 0 ) + sup z∈∆κ\BR R L (z) B(L 2 ;H 1 0 ) < ∞, (ii) Σ(L) ∩ ∆ κ ⊂ Σ d (L) ∩ B R ,
where B := L -A and z = x + iy, x, y ∈ R.

Proof of Lemma 7.2. Let us consider an a priori solution to the stationary problem

f ∈ H 1 0 , z = x + iy ∈ ∆ κ , (L + z)f = g ∈ L 2
. This one satisfies

-(a∇f + βf ) • ∇ f + b • ∇f f + (c + z)|f | 2 = g f .
Using the elliptic condition, the Cauchy-Schwarz inequality and triangular inequalities, we get

g f ≥ a∇f ∇ f + ((c + x) + + iy)|f | 2 - b • ∇f f -βf • ∇ f + (c + x) -|f | 2 ≥ ν 2 ∇f 2 L 2 + |y| 2 -x -f 2 L 2 -(|b| + |β|)f L 2 ∇f L 2 - √ c -f 2 L 2 .
Using next similar arguments and those introduced in the paragraph dedicated to condition (H1) and with similar definition for the constant M := M (b, β, c) > 0, we deduce

g f ≥ |y| 2 -x --M f 2 L 2 + ν 4 ∇f 2 L 2 .
Defining the sectorial set

S := z = x + iy ∈ C; |y| > 2x -+ M ,
we have established the a priori estimates

f L 2 ≤ |y| 2 -x --M -1/2 g L 2 , ∇f L 2 ≤ 2ν -1/2 |y| 2 -x --M -1/4 g L 2 ,
for any z ∈ S. We classically and immediately deduce that ρ(L) ⊃ S and the resolvent estimate

R L (z) B(L 2 ,H 1 0 ) |y| 2 -x --M -1/2 + |y| 2 -x --M -1/4
for any z ∈ S, and in particular the estimate (i) holds true.

On the other hand, because L has compact resolvent as established just above or during the proof of (H3) and using the Fredholm alternative, we have Σ(L) = Σ d (L) and Σ(L) ∩ ∆ κ is finite for any κ ∈ R, what is nothing but the property (ii).

From the above lemma and Theorem 5.30 or Theorem 5.32, we deduce that S(t)f → f, φ 1 f 1 in the L 2 norm sense as t → ∞ for any f ∈ L 2 (Ω) with exponential rate.

We may summarize our analysis in the following result.

Theorem 7.3. Consider the elliptic operator (7.1) in a bounded domain and assume that the coefficients satisfy (7.2), (7.5) and (7.18). Then the conclusions (C3) holds as well as (E2) in L 1 φ1 norm and (E3 1 ) in L 2 with non constructive rate. It is however worth emphasizing again that the above approach is definitively not constructive. We propose now an alternative approach which is constructive.

Quantitative estimate of stability. Using the Doblin-Harris type approach presented in Section 6, we are able to establish a rate of convergence to the principal dynamic, at least in a regular framework. We thus make some regularity assumptions on Ω and additional regularity assumptions on the coefficients.

-For the domain, we assume that there exists a constant r Ω > 0 such that for any x ∈ Ω there is y ∈ Ω such that x ∈ B(y, r Ω ) ⊂ Ω, in particular, for any x ∈ ∂Ω there is y ∈ Ω such that x ∈ ∂B(y, r Ω ), B(y, r Ω ) ⊂ Ω. We also assume that Ω is C 1,1 .

-For the coefficients, we assume a ij ∈ C( Ω), bi , c ∈ L ∞ (Ω), where bi and c are defined in (7.25).

Theorem 7.4. Consider the elliptic operator (7.1) in a bounded domain and assume that the assumptions of Theorem 7.3 hold together with the above additional regularity assumptions on the coefficients and the boundary. Then the conclusion (E3 1 ) holds with constructive exponential rate.

The proof of Theorem 7.4 follows from Theorem 6.3. We split the proof into several steps.

-Step 1. Regularity estimates. Thanks to De Giorgi-Nash-Moser regularity technique for parabolic equations developed for instance in [START_REF] Ladyženskaja | ceva. A boundary-value problem for linear and quasi-linear parabolic equations. I, II, III[END_REF] (in Russian), [346, Thm. 1.3, Thm. 2.2] as well as more recently in [230, Lem. 2.7] and [START_REF] Guerand | Quantitative regularity for parabolic De Giorgi classes[END_REF]Thm. 1.1], there exists α = α(a ij ) ∈ (0, 1) and for any T 1 > T 0 > 0 and any ̺ ∈ (0, 1), there exist constructive constants

C i = C i ( f L ∞ t L 2 x , T, τ, r) such that any solution f ∈ L ∞ (0, ∞; L 2 (Ω)) to the parabolic equation ∂ t f = Lf satisfies (7.30) f L ∞ ([T0,T1]×Ω) ≤ C 1 , f C α ([T0,T1]×ω̺) ≤ C 2 ,
with ω r := {x ∈ Ω; d(x, ∂Ω) > r}. More precisely, in order to establish the second estimate in (7.30) with constructive constant, one may observe that the proof of [191,

u C 0,1 (Ω) u W 2,d+1 (Ω) (κ 1 -L)u L d+1 (Ω) ,
see for instance Theorem 7.10, Theorem 7.25 and Lemma 9.17 in [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]. Iterating the same kind of arguments, we get

(7.32) u C 0,1 (Ω) ≤ C (κ 1 -L) k u L 2 (Ω) ,
with constructive constants C and k.

-Step 2. Harnack estimate. We claim that for any T > t 0 > 0 and ̺ > 0, there exist a constant C H > 0 such that, for any f 0 ∈ L 2 , the associated solution f := S L f 0 satisfies

(7.33) sup ω̺ f t0 ≤ C H inf ω̺ f T .
The proof mainly follows form Aronson-Serrin [START_REF] Aronson | Local behavior of solutions of quasilinear parabolic equations[END_REF] (see also [START_REF] Moser | A Harnack inequality for parabolic differential equations[END_REF][START_REF] Ivanov | A priori estimates for solutions of linear second-order equations of elliptic and parabolic types[END_REF][START_REF] Ivanov | Certain properties of generalized solutions of parabolic equations of the second order[END_REF][START_REF] Kružkov | A priori bounds and some properties of solutions of elliptic and parabolic equations[END_REF][START_REF] Ivanov | The Harnack inequality for generalized solutions of second order quasilinear parabolic equations[END_REF][START_REF] Trudinger | On Harnack type inequalities and their application to quasilinear elliptic equations[END_REF][START_REF] Trudinger | Pointwise estimates and quasilinear parabolic equations[END_REF][START_REF] Kurihara | On a Harnack inequality for nonlinear parabolic equations[END_REF] for similar results). First, we know from [START_REF] Aronson | Local behavior of solutions of quasilinear parabolic equations[END_REF]Thm. 3] that (7.34) max

Q * (ρ) f ≤ C min Q(ρ) f, for any ρ > 0, t > 0 such that Q * (3ρ) ⊂ (0, ∞) × Ω, where Q(ρ) := [t -ρ 2 , t] × C(ρ), Q * (ρ) := [t -8ρ 2 , t -7ρ 2 ] × C(ρ)
and C(ρ) is a cube with length ρ. To avoid technical issues we assume that w ̺ is convex. In other case, the geometrical condition given above implies that there is N ∈ Z + such that any two points x, y ∈ Ω can be connected by a polygonal path of at most N segments, and we can argue as follows for any segment. We define D := sup a,b∈Ω d(a, b) the diameter of Ω and we choose r ′ < ̺/7 such that

7(⌊ D 2r ′ ⌋ + 1)(r ′ ) 2 < T -t 0 .
For any x, y ∈ ω ̺ , we also define

N c = ⌊ |x-y| r ′ ⌋. Since ω ̺ is convex, r ′ < ̺/7
, we have that the family of cubes {C(x i , 2r ′ )} i=0,Nc of center x i and length 2r ′ for x i = x + (x-y)i Nc satisfy that C(x i , 6r ′ ) ⊂ Ω and C(x i , 2r ′ ) ∩ C(x i+1 , 2r ′ ) = ∅ for any i = 0, . . . , N c . As a consequence, we can apply Aronson-Serrin estimate (7.34) for each cube to obtain max

C(xi,2r ′ ) f ti ≤ C 2r ′ min C(xi,2r ′ ) f ti+1 , with t i = t 0 + 7i(2r ′ ) 2 . Taking y i ∈ C(x i , 2r ′ ) ∩ C(x i+1 , 2r ′ ), we deduce max C(xi,2r ′ ) f ti ≤ C 2r ′ min C(xi,2r ′ ) f ti+1 ≤ C 2r ′ f ti+1 (y i ) ≤ C 2r ′ max C(xi+1,2r ′ ) f ti+1 ≤ C 2 2r ′ min C(xi+1,2r ′ ) f ti+2 .
By induction, we obtain

f t0 (x) ≤ max C(x1,2r ′ ) f t1 ≤ C Nc 2r ′ min C(xN c ,2r ′ ) f tN c ≤ C Nc 2r ′ f tN c (y),
with t Nc = t 0 + 7N c r ′2 ≤ T. Note that in any case the constant C 2r ′ is the same since it only depends on the length 2r ′ and the coefficient of the equation. We have thus established (7.33) with

C H := C ⌊ D 2r ′ ⌋+1 2r ′ .
On the other hand, we state an improved version of the already mentioned stationary Harnack inequality. Because of the interior ball condition the Hopf Lemma (see for instance the proof of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Lem. 3.4]) claims that for any ̺ ∈ (0, r Ω /2] there exists a constructive constant α > 0 such that if u ∈ W 2,p (Ω), p > d, is such that

u ≥ 1 ω̺ , (κ 1 -L * )u ≥ 0, then u satisfies (7.35) u ≥ χ(x) := e -α(2̺-δ(x)) 2 -e -α(2̺) 2 on ω c ̺ .
Let us give two applications of the above sharp regularity and positivity estimates. First, recalling (7.24) and using (7.31) and (7.35), we deduce that there exist two constructive constants c i ∈ (0, ∞) such that (7.36) c 0 δ ≤ φ 0 ≤ c 1 δ on Ω.

Consider now f 1 ∈ H 1 0 (Ω) the positive first eigenfunction with normalization f 1 L 2 = 1. Using the estimate of regularity (7.32) on the iterated equation (κ

1 -L) k f 1 = (κ 1 -λ 1 ) k f 1 , we have f 1 L ∞ (Ω) ≤ f 1 C 0,1 (Ω) ≤ C 1 ,
for some constructive constant C 1 ∈ (0, ∞). Next using the elementary inequality

1 = Ω f 2 1 ≤ f 1 L ∞ f 1 L 1 ≤ C 1 f 1 L 1 ,
we deduce

|Ω| sup ω̺ f 1 ≥ ω̺ f 1 = f 1 L 1 - ω c ̺ f 1 ≥ 1/C 1 -C 1 |ω c ̺ | ≥ 1/(2C
f 1 ≥ C H sup ω̺ f 1 ≥ C H (2C 1 |Ω|) -1 .
Finally, from the above Hopf lemma and the above Lipschitz continuity, we have established

(7.37) c 0 δ ≤ f 1 ≤ c 1 δ on Ω,
for two constructive constants c i ∈ (0, ∞). The same arguments on the normalized and positive first dual eigenfunction φ 1 lead to the same estimate

(7.38) c 0 δ ≤ φ 1 ≤ c 1 δ on Ω.
In particular, for any such ̺ ∈ (0, r Ω /2), we have

(7.39) φ 1 , 1 ω̺ ≥ r ̺ ,
with constructive constant r ̺ , what is nothing but condition (6.9) in the Harris theorem that we will use below.

-Step 3. Splitting of L. We introduce the splitting L = A + B, with Af = M 1 ω̺ f , M ≥ 0 large enough and ̺ > 0 small enough that we fix just below. Using (7.8), we observe that

(Bf, f ) L 2 = (Lf, f ) L 2 -M f L 2 + M ω c ̺ f 2 ≤ - ν 4 ∇f 2 L 2 + (κ 1 -M ) f 2 L 2 + M |ω c ̺ | 4/d f 2 L 2 * ≤ κ 0 f 2 L 2 ,
by choosing first M ≥ κ 1κ 0 and next ̺ > 0 small enough in order to be able to throw away the last term using the negative first term and the Sobolev inequality. We deduce

(7.40) S B (t) : L 2 → L 2 with bound O(e κ0t ).
On the other hand, denoting f t := S L (t)f for f ∈ L 2 (Ω) and recalling that φ 0 defined by (7.22) satisfies (7.23), we have

d dt |f t |φ 0 ≤ L|f t |φ 0 ≤ |f t |L * φ 0 ≤ κ 1 |f t |φ 0 , so that (7.41) |f t |φ 0 ≤ e κ1t |f 0 |φ 0 .
Arguing in the same way for S B and using (7.36), we have established

(7.42) S L (t), S B (t) : L 1 δ → L 1 δ with bound O(e κ1t
). For a solution to the evolution equation

∂ t f = Cf , C = L or C = B, we also classically compute d dt f 2 φ 0 = 2 (Cf )f φ 0 = -2 (∇f • a∇f )φ 0 + f 2 C * φ 0 .
Thanks to (7.23) again, we have

(7.43) d dt f 2 φ 0 ≤ -2ν |∇f | 2 φ 0 + κ 1 f 2 φ 0 ,
from what we deduce

S L (t), S B (t) : L 2 (δ) → L 2 (δ) with bound O(e κ1t/2
).

In the sequel, we will need the following version of Nash inequality.

Lemma 7.5 (weighted Nash inequality).

There exists a constructive constant C N such that

(7.44) f L 2 (δ) ≤ C N ∇f d+1 d+2 L 2 (δ) f 1 d+2 L 1 δ , ∀ f ∈ H 1 (δ).
Proof of Lemma 7.5. For ε > 0, we define

f ε (x) := 1 δ ε (x) B(x,ε) f (y) δ(y)dy, δ ε (x) = δ(B(x, ε)) := B(x,ε)
δ(y)dy, and B(x, ε) := {y ∈ Ω; |x -y| < ε}. It is worth emphasizing that (7.45)

ε d+1 δ ε (x) ε d , ∀ ε > 0.
For f ∈ H 1 (δ), we compute

f -f ε 2 L 2 (δ) = Ω 1 δ ε (x) B(x,ε) (f (y) -f (x)) δ(y)dy 2 δ(x)dx ≤ Ω Ω 1 |y-x|≤ε |f (y) -f (x)| 2 δ(y) δ ε (x) δ(x)dxdy ≤ ε 2 1/2 0 Ω Ω |∇f ((1 -t)x + ty)| 2 δ(y) δ ε (x) δ(x)dxdydt +ε 2 1 1/2 Ω Ω |∇f ((1 -t)x + ty)| 2 δ(y) δ ε (x) δ(x)dxdydt ε 2 1/2 0 Ω Ω |∇f (z)| 2 δ(y) ε d+1 dy2δ(z) dz (1 -t) d dt +ε 2 1 1/2 Ω Ω |∇f (z)| 2 2δ(z) ε d+1 δ(x) dz t d dtdx
where for the last inequality we have used the first inequality in (7.45), the fact that δ(x) ≤ 2δ(z) when 0 < t < 1/2 and the fact that δ(y) ≤ 2δ(z) when 1/2 < t < 1. Using the second inequality in (7.45), we straightforwardly obtain

f -f ε 2 L 2 (δ) ≤ C 1 ε ∇f 2 L 2 (δ) , ∀ ε > 0, for a constant C 1 > 0.
On the other hand, we also observe that

f ε L ∞ ≤ C 2 ε d+1 f L 1 δ . Writing now f 2 = f (f -f ε ) + f f ε
and using the above two estimates, we deduce

f 2 L 2 δ ≤ f L 2 δ f -f ε L 2 δ + f L 1 δ f ε L ∞ . ≤ f L 2 δ C 1 ε 1/2 ∇f L 2 δ + C 2 ε -d-1 f 2 L 1 δ ≤ 1 2 f 2 L 2 δ + C 1 2 ε ∇f 2 L 2 δ + C 2 ε -d-1 f 2 L 1 δ ,
and we obtain the weighted Nash inequality (7.46) by choosing ε := ( f 2

L 1 δ / ∇f 2 L 2 δ ) 1/(d+2) .
Defining

u := |f t |φ 0 dxe -2κt , v := f 2 t φ 0 dxe -2κt ,
with κ := κ 1+ , coming back to (7.43) and using (7.36), the Nash inequality (7.46) and the estimate (7.41), we get

v ′ (t) ≤ -2νc 0 |∇f t | 2 δe -2κt ≤ -2νc 0 C -2 d+2 d+1 N f t 2 L 2 (δ) e -2κt d+2 d+1 f t 2 L 1 (δ) e -2κt 1 d+1 ≤ -C v(t) 1+α u(0) 2α , with C := 2νC -2 d+2 d+1 N c 1+2 d+2 d+1 0 c - 2 d+1 1
and α := 1/(d + 1). Integrating in time, we deduce

v(t) ≤ α 1/α C 1/α u(0) 2 t 1/α , ∀ t > 0.
We have thus established that there exist constructive constants K > 0 and κ ≥ 0 such that

(7.46) S C (t)f L 2 (φ0) ≤ K e κt t (d+1)/2 f L 1 (φ0) , ∀ f ∈ L 1 (φ 0 ).
From that last result, the estimates (7.36) and the properties of A, we deduce that for N ≥ 1 large enough

(7.47) (S B A) ( * N ) : L 1 (δ) → L 2 (δ) with bound O(e κt ).
We refer to [190,

S L = V + W * S L , with V := S B + • • • + ( S B A) ( * (N -1)) , V := ( S B A) ( * N ) .
On the one hand, using that A : L 2 → L 2 is bounded and (7.40), we deduce that

V : L 2 → L 2 , with bound O(e κt ),
for any κ ∈ (κ 0κ 1 , 0). On the other hand, using that A : L 2 δ → L 2 is bounded as well as (7.42) for S L , (7.47), (7.38), (7.36) and (7.40), we deduce that

W * S L : L 1 φ1 → L 2 , with bound O(e κ ′ t
), for any κ ′ > κ 1κ 0 . We may thus fix t = T large enough such that the following Lyapunov inequality holds

(7.48) S L (T )f L 2 ≤ 1 2 f L 2 + M T f L 1 φ 1
, which is nothing but (6.7) in the hypothesis of the Harris theorem.

-Step 5. Harris condition Let A > 0 and consider 0 ≤ f 0 ∈ L 2 such that f 0 2 ≤ A f 0 , φ 0 . We set f t := e -λ1t S L (t)f 0 . From the first inequality in (7.23), we have

d dt f t , φ 0 = f t , (L * -λ 1 )φ 0 ≥ -(λ 1 -κ 0 ) f t , φ 0 ,
and then, thanks to Gronwall lemma again, we obtain,

f t , φ 0 ≥ e -(λ1-κ0)t f 0 , φ 0 .
This estimate, together with the previous step, shows that

ω̺ f t0 (x)φ 0 dx = Ω f t0 (x)φ 0 dx - ω c ̺ f t0 (x)φ 0 dx ≥ e -(λ1-κ0)t0 f 0 , φ 0 -f t0 2 φ 0 ∞ |ω c ̺ | 1/2 ≥ e -(κ1-κ0)t0 f 0 , φ 0 -e (κ1-κ0)t0 f 0 2 φ 0 ∞ |ω c ̺ | 1/2 ≥ e -(κ1-κ0)t0 -Ae (κ1-κ0)t0 φ 0 ∞ |ω c ̺ | 1/2 f 0 , φ 0 .
Choosing ̺ > 0 small enough, we get

ω̺ f t0 (x)φ 0 dx ≥ γ f 0 , φ 0 , γ := 1 2 e -(λ1-κ0)t0 .
As a consequence, there is

x f t0 ∈ ω ̺ such that f t0 (x f t0 ) ≥ 1 |ω ̺ | ω̺ f t0 (x)dx ≥ 1 |Ω|c 1 ̺ ω̺ f t0 (x)φ 0 dx ≥ γ |Ω|c 1 ̺ f 0 , φ 0 .
On the other hand, from the Harnack inequality (7.33) established in Step 2, we know that for any T > t 0 , there exits C H such that

f t0 (x f t0 ) ≤ sup ω̺ f t0 ≤ C H inf ω̺ f T .
The two last estimates together with (7.38) and (7.36) imply the Harris type estimate (7.49)

f T = S(T )f 0 ≥ g A f 0 , φ 1 , with g A := c0γ CH |Ω|c 2 1 ̺ 1 w̺ ,
Lf := ∆f + b • ∇f + cf, f ∈ H 1 (R d ), with b ∈ L ∞ loc (R d ), c ∈ L 2 loc (R d
) and a confinement condition that we impose through the properties of the potential function c, which is roughly speaking c → -∞ as |x| → ∞. More precisely, we assume (7.51)

σ i+ ∈ L d/2 , meas{σ i ≥ K} < ∞, ∀ K < 0,
with either σ 1 := c + |b| 2 /κ for some constant κ ∈ (0, 4) or either σ 2 := c + divb/2. When we assume that c ∼ -|x| γ and b ∼ x|x| β-1 as |x| → ∞, the condition (7.51) for σ 1 is reached when γ > max(0, 2β) or γ = 2β > 0 and some conditions on the constants involved in the behavior of the coefficients. In that context, the condition (7.51) for σ 2 is more general since it is reached when γ > max(0, β -1) or γ = β -1 > 0 and some conditions on the constants involved in the behavior of the coefficients. A similar framework is considered in [START_REF] Lions | Premières valeurs et fonctions propres. Cours du Collège de France 2020-2021[END_REF] and for the reader convenance we just briefly check that it falls in the framework developed before by slightly modifying the arguments presented in the previous section. The integrability conditions on b and c may be probably weaken. For the sake of clarity we do not follow this line of research but rather focus on the new arguments which are necessary in order to deal with the unbounded domain Ω = R d . Condition (H1). The definition of the operator is still made through the formula (7.7). Under assumption (7.51) on σ 1 , denoting θ 1 := 1κ/4 and proceeding exactly as in the previous section during the proof of (7.8), for any f ∈ H 1 (R d ) and λ ∈ R, we have

(λ -L)f, f = R d |∇f | 2 + R d f b • ∇f + R d (λ -c)f 2 ≥ θ 1 R d |∇f | 2 + R d (λ -σ 1 )f 2 ,
by using successively the Cauchy-Schwarz inequality and the Young inequality. On the other hand, under assumption (7.51) on σ 2 , denoting θ 2 := 1, for any f ∈ H 1 (R d ) and λ ∈ R, we write

(λ -L)f, f = θ 2 R d |∇f | 2 + R d (λ -σ 2 )f 2 ,
by performing one integration by part in the previous equation. In both cases, for and any M > 0, proceeding again as in the previous section during the proof of (7.8), and denoting from now on σ = σ i , θ = θ i we have

(λ -L)f, f ≥ θ 2 ∇f 2 L 2 + √ σ -f 2 L 2 + (λ -M ) f 2 L 2 + ( θC S 2 -σ1 σ≥M L d/2 ) f 2 L 2 *
, by using the Sobolev inequality (with associated constant C S ) and the Holder inequality. Taking M > 0 large enough, and next κ 1 > 0 large enough, we finally obtain

(7.52) (λ -L)f, f ≥ θ 2 ∇f 2 L 2 + √ σ -f 2 L 2 + f 2 L 2 , ∀ λ ≥ κ 1 .
With the same arguments as in the previous section, we conclude that L is the generator in L 2 of a positive semigroup S L , so that (H1) holds.

Condition (H2). We may for instance use the third constructive argument (which is local) presented in section 7.1 and we establish [START_REF] Agoshkov | Some trace and extension theorems for functions from W 1 x i ,p (D) in the case of a bounded domain[END_REF], so that condition (H2) holds.

∃ f 0 ∈ H 1 0 \{0}, f 0 ≥ 0, ∃ κ 0 ∈ R, Lf 0 ≥ κ 0 f 0 . That is condition (ii) in Lemma 2.
Condition (H3). We introduce again the splitting L = A + B with A := κ 1κ 0 + 1, so that from (7.52), the operator λ -B = (λκ 0 + 1) + (κ 1 -L) is invertible for any λ ≥ κ B := κ 0 -1. We claim that the operator (λ -B) -1 is compact for any λ ≥ κ B . For that purpose, let us consider a sequence (f n ) such that (λ -B)f n is bounded in L 2 and we have to prove that (f n ) is relatively strongly compact. When condition (7.51) holds and because of the estimate (7.52) and the very definition of B, we have

(7.53) θ 2 ∇f n 2 L 2 + √ σ -f n 2 L 2 + f n 2 L 2 ≤ C, for some constant C ∈ R + . Because of the Rellich-Kondrachov theorem, we just have to show that lim R→∞ sup n B c R f 2 n = 0.
But that last convergence may be established using the assumption (7.51) in the following way. We write

B c R f 2 n = B c R ∩{σ≥K} f 2 n + B c R ∩{σ<K} f 2 n ≤ f n d-2 d L 2 * [meas(B c R ∩ {σ ≥ K})] 2 d + 1 |K| σ -f 2 n ,
for any K < 0, by using the Holder inequality. Using next the Sobolev inequality, the estimate (7.53) and the assumption (7.51), we deduce lim sup

R→∞ B c R f 2 n lim sup R→∞ inf K<0 [meas(B c R ∩ {σ ≥ K})] 2 d + 1 |K| = 0,
and the claim is proved. As a consequence, we may apply Lemma 2.8-(2) and we deduce that (H3) holds for both the primal and the dual problems.

Condition (H4).

As in [231, Prop. 5.4], we establish the strong maximum principle by exhibiting a barrier function and using Lemma 4.11. An alternative argument should be to adapt the proof based on the Harnack inequality as presented in the previous section. Let us then consider f ∈ D(L k ) ∩ X + \{0} such that (λ -L)f ≥ 0 with k large enough (k > d/2 must be suitable) and λ ≥ λ 1 large enough but fixed (λ ≥ κ 1 is suitable). Using a very classically bootstrap argument based on iterated application of the Calderon-Zygmond elliptic regularity theorem and the Morrey estimate, we have f ∈ C(R d ). By assumption, there thus exist x 0 ∈ R d , and two constants τ, r > 0 such that f ≥ τ on B(x 0 , r) and we take choose x 0 = 0 in order to simplify the notations. We next fix R > r and we observe that the function

g(x) := τ * (g 0 (|x|) -g 0 (R)), g 0 (s) := exp(σr 2 /2 -σs 2 /2), satisfies (τ * ) -1 (λ -L)g = (λ -c)(g 0 -g 0 (R)) + (dσ -σb • x -σ 2 r 2 ) g 0 ≤ [2(|λ| + c L ∞ (BR) ) + σ(d + b • x L ∞ (BR) ) -σ 2 r 2 ] g 0 ≤ 0
on O := B(0, R)\B(0, r) for σ > 0 large enough. We next fix τ * such that g = τ on ∂B(x 0 , r). We also observe that from (7.52), λ -L is coercive on O, in the sense that

∀ h ∈ H 1 0 (O) ((λ -L)h, h) L 2 (O) ≥ h L 2 (O) .
In particular, λ -L satisfies the weak maximum principle as explained in the proof of (7.9). Arguing as in the proof of Lemma 4.11, we deduce that f ≥ g > 0 on O, what we also see directly by observing that h := (gf ) + ∈ H 1 0 (O), (λ -L)h ≤ 0 and using that the weak maximum principle implies h ≤ 0, thus h ≡ 0 and finally f ≥ g. Because R > r can be chosen arbitrarily large, we conclude with f > 0 on R d .

Condition (H5). The reverse Kato's inequality condition is proved by using local arguments, so that it holds for the same reasons as in the previous section. Similarly, because the argument are local, the conclusion of Lemma 7.1 holds here.

As a consequence, using Theorem 2.21, Theorem 4.13, Theorem 5.16 and Theorem 5.23, we may summarize our analysis in the following result.

Theorem 7.6. Consider the elliptic operator (7.50) in the whole space and assume that the coefficients satisfy (7.51). Then the conclusions (C3) holds as well as (E2) in L 1 φ1 . We do not present an exponential constructive estimate, which we believe is possible to prove, but would require significantly more development.

7.3. Diffusion in R d with weak potential confinement. We consider in this section the same elliptic operator (7.50) with now a weak confinement condition assuming that c converges to a constant. With no loss of generality, we may assume c → 0. More precisely, we consider the elliptic operator (7.54)

Lf := ∆f + b • ∇f + rcf, with c ∈ C 0 (R d ), b ∈ C 0 (R d
) and r ∈ R + a parameter. When not necessary in the discussion we will take r = 1. The associated first eigenvalue problem in such a situation has been studied in [252, 8th and 9th courses] to which we refer for more details. We define

λ 1 = λ 1 (r) := inf{κ ∈ R; (λ -L) -1
well defined and positive for any λ ≥ κ}.

Proceeding exactly as in the proof of (H1) in the preceding section, we see that the operator λ -L is invertible for any λ > c + L ∞ , and then its inverse is positive. Because the proof of (H2) in the preceding section also applies here, we deduce that the infimum λ 1 of the set I of real resolvent values is well defined with λ 1 ∈ (κ 0 , κ 1 ), for some constructive constants κ i ∈ R.

We split now the discussion into two cases. Case 1. We start considering the case b = 0. In that case, L is self-adjoint so that λ 1 is also characterized by

λ 1 = sup f L 2 =1 E(f ), with E(f ) := (Lf, f ) = r cf 2 -|∇f | 2 .
We make the following elementary observations :

• We claim that λ 1 ≥ 0. Taking f n (x) := n -d/2 u(x/n) for some function u ∈ H 1 (R d ), u L 2 = 1, we compute -E(f n ) = |∇f n | 2 - BR rcf 2 n - B c R rcf 2 n ≤ 1 n 2 |∇u| 2 + rc L ∞ (BR) B R/n u 2 + rc L ∞ (B c R ) ,
for any R > 0, so that -λ 1 ≤ lim sup(-E(f n )) ≤ 0.

• We claim that λ 1 = 0 when c ≤ 0. In that case, we have E(f ) ≤ 0 for any f ∈ H 1 (R d ), and we deduce the reverse inequality λ 1 ≤ 0. In particular, as a function λ 1 = λ 1 (r) of r ≥ 0, we have λ 1 (0) = 0. We also claim that λ 1 (r) → ∞ as r → ∞ when c + ≡ 0. We may indeed fix f ∈ H 1 (R d ), f L 2 = 1, supp f ⊂ supp c + , and we compute

E(f ) = r R d c + f 2 -|∇f | 2 → ∞, as r → ∞.
• We finally observe that λ 1 : R + → R + is convex since it is defined as the supremum of linear functions r → E(f ) for any fixed f ∈ H 1 (R d ). As a consequence, we have the following alternative:

-λ 1 ≡ 0; -∃r 0 ∈ [0, ∞) such that λ 1 (r) = 0 for r ≤ r 0 and λ 1 (r) > 0 for r > r 0 .

Concerning the value of r 0 , it may happen that r 0 > 0, and that is the case when c ∈ L d/2 because of the Sobolev inequality, or that r 0 = 0, and that is the case for instance when c ≥ 0, c(x) = |x| -m for x ∈ B c R , m ∈ (0, 2), R > 0. To prove that last claim, we may take the same sequence (f n ) as above, and we compute

E(f n ) ≥ B c R r|x| -m f 2 n -|∇f n | 2 dx = r n m B c R/n |x| -m u 2 - 1 n 2 |∇u| 2 dx > 0,
for n large enough (whatever is the value of r > 0).

About condition (H3).

It is established in [START_REF] Lions | Premières valeurs et fonctions propres. Cours du Collège de France 2020-2021[END_REF] that when λ 1 = 0, the condition (H3) is not satisfied and there does not exist a first eigenfunction f 1 ∈ L 2 (R d ) to the operator L defined by (7.54). We refer to [252, 8th course] for a proof of that result. On the other hand, we claim that the condition (H3) is satisfied when λ 1 > 0. Consider indeed three sequences (λ

n ) of R, (f n ) of H 1 (R d ) and (ε n ) of L 2 (R d ) such that (λ n -L)f n = ε n , ε n , f n ≥ 0, f n L 2 = 1, for any n ≥ 1, λ n → λ 1 and ε n → 0 in L 2 as n → ∞.
We then have

λ n -E(f n ) = (λ n -L)f n , f n = ε n , f n → 0,
as n → ∞. By definition of E and boundedness of c, we see that (f n ) is bounded in H 1 . As a consequence, up to the extraction of a subsequence, we have f n → f 1 ≥ 0 in L 2 loc and thus next (λ 1 -L)f 1 = 0 in the variational sense and

cf 2 n → cf 2 1 , ∇f 1 L 2 ≤ lim inf ∇f n L 2 ,
where we have used the dominated convergence theorem of Lebesgue and the fact that c → 0 at infinity in order to get the first convergence. We finally deduce

E(f 1 ) ≥ lim sup E(f n ) = λ 1 > 0,
so that f 1 ≡ 0, and the condition (H3) is verified.

As a conclusion, for a self-adjoint operator, condition (H3) is automatically fulfilled by its adjoint and the conditions (H4) and (H5) have been proved in a general situation, including the present framework. The same conclusions of existence, uniqueness and asymptotic stability of the first eigentriplet solution (λ 1 , f 1 , φ 1 ) as in section 7.2 hold true when λ 1 > 0.

Case 2. We consider the general case b ∈ C 0 (R d ).

• We claim that λ 1 ≥ 0. Adapting the second constructive argument in the proof of (H2) in Section 7.1, we consider

χ ∈ C 1 c (R + ) ∩ W 2,∞ (R + ) such that 1 [0,1/2] ≤ χ ≤ 1 [0,1] , χ ′ ≤ 0 on [0, 1], χ(s) := (1 -s) 2 /2 on [η, 1] with η ∈ (1/2, 1) large enough in such a way that (7.55) χ ′′ (s) + (d -1)χ ′ (s)/s ≥ 1/2, ∀ s ∈ (η, 1),
and define f 0 (x) := χ(|x-x 0 |/n) for |x 0 | large enough to be chosen later. We have supp f 0 ⊂ B n (x 0 ) for any n ≥ 1 and we compute

Lf 0 (x) = 1 n 2 χ ′′ (r/n) + d -1 r/n χ ′ (r/n) + 1 n b(x) • ŷ χ ′ (r/n) + c(x)χ(r/n)
where y = xx 0 and r = |y|. On B nη (x 0 ), we have

Lf 0 (x) ≥ - χ ′′ ∞ n 2 - d -1 n 2 χ ′ (r) r ∞ - χ ′ ∞ n sup Bn(x0) |b| -χ ∞ sup Bn(x0)
|c|.

On B n (x 0 ) \ B ηn (x 0 ), thanks to (7.55), we have

Lf 0 (x) ≥ 1 2n 2 - χ ′ ∞ n sup Bn(x0) |b| -χ ∞ sup Bn(x0)
|c|.

Let now fix ε > 0 and choose first n large enough so that

- χ ′′ ∞ n 2 - d -1 n 2 χ ′ (r) r ∞ ≥ - ε 2 inf (0,η) χ.
Then, using that b, c ∈ C 0 (R d ), we can take |x 0 | large enough so that

- χ ′ ∞ n sup Bn(x0) |b| -χ ∞ sup Bn(x0) |c| ≥ - ε 2 inf (0,η) χ and χ ′ ∞ n sup Bn(x0) |b| + sup Bn(x0) |c| ≤ 1 2n 2 .
Gathering the above inequalities, we obtain

Lf 0 ≥ -εf 0 ,
and the condition (H2) is verified with κ 0 = -ε. Because ε > 0 can be choose arbitrarily small, we conclude with λ 1 ≥ 0.

• We claim that λ 1 = 0 when σ 2 ≤ 0. Indeed, we have already seen that

Lf, f = - R d |∇f | 2 + R d σ 2 f 2 , from which we deduce that d dt S t f 2 = 2 Lf, f ≤ 0.
This ensures that (H1) is verified with κ 1 = 0 and so λ 1 ≤ 0.

• We claim that λ 1 > 0 when c + ≡ 0 and r > 0 is large enough. For simplifying notations and up to translation and dilatation, we may reduce to the case c ≥ c 0 1 B(0,1) with c 0 > 0.

Adapting the second constructive argument in the proof of (H2) in Section 7.1, we consider

χ ∈ C 1 c (R + ) ∩ W 2,∞ (R + ), 1 [0,1/2] ≤ χ ≤ 1 [0,1] , suppχ = [0, 1], χ ′′ (1) = 1, χ ′ ≤ 0 on [0, 1]
and we set f 0 (x) := χ(|x|). We compute

Lf 0 = χ ′′ (|x|) + χ ′ (|x|)((d -1)/|x| + b • x) + rc(x)χ(|x|).
On the one hand, we fix η ∈ (1/2, 1), 1η small enough, in such a way that

χ ′ L ∞ (η,1) 2(d -1) + b L ∞ ≤ 1/4, 1/2 ≤ χ ′′ L ∞ (η,1) , and thus Lf 0 ≥ 1 4 ≥ 1 4 f 0 on B(0, η) c .
On the other hand, we fix r > 0, large enough, in such a way that

χ ′′ L ∞ + χ ′ L ∞ 2(d -1) + b L ∞ ≤ κ(r) := 1 2 rc 0 inf [0,η) χ,
and thus Lf 0 ≥ κ(r) ≥ κ(r)f 0 on B(0, η).

As a conclusion, we have established that condition (ii) in the statement Lemma 2.4 holds with κ 0 := min(1/4, κ(r)), and that ends the constructive proof of condition (H2) by using Lemma 2.4. That implies in particular the claim since then λ 1 ≥ κ 0 > 0.

• We finally claim that (H3) holds when λ 1 > 0. To see that, we consider again three sequences

(λ n ) of R, (f n ) of H 1 (R d ) and (ε n ) of L 2 (R d ) such that (λ n -L)f n = ε n , ε n , f n ≥ 0, f n L 2 = 1, for any n ≥ 1, λ n ց λ 1 and ε n → 0 in L 2 as n → ∞.
As a consequence, we have

λ n + |∇f n | 2 -f n b • ∇f n -cf 2 n = ((λ n -L)f n , f n ) = ε n , f n → 0,
as n → ∞. Using the boundedness of c, b and λ n , we see that (f n ) is bounded in H 1 . As a consequence, up to the extraction of a subsequence, we have f n → f 1 ≥ 0 in L 2 loc . We assume by contradiction that f 1 ≡ 0. We deduce that

cf 2 n → 0, f n b • ∇f n → 0,
where we have used the dominated convergence theorem of Lebesgue and the fact that b, c → 0 at infinity. We thus obtain

0 < λ 1 ≤ λ n + |∇f n | 2 = f n b • ∇f n + cf 2 n + ε n , f n → 0,
and our contradiction. So that f 1 ≡ 0, and the condition (H3) is verified.

For the dual problem, from the above analysis, we know that there exist two sequences (φ n ) of

H 1 (R d ), (ε n ) of L 2 (R d ) such that (λ n -L * )φ n = ε n , ε n , φ n ≥ 0 and φ n L 2 = 1
, for any n ≥ 1, and ε n → 0 in L 2 as n → ∞. But we face the same situation as previously, since again

λ n + |∇φ n | 2 -φ n b • ∇φ n -cφ 2 n = ((λ n -L * )φ n , φ n ) = (ε n , φ n ) → 0,
and thus the same conclusion, namely

φ n → φ 1 , with φ 1 ∈ H 1 (R d ), φ 1 ≥ 0, φ 1 ≡ 0.
Conclusion. The conditions (H4) and (H5) have been proved in a general situation, including the present framework. The same conclusions as in section 7.2 hold true when r > 0 is large enough (and thus λ 1 > 0).

7.4. Diffusion in R d with drift confinement. We now consider the elliptic operator

Lf := ∆f + b • ∇f + cf,
with a drift confinement as it is the case for the Fokker-Planck operator. More precisely, and for the sake of simplicity, we assume here

(7.56) b = ∇U, U (x) = 1 γ x γ , γ > 0.
When γ = 2 and c = x, that operator corresponds to the classical harmonic Fokker-Planck operator which is known to be related to the standard Poincaré inequality and to the standard log-Sobolev inequality, see [START_REF] Bakry | Diffusions hypercontractives[END_REF][START_REF] Bakry | L'hypercontractivité et son utilisation en théorie des semigroupes[END_REF][START_REF] Toscani | Entropy production and the rate of convergence to equilibrium for the Fokker-Planck equation[END_REF] or more recently [START_REF] Bakry | A simple proof of the Poincaré inequality for a large class of probability measures including the log-concave case[END_REF][START_REF] Kavian | The Fokker-Planck equation with subcritical confinement force[END_REF] and the references therein. When c = divb, the operator L is on divergence form and L * 1 = 0, so that (0, 1) ∈ R × L ∞ (R d ) is a solution to the dual first eigenvalue problem. Existence of stationary solution f 1 (which is also the first eigenfunction) and its stability have been widely studied. We refer for instance to [START_REF] Toscani | On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds[END_REF][START_REF] Röckner | Weak Poincaré inequalities and L 2 -convergence rates of Markov semigroups[END_REF][START_REF] Gallay | Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on R 2[END_REF][START_REF] Bakry | Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré[END_REF] as well as to [START_REF] Kavian | The Fokker-Planck equation with subcritical confinement force[END_REF][START_REF] Mischler | Exponential stability of slowly decaying solutions to the kinetic-Fokker-Planck equation[END_REF][START_REF] Gualdani | Factorization of non-symmetric operators and exponential Htheorem[END_REF]] which techniques will be adapted here.

In the present situation, we impose that the contribution of c has lower influence at the infinity that the drift term b and we assume

(7.57) c ∈ L ∞ loc (R d ), ∃ C 0 , R 0 > 0, ∀ x ∈ B c R0 , |c(x)| = o(|x| 2(γ-1)
). We further assume that (7.58) c ≥ divb when γ ∈ (0, 1].

The action of the drift term will be revealed through the choice of a convenient "confining space". More precisely, for a weight function m :

R d → [1, ∞),
(7.59) (Lf ) f |f | p-2 m p = -(p -1) |∇f | 2 |f | p-2 m p + |f | p m p ϕ 1 , with (7.60) ϕ 1 := (p -1) |∇m| 2 m 2 + ∆m m + c - 1 p div b -b • ∇m m as well as (7.61) (Lf ) f |f | p-2 m p = -(p -1) |∇(f m)| 2 |f m| p-2 + |f | p m p ϕ 2 , with (7.62) ϕ 2 := 2(1 - 1 p ) |∇m| 2 m 2 + ( 2 p -1) ∆m m + c - 1 p div b -b • ∇m m .
In order to simplify the discussion, we restrict ourself to the exponent p = 2 and to the exponential weight function m = e a x s , s ∈ (0, γ], a > 0. We thus work in the Banach lattice X := L 2 m . We observe that ∇m m = sax x s-2 ∼ sa|x| s-1 ,

∆m m = sad x s-2 + s(s -2)a|x| 2 x s-4 + (sa) 2 |x| 2 x 2s-4 ∼ (sa) 2 |x| 2s-2 , divb = d x γ-2 + (γ -2)|x| 2 x γ-4 ∼ (d + γ -2)|x| γ-2 , b • ∇m m = sax x s-2 • x x γ-2 ∼ sa|x| s+γ-2 ,
so that the contribution of (cdivb/2) is always negligible at infinity, and we get (7.63)

ϕ i ∼ (sa) 2 |x| 2s-2 -sa|x| s+γ-2 .
We denote

a ′ := sa > 0 if s ∈ (0, γ), a ′ := aγ -2(aγ) 2 > 0 if s = γ and a ∈ (0, 1/( √ 2γ)).
We then face to three cases :

(i) γ > 1 : taking s ∈ ((2 -γ) + , γ), we have ϕ i ∼ -a ′ |x| s+γ-2 → -∞ with s + γ -2 > 0; (ii) γ = 1 : taking s = γ, a < 1/( √ 2γ), we have ϕ i → -a ′ ; (iii) γ ∈ (0, 1) : taking s = γ, a < 1/( √ 2γ), we have ϕ i ∼ -a ′ |x| 2γ-2 → 0 with 2γ -2 < 0.
Condition (H1). In any of the above cases, we have from (7.59)

((λ -L)f, f ) = |∇f | 2 m 2 + (λ -ϕ 1 )f 2 m 2 ,
for λ ∈ R, with inf(λϕ 1 ) > 0 for λ ≥ κ 1 and κ 1 > 0 large enough. We deduce that λ -L is coercive for λ ≥ κ 1 . With the same arguments as in section 7.1, we conclude that L is the generator in L 2 m of a positive semigroup S L , so that (H1) holds. Condition (H2). When γ > 1, the same arguments as in Section 7.2 imply that condition (H2) holds for some κ 0 ∈ R. When γ ∈ (0, 1], we have L * 1 = cdivb ≥ 0 from (7.58) and (H2) holds with κ 0 = 0.

Conditions (H4) and (H5).

The strong maximum principle holds here because for instance we may apply the same barrier function argument as presented in Section 7.2. The reverse Kato's inequality condition is proved by using local arguments, so that it holds for the same reasons as in the previous section.

Condition (H3). We define the multiplication operator A and the elliptic operator B by

A := M χ R , B := L -A, for M, R > 0 and χ R (x) := χ(x/R) with χ ∈ D(R d ), 1 B1 ≤ χ ≤ 1 B2 . We fix κ B < κ 0 in case (i), κ B := -a ′ /
4 in case (ii) and κ B := 0 in case (iii), and we set a ′′ := a ′ /2. Choosing M, R > 0 large enough, from Lemma 7.7 and the discussion which follows, we deduce that

(7.64) ((B -α)f, f ) ≤ -|∇f | 2 m 2 -a ′′ f 2 (1 B1 + 1 B c 1 |x| s+γ-2 )m 2 ,
for any α ≥ κ B and any nice function f . We classically deduce that α -B is coercive and thus invertible. We discuss the three different cases.

-In the first case γ > 1, so that s + γ -2 > 0, we see that the operator R B (α) is compact from Rellich-Kondrachov theorem, so that also W(α) := R B (α)A for any α ≥ κ B . We may thus apply Lemma 2.8-( 2) and we deduce that (H3) holds for both the primal and the dual problems.

-In the case γ = 1, so that 2γ -2 ≤ 0, the operator R B (α) is not compact anymore. However, for any sequence (f n ) which is bounded in L 2 m , we define the sequence (g n ) by g n := Af n , and (g n ) is bounded in L 2 m with m := e ã x γ , ã ∈ (a, 1/ √ 2γ). Using the dissipativity estimate (7.64) in L 2 m, we see that Bα is dissipative in L 2 m for any α ≥ κ B , and more precisely the sequence (h n ) defined by h n := R B (α)g n satisfies

|∇h n | 2 m 2 + ã′′ h 2 n (1 B1 + 1 B c 1 |x| 2γ-2 ) m2 ≤ g 2 n m2
.

Using that |x| 2γ-2 m2 /m 2 → ∞ as |x| → ∞, that implies that (h n ) is relatively compact in L 2 m . More precisely, the above estimates show that W(α)

:= R B (α)A : L 2 m → H 1 m ∩ L 2 m ♯ with m ♯ := m 1/2 m1
/2 and in particular we have established that W(α) := R B (α)A is a compact operator in L 2 m uniformly on α ≥ κ B because of the Rellich-Kondrachov theorem and the fact that

m = o(m ♯ ). Since R B (α) is bounded in B(L 2
m ) uniformly for any α > κ B , the operator L satisfies the splitting structure (HS1) and, applying Lemma 2.8-(2), we deduce that (H3) holds for both the primal and the dual problems.

At this stage, when γ ≥ 1, we obtain a solution (λ 1 , f 1 , φ 1 ) to the first eigentriplet problem (7.3) by using Theorem 2.21.

Condition (HS3).

In the case γ ∈ (0, 1), the same as in the case γ = 1 holds except that R B (α) is not uniformly bound in B(L 2 m ) for α ≥ κ B because we are in the critical case κ B = κ 0 . We do not know how to adapt the stationary approach in that situation and we thus aim to use a dynamical approach through the use of Theorem 3.4 with the above splitting L = A + B and N := [d/4] + 1. We set X = X 1 := L 2 m and X 0 := L 1 . The proof of condition (HS3) is an immediate consequence of the following estimate. Proposition 7.8. We define Θ ζ (t) := e -ζt γ/(2-γ) . For N := [d/4] + 1, there hold For the sake of completeness we however present the main lines of the proof. We start with a technical result that we will use during the proof of Proposition 7.8.

(i) S B ∈ L ∞ t (B(X 1 )); (ii) S B AΘ -1 ζ ∈ L ∞ t (B(X i )) for i = 0, 1 and any ζ ∈ (0, ζ * ); (iii) (S B A) ( * N ) Θ -1 ζ ∈ L ∞ t (B(X 0 , X 1 )) for any ζ ∈ (0, ζ * /2
Lemma 7.9. Consider two Banach spaces X 0 , X 1 and a function u : R + → B(X 0 ) + B(X 1 ) which satisfies

(a) uΘ -1 ∈ L ∞ (0, ∞; B(X 0 ) ∩ B(X 1 )); (b) u℘ ∈ L ∞ (0, ∞; B(X 0 , X 1 ));
for any exponentially decaying function Θ = Θ ζ = e -ζt ς , ∀ζ ∈ (0, ζ * ), and for the power function ℘ := t -α , with ζ * > 0, ς ∈ (0, 1] and α ≥ 0 fixed. Then (c) there exists N such that u ( * N ) Θ ∈ L ∞ (0, ∞; B(X 0 , X 1 )),

for any Θ = Θ ζ , ζ ∈ (0, ζ * /2).
Proof of Lemma 7.9. Step 1. Consider two functions v and w which satisfy the estimate (a). For X = X 0 or X = X 1 , we compute

A
v * w(t) X →X ≤ t 0 v(t -s)w(s) X →X ds ≤ t 0 C v X Θ(t -s) C w X Θ(s) ds ≤ C v X C w X t Θ(t),
with obvious notation and where we have used that Θ(ts) Θ(s) ≤ Θ(t) for any 0 < s < t. Since for any ζ ′ ∈ (0, ζ), there exists a constant C such that tΘ ζ (t) ≤ C Θ ζ ′ (t) for any t ≥ 0, we see that the function v * w satisfies the same estimate (a) for any Θ = Θ ζ , ζ ∈ (0, ζ * ).

Step 2. Consider two functions v and w which satisfy the estimates (a) and (b) with α ≥ 1. We compute

v * w(t) X0→X1 ≤ t/2 0 v(t -s)w(s) X0→X1 ds + t t/2 v(t -s)w(s) X0→X1 ds ≤ t/2 0 C v 01 (t -s) -α C w 0 Θ(s) ds + t t/2 C v 1 Θ(t -s) C w 01 s -α ds = [C v 1 C w 01 + C v 01 C w 0 ] Θ(0) t -α+1 1/2 0 (1 -τ ) -α dτ,
with obvious notation and we have used that Θ is a decaying function. As a consequence, the function v * w satisfies estimate (b) with an exponent α -1 instead of α.

Step 3. Consider two functions v and w which satisfy the estimates (a) and (b) with α ∈ [0, 1). We compute

v * w(t) X0→X1 ≤ t/2 0 v(t -s)w(s) X0→X1 ds + t t/2 v(t -s)w(s) X0→X1 ds ≤ t/2 0 C v 1 Θ(t -s) C w 01 s -α ds + t t/2 C v 01 (t -s) -α C w 0 Θ(s) ds ≤ C v 1 C w 01 Θ(t/2) t/2 0 s -α ds + C v 01 C w 0 Θ(t/2) t t/2 (t -s) -α ds = [C v 1 C w 01 + C v 01 C w 0 ] Θ(t/2) t 1-α 1 -α ,
with the same obvious notation and we have used again that Θ is a decaying function.

Step 4. Iterating n := [α] times steps 1 and 2, we get that u ( * n) still satisfies estimate (a) and satisfies the estimate (b) for the exponent α -[α] ∈ (0, 1). We then conclude that (c) holds with N := n + 1 and any ζ ∈ (0, ζ * /2) by using the third step.

Proof of Proposition 7.8. We classically establish that B generates a positive semigroup S B in both spaces X i and we thus concentrate on the announced estimates. On the one hand, proceeding as for the proof of (7.64), we have

(7.65) (Bf )(sign f )m ≤ -a ′′ |f |(1 B1 + 1 B c 1 |x| s+γ-2 )m,
for any nice function f and any weight function m = m a , with m a (x) := e a x γ , a ∈ (a 1 , a 2 ), 0 < a 1 < a 2 < 1/( √ 2γ), where we define a ′′ := aγ/2 -(aγ) 2 . That exactly means that B is weakly dissipative in L 1 m as defined in (3.19). From the discussion in Section 3.3, we deduce that S B is a semigroup of contractions and satisfies the associated decay estimate (3.23), (3.24), and more precisely -γ) . We refer to [231, Lem. 2.1] for details. Using that A : L 1 → L 1 m is bounded, that establishes (ii) in X 0 . Similarly, starting from (7.61) and proceeding as in the proof of (7.64), we get

(7.66) S B (t)f L 1 ma ≤ f L 1 ma , S B (t)f L 1 ma ≤ Θ ζ (t) f L 1 m a ′ , for any a, a ′ ∈ (a 1 , a 2 ), a < a ′ , ζ ∈ (0, ζ * ), ζ * := (a ′ -a) (2-2γ)/(2-γ) (a ′ γ(1 -a ′ γ)) γ/(2
(7.67) (Bf, f ) L 2 m ≤ -|∇(f m)| 2 -a ′′ f 2 (1 B1 + 1 B c 1 |x| s+γ-2 )m 2 ,
for any nice function f and any weight function m = m a as above. Throwing away the first term at the RHS and arguing as we did in L 1 m , we obtain that S B satisfies (7.68)

S B (t)f L 2 ma ≤ f L 2 ma , S B (t)f L 2 ma ≤ Θ ζ (t) f L 2 m a ′ ,
for any a, a ′ ∈ (a 1 , a 2 ). Using that A :

L 2 ma → L 2 m ′
a is bounded, that establishes (i) and (ii) in X 1 .

On the other hand, throwing away the second term at the RHS in (7.67), for any trajectory

f t = S B (t)f 0 , f 0 in the domain of B in L 2 m , we have 1 2 d dt R d f 2 t m 2 dx ≤ - R d |∇(f t m)| 2 dx.
Using Nash's inequality which for some constant C N ∈ (0, ∞) stipulates that

R d g 2 dx ≤ C N R d |∇g| 2 dx d d+2 R d |g|dx 4 d+2
, ∀ g, with g := f t m and the first estimate in (7.68), we deduce (7.69)

F ′ (t) ≤ -2 C ′ N F (t) -4/d G(t) 1+ 2 d ≤ -2 C ′ N F (t) -4/d G(0) 1+ 2 d , with C ′ N := C -1-2/d N
and where for brevity of notations we have set

F (t) := f t 2 L 2 (m) , G(t) := f t L 1 (m) .
Integrate the differential inequality (7.69), we find

S B (t)f 0 2 L 2 m t -d/4 f 0 L 1 m , ∀ t > 0, and using that A : L 1 → L 1 m , we next obtain (7.70) S B (t)At d/4 ∈ L ∞ (0, ∞; B(X 0 , X 1 )).
Setting with u(t) := S B (t)A, we see that u satisfies (a) in Lemma 7.9 thanks to (ii) in X 0 and X 1 . Furthermore, u satisfies (b) in Lemma 7.9 thanks to (7.70). Using Lemma 7.9, we conclude that condition (iii) holds.

We come back to the proof of (HS3). Gathering (i) and (ii) in X 1 in Proposition 7.8, we get that (S B A)

( * ℓ) * S B ∈ L ∞ t (B(X 1 )) for any ℓ ∈ {0, . . . , N -1}, N := [d/4] + 1.
Using that Θ ∈ L 1 (0, ∞) and (iii) in Proposition 7.8, we deduce that (S B A) ( * N ) ∈ L 1 (0, ∞; B(X 0 , X 1 )). We may now handle the existence part of the first eigenvalue problem. On the one hand, recalling (H2), we have L * φ 0 ≥ 0 with φ 0 = 1 so that the condition (i) in Theorem 3.4 holds. On the other hand, the condition (ii) in Theorem 3.4 is an immediate consequence of (HS3) as emphasized in Remark 3.5-(1). As a conclusion, the hypotheses of Theorem 3.4 are thus met, and we deduce that there exists (λ 1 , f 1 ) ∈ R + × L 2 m+ solution to the first eigenvalue problem. Because the strong maximum principle (H4) holds, we have f 1 > 0 on R d . In order to prove the existence of a first positive eigenfunction for the dual problem, we argue in the following way. We start observing that we have the alternative: λ 1 = 0 or λ 1 > 0. -In the first case, we may argue as in Remark 4.19. We indeed have in the same time L * φ 0 ≥ 0 and L * φ 0 , f 1 = φ 0 , Lf 1 = 0, so that L * φ 0 = 0 because f 1 > 0. The function φ 1 := φ 0 is thus a solution to the first dual eigenvalue problem. -In the second case λ 1 > 0, we may argue as in the case γ = 1 above. On the one hand, the operator R B (α) is uniformly bounded in L 2 m for any α ≥ κ B := λ 1 /2 > 0 and on the other hand the operator W(α)

:= R B (α)A : L 2 m → H 1 m ∩ L 2 m ♯ is uniformly bounded for any α ≥ κ B with m = o(m ♯ ), so that H 1 m ∩ L 2 m ♯ ⊂ L 2
m is compact. We may thus apply Theorem 2.21 and we conclude to the existence of a solution (λ ′ 1 , f ′ 1 , φ ′ 1 ) to the eigentriplet problem. The conditions (H4) and (H5) being true in a general situation as well as the conclusions of Lemma 7.1, as an intermediate conclusion, we have established under the general condition γ > 0 in (7.56) that yet the same conclusions as in section 7.2 hold true.

Quantitative stability. We now establish a quantitative stability estimate using the Doblin-Harris approach presented in Section 6 and yet used in the case of a bounded domain in Section 7.1. We first consider the more difficult case γ ∈ (0, 1), so that λ 1 ≥ κ 0 = 0, and then explain the modifications to be made in order to deal with the case γ ≥ 1. As explained just above, λ 1 = 0 corresponds to the conservative case (λ 1 , φ 1 ) = (0, 1) which has been considered in [START_REF] Kavian | The Fokker-Planck equation with subcritical confinement force[END_REF]. We thus focus on the case λ 1 > 0 for which an adapted version of Theorem 7.3 already imply the exponential asymptotic stability (E3 1 ) in L 2 m with non constructive rate. We do not develop further this argument but rather establish a a constructive sub-exponential asymptotic stability.

Step1 -Lyapunov condition. We take m = e a|x| γ with 0 < a < γ -1 . From Lemma 7.7 or a direct computation, we have

L * m = ∆m + (c -divb)m -b • ∇m ≤ (C x γ-2 + |c| -2a * x 2γ-2 )m ≤ C 0 1 B̺ 0 -a * x 2γ-2 m,
for three positive constants C = C(d), C 0 = C 0 (c, C, a, γ), ̺ 0 = ̺ 0 (c, C, a, γ) and with a * := (aγ -(aγ) 2 )/2 > 0. We now set m 1 := m and m 0 := a * x 2γ-2 m. We fix T > 0 and for 0 ≤ f 0 ∈ L 1 m , we denote f t := S L (t)f 0 . Recording that λ 1 ≥ 0 and using the above pointwise estimate, we deduce (7.71)

f T m 1 + T 0 f t m 0 dt ≤ f 0 m 1 + C 1 T 0 B̺ 0 f t dt.
Because the same kind of pointwise estimate holds for L * m 0 , we have

f T m 0 ≤ f t m 0 + C 1 T t B̺ 0 f s ds
and integrating in time, we get

T f T m 0 ≤ T 0 f t m 0 dt + C 0 T 0 s B̺ 0 f s ds.
Coming back to the first estimate, we deduce

(7.72) f T m 1 + T f T m 0 ≤ f 0 m 1 + (C 1 + C 0 T ) T 0 B̺ 0 f t dt.
Step 2 -Pointwise estimates on φ 1 . We define B := L -C 0 χ ̺0 which is the generator of a positive semigroup of contraction in L 1 m because of the above discussion. For λ > 0, 0 ≤ g ∈ L 1 m and 0 ≤ f ∈ L 1 m the solution to (λ -B)f = g, we compute

gm = f (λ -B * )m ≥ f (λm + m 0 ) ≥ f m 0 , from what we deduce R B (λ)f L 1 m 0 ≤ f L 1 m , ∀ f ∈ L 1 m
. Now, we consider two weight functions m 1 and m 3 with m i := e ai|x| γ , 0 < a 1 < a 3 < γ -1 , we denote m 0 := a * 1 x 2γ-2 m 1 and we compute

AR B (λ)f L 1 m 3 ≤ C 0 R B (λ)f L 1 m 3 (B2̺ 0 ) ≤ C 1 R B (λ)f L 1 m 0 ≤ C 1 f L 1 m 1
.

By duality, we obtain

(7.73) R B * (λ)φ L ∞ m -1 1 ≤ φ L ∞ m -1 0 and R B * A * (λ)φ L ∞ m -1 1 ≤ C 1 φ L ∞ m - 1 3
,

for any λ > 0 and φ ∈ L ∞ m -1 0
. We also deduce from Proposition 7.8 the regularization estimate

(A * R B * ) N : L 2 m -1 → L ∞ . Let us now consider 0 ≤ φ 1 ∈ L 2 m -1 1
the first eigenvector for the dual problem built in the preceding paragraph. From the eigenvalue equation

B * φ 1 + A * φ 1 = L * φ 1 = λ 1 φ 1 ,
we deduce that φ 1 = (R B * A * )φ 1 , and iterating

φ 1 = (R B * A * ) N +1 φ 1 = R B * (A * R B * ) N A * φ 1 .
From the above regularization estimate and the first estimate in (7.73), we thus deduce that

φ 1 ∈ L ∞ m -1 1
. Moreover, normalizing φ 1 and using the second estimate in (7.73), we may obtain

(7.74) φ 1 L ∞ m 3 -1 = 1 and φ L ∞ m -1 1 ≤ C 1 .
We deduce

1 = max sup B̺ 2 |φ 1 | m 3 , sup B c ̺ 2 |φ 1 | m 3 ≤ max sup B̺ 2 |φ 1 | m 3 , C 0,1 sup B c ̺ 2 m 1 m 3 ,
so that sup B̺ 2 |φ1| m3 = 1 by choosing ̺ 2 := max(̺ 0 , ̺ 1 ) with C 1 e (a1-a3)̺ γ 1 = 1. As a consequence, there exists x 0 ∈ B R such that φ 1 (x 0 ) ≥ 1. On the other hand, using standard regularity result for elliptic equation in the ball B 2R , we obtain that φ 1 ∈ C 0,1 (B R ) ∩ W 2,p (B R ) for any p ∈ [1, ∞) with constructive bound. Making use next of the Harnack inequality as at the end of Section 7.1 or using barrier functions as in in the proof of [START_REF] Kavian | The Fokker-Planck equation with subcritical confinement force[END_REF]Lem. 6.2], we classically deduce that (7.75)

φ 1 ≥ z ̺ 1 B̺ , ∀ ̺ > 0,
for a constructive constant z ̺ > 0 (where we emphasize here and below the φ 1 always denote the normalized by (7.74) dual eigenvector).

Step 3 -Doblin-Harris estimate. We fix T > 0 (for instance T := 1) and A > 0 arbitrary. For 0

≤ f 0 ∈ L 1 m such that f 0 L 1 m ≤ A f 0 L 1 φ 1
, we denote f t := S L (t)f 0 . On the one hand, we have

f t φ 1 = f 0 φ 1 and f t m ≤ C T f 0 m,
for any t ∈ [0, T ], the second estimate being an immediate consequence of (7.71). On the other hand, we define ε(r) := sup |x|≥r (m(x)/φ 1 (x)) and we compute Bρ

f t φ 1 = f t φ 1 - B c ρ f t φ 1 ≥ f t φ 1 -ε(ρ) f t m ≥ f 0 φ 1 -C T ε(ρ) f 0 m ≥ 1 - C T A ε(ρ) f 0 φ 1 ≥ 1 2 f 0 φ 1 ,
for any t ∈ (0, T ), by choosing ρ := ρ(T, A) > 0 large enough. In particular, there exists x 0 (t) ∈ B ρ such that f (t, x 0 (t)) ≥ ϑ := 1 2

1 φ 1 L 1 (Bρ) f 0 φ 1 .
Next, arguing exactly as in Section 7.1 or as in the proof of [231, Lem. 6.2], we deduce (7.76)

S T f 0 ≥ η T,A 1 B1 [f 0 ] L 1 φ 1 ,
for some constructive constant η T,A > 0.

Transport equations

The main aim of this part is to analysis the long time asymptotic of the solutions to the transport equation (8.1) When O = R D , the equation is complemented with a boundary condition which imposes the value of the trace γ -f of f on the incoming subsets of the boundary and takes the form

∂ t f + div y (af ) = K [f ] -Kf in (0, ∞) × O, on the function f = f (t, y), t ≥ 0, y ∈ O, with O ⊂ R D , D ≥ 1,
(8.3) (γ -f )(t, y) = R[f (t, •), γ + f (t, .)](y) on (0, ∞) × Σ -.
Let us explain the meaning of the different terms involved in (8.3). We denote by Σ := ∂O the boundary set, by dσ y the Lebesgue measure on Σ, by n : Σ → S D-1 the normal outward vector field, we write n = n y = n(y), and by Σ -the incoming, Σ + the outgoing and Σ 0 the singular subsets of the boundary defined by Σ ± := {y ∈ Σ; ±a(y) • n y > 0}, Σ 0 := {y ∈ Σ; a(y) • n y = 0}.

We denote γf = f |(0,∞)×Σ the trace of f and γ ± f := 1 (0,∞)×Σ± γf the trace restrictions on the incoming and outgoing sets. We then assume that the boundary operator R splits into two pieces R(g, h) = R O (g) + R Σ (h), where

(8.4) (R O g)(y) = O g(y * )r O (y, dy * ), (R Σ h)(y) = Σ+ h(y * )r Σ (y, dy * ), for a domain transition kernel r O : Σ -× B O → [0, ∞], a boundary transition kernel r Σ : Σ -× B Σ+ → [0, ∞]
and for any (conveniently) bounded functions g : O → R and h : Σ + → R, where B E stands for the set of Borel subsets of E.

In the next sections we will first consider the trace problem for a general force field a and next the well-posedness for the transport equation with given inflow at the boundary and with reflection condition at the boundary. We will also revisit the characteristic method for general force field a. We will next consider the Krein-Rutman problem still for a general force field a, but making strong simplification assumptions on the kernel operators K and R. We will next explain how the classical age structured equation falls into the present framework. We will come back to more specific physical situations concerning the growth-fragmentation equation and the kinetic relaxation equation with more general and physically relevant hypothesis on the kernel in parts 9 and 10.

The trace problem.

In this section, we are concerned with the trace problem associated to a (mainly stationary) transport equation for a general vector field a : O → R D for which we only assume (8.5) a ∈ W 1,1 loc ( Ō), where we recall that O ⊂ R D , D ≥ 1, is a smooth open connected set. The regularity needed on the domain is formulated in the following way: we assume that there exists n : O → R D , y → n(y) a vector field belonging to W 1,∞ (O) and which coincides with the previously defined unit outgoing normal vector field on Σ and satisfies n L ∞ = 1. In that situation, it is well-known that the above vector field is the restriction of a vector field a ∈ W 1,1 loc (R D ) (where we abuse notations denoting the restriction and the extension in the same way). We also consider the associated differential equation (8.6)

dY dt = a(Y ), Y (0) = y,
and then define the characteristic flow Y t = Y (t, y), for any y ∈ O, which is the solution to (8.6) on a maximal time interval (t -(y), t + (y)) where t -(y) < 0 < t + (y) are defined by t -:=t b and t + := t f , the backward exit time is defined by

(8.7) t b (y) := sup{τ > 0; Y -t (y) ∈ O, ∀ t ∈ [0, τ ]} ∈ (0, +∞]
and the forward exit time is defined by

(8.8) t f (y) := sup{τ > 0; Y t (y) ∈ O, ∀ t ∈ [0, τ ]} ∈ (0, +∞].
The real number t ℓt (y) := t b (y) + t f (y) ∈ (0, ∞] corresponds to the "life time" of the characteristic flow in O going by y. The construction of the flow (Y t ) is classical when a is a Lipschitz function and we refer to [139, Thm. II.3] for a more general situation which corresponds to the assumptions we will make in the present work (see also Lemma 8.14 below). For a solution g : O → R to the transport equation

(8.9) a • ∇ y g = G in O,
for a given source term G : O → R, we wish to define the trace γg of g on the boundary set Σ.

Similarly, for a solution g : (0, T ) × O → R, T ∈ (0, +∞], to the transport equation (8.10)

∂ t g + a • ∇ y g = G in (0, T ) × O,
for a given source term G : (0, T ) × O → R, we wish to define the trace γg of g on the boundary set (0, T ) × Σ. It is worth emphasizing that the trace will be in fact only defined out of the singular set Σ 0 and thus only on the boundary set Σ\Σ 0 . We start by recalling several possible definitions of the trace of a function g satisfying (8.9) when

(8.11) a ∈ W 1,s loc ( Ō), g ∈ L p loc ( Ō), G ∈ L q loc ( Ō), s, p, q ∈ [1, ∞].
Here and below, we denote by L(E) the Lebesgue space of measurable functions g : E → R := [-∞, +∞] with typically E = O or E ⊂ Σ, and by L 0 (E) = L 0 (E, µ) ⊂ L(E) the subset of almost everywhere finite measurable functions on a measurable space (E, A, µ). Definition 8.1. We say that a function g on O satisfying (8.9) and (8.11) admits a trace if one of the following assertions holds true:

• Extension of the restriction on the boundary. There exists γg ∈ L r loc (Σ\Σ 0 ), r ∈ [1, ∞], such that g n|Σ\Σ0 → γg in L r loc (Σ\Σ 0 ) for any sequence (g n ) satisfying (8.12)

g n ∈ C 1 c ( Ō), g n → g in L p loc ( Ō), a(y) • ∇ y g n → G in L q loc ( Ō). • Characteristics.
O β ′ (g) G ϕ + β(g) div(aϕ) dy = Σ β(γg) ϕ a • n dσ y , for any ϕ ∈ C 1 c ( Ō) and any β ∈ C 1 (R) such that β ′ ∈ L ∞ (R). Remark 8.2. (1)
In order that the first definition makes sense, we implicitly assume that there exists at least one sequence (g n ) which satisfies (8.12). That last fact corresponds to the density of C 1 c ( Ō) in the Sobolev space {g ∈ L p (O); a(y) • ∇ y g ∈ L q (O)}, which is true as we will see in Lemma 8.5 below under the regularity assumptions made on a and O. It is worth emphasizing that the last convergence in (8.12) may require additional integrability assumption, typically a ∈ W 1,s (O) with 1/r ≥ 1/p + 1/s. Such a definition has been introduced in [START_REF] Bardos | Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d'approximation; application à l'équation de transport[END_REF] for a C 1 vector field a. It is also the point of view adopted dy Cessenat in [START_REF] Cessenat | Théorèmes de trace L p pour des espaces de fonctions de la neutronique[END_REF][START_REF] Cessenat | Théorèmes de trace pour des espaces de fonctions de la neutronique[END_REF] in the case of the neutronic operator, see also [126, chap. XXI] or Agoshkov [START_REF] Agoshkov | Problema sledov funktsiȋ iz prostranstv H 1 p (Ω × D) i issledovanie nekotorykh variatsionnykh kraevykh zadach perenosa[END_REF][START_REF] Agoshkov | The existence of traces of functions in spaces used in problems of transport theory[END_REF][START_REF] Agoshkov | Some trace and extension theorems for functions from W 1 x i ,p (D) in the case of a bounded domain[END_REF].

(2) In order that the second definition makes sense, we implicitly assume that the set of points y ∈ O such that the characteristic Y t (y) hits the boundary on Σ 0 has zero measure in O. It is indeed the case thanks to the Sard theorem under enough regularity assumption on a and O, see [37, Prop. 2.3]. It is worth emphasizing that what we really need in order to write (8.13) and (8.14) is that t → G(Y (t, y)) ∈ L 1 (t -(y), t + (y)) for a.e. y ∈ O. We also mention that this characteristics description leads to a layer cake formula linking the integral of a function on the domain to the integral of its trace on the boundary. Such a definition has been widely used in kinetic theory for constructing DiPerna-Lions renormalized solution, see [START_REF] Diperna | On the Cauchy problem for Boltzmann equations: global existence and weak stability[END_REF][START_REF] Hamdache | Initial-boundary value problems for the Boltzmann equation: global existence of weak solutions[END_REF][START_REF] Arkeryd | On diffuse reflection at the boundary for the Boltzmann equation and related equations[END_REF] and the references therein. For the classical kinetic operator this trace approach is developed by Arkeryd, Cercignani and coauthors in [START_REF] Cannone | A trace theorem in kinetic theory[END_REF][START_REF] Cercignani | On the initial-boundary value problem for the Boltzmann equation[END_REF][START_REF] Arkeryd | A global existence theorem for the initial-boundary value problem for the Boltzmann equation when the boundaries are not isothermal[END_REF][START_REF] Arkeryd | On the solvability and asymptotics of the Boltzmann equation in irregular domains[END_REF] while for more general (but still regular) vector fields, the approach has been developed in [START_REF] Bardos | Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d'approximation; application à l'équation de transport[END_REF][START_REF] Ukai | Solutions of the Boltzmann equation[END_REF][START_REF] Beals | Abstract time-dependent transport equations[END_REF][START_REF] Greenberg | Boundary value problems in abstract kinetic theory[END_REF] and more recently by Arlotti et al. in [START_REF] Arlotti | On transport equations driven by a non-divergence-free force field[END_REF][START_REF] Arlotti | A new approach to transport equations associated to a regular field: trace results and well-posedness[END_REF][START_REF] Arlotti | On general transport equations with abstract boundary conditions. The case of divergence free force field[END_REF][START_REF] Arlotti | An L p -approach to the well-posedness of transport equations associated to a regular field: Part II[END_REF][START_REF] Arlotti | An L p -approach to the well-posedness of transport equations associated with a regular field: Part I[END_REF].

(3) In order that the third definition makes sense, we need that a • n γg ∈ L 1 loc (Σ) and p ≥ s ′ . In some situation, this third definition is in some sense the weakest: it makes sense also when γg ∈ M 1 loc (Σ\Σ 0 ) for instance and can be relevant under the weak assumption a, diva ∈ L p ′ loc ( Ō) as it is the case in the early works on weak solution to the Vlasov-Poisson equation in [START_REF] Poupaud | Solutions stationnaires des équations de Vlasov-Poisson[END_REF][START_REF] Guo | Global weak solutions of the Vlasov-Maxwell system with boundary conditions[END_REF][START_REF] Abdallah | Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system[END_REF][START_REF] Weckler | On the initial-boundary-value problem for the Vlasov-Poisson system: existence of weak solutions and stability[END_REF]. It is also easier to handle than the two first definitions because of the way it connects the function g and its trace. (4) We will adopt the last definition which extends up to the boundary the renormalization technique introduced in [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF]. It is more general and adapted to the weak regularity assumption made on the vector field a than the two first definitions and we recover the third definition by just letting β(s) → s when the conditions of integrability make the limit well defined. Such a kind of definition has been introduced in [START_REF] Mischler | On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system[END_REF][START_REF] Mischler | Kinetic equations with Maxwell boundary conditions[END_REF] for kinetic equations and in [START_REF] Boyer | Trace theorems and spatial continuity properties for the solutions of the transport equation[END_REF][START_REF] Ambrosio | Traces and fine properties of a BD class of vector fields and applications[END_REF] for transport equations.

We start with a trace result in a L ∞ framework. We denote by C 1 pw (R) the space of continuous functions β : R → R with piecewise continuous derivative.

Theorem 8.3. Assume that g ∈ L ∞ (O), a ∈ W 1,1
loc ( Ō) and G ∈ L 1 loc ( Ō) satisfy the transport equation (8.9) in the distributional sense. Then, there exists a unique function

γg ∈ L ∞ (Σ\Σ 0 ; dσ y ), γg L ∞ ≤ g L ∞ ,
which satisfies the renormalized Green formula

(8.17) O β ′ (g) G ϕ + β(g) div(aϕ) dy = Σ β(γg) ϕ a • n dσ y ,
for any ϕ ∈ C 1 c ( Ō) and any β ∈ C 1 pw (R). As a consequence, renormalization and trace operations commute: (2) An alternative approach has been developed by Ambrosio and co-authors by assuming weaker bound on Da but stronger bound on a. More precisely, denoting by M ∞ the set of vector fields a ∈ L ∞ (O) such that diva ∈ M 1 (O), it is established in [START_REF] Ambrosio | Traces and fine properties of a BD class of vector fields and applications[END_REF]Prop. 3.2] that there exists a linear and bounded mapping Tr :

(8.18) γ β(g) = β(γ g), ∀ β ∈ C 1 pw (R).
M ∞ (O) → L ∞ (∂O) such that Tra = n • a |∂O when a ∈ C 1 ( Ō).
The proof relies on Ambrosio's extension to a BV framework in [START_REF] Ambrosio | Transport equation and Cauchy problem for BV vector fields[END_REF] of the famous Di Perna-Lions improvement [139, Lem. II.1] of Freidrichs' type Lemma on the estimate of the commutator between directional derivative and convolution (see Lemma 8.5 below). Moreover, it is also established in [START_REF] Ambrosio | Traces and fine properties of a BD class of vector fields and applications[END_REF] (see in particular [START_REF] Ambrosio | Traces and fine properties of a BD class of vector fields and applications[END_REF]Thm. 4.2]) that

Tr(aβ(g)) = β Tr(ag) Tra Tr(a), ∀ β ∈ C 1 (R), for any a ∈ BV (O) ∩ L ∞ (O) and g ∈ L ∞ (O) such that ag ∈ M ∞ . The above formula is then nothing but (8.18) when a ∈ W 1,1 (O) ∩ L ∞ (O).
Before coming to the proof of Theorem 8.3, we state one technical but fundamental result. We define the mollifier (ρ ε ) ε>0 by (8.19)

ρ ε (z) = 1 ε d ρ(z/ε), 0 ≤ ρ ∈ D(R d ), supp ρ ⊂ B 1 , R N ρ(z) dz = 1,
and for any u

∈ L 1 loc ( Ō), v ε ∈ C c (R D ), supp v ε ⊂ B ε , we introduce the convolution-translation function u * ε v ε defined by (8.20) (u * ε v ε )(y) := O u(z) v ε (y -2ε n(y) -z) dz. Lemma 8.5. For g ∈ L p loc ( Ō), p ∈ [1, ∞], a ∈ W 1,p ′ loc ( Ō)
and G ∈ L 1 loc ( Ō) satisfying (8.9) in the distributional sense, the sequence (g ε ) defined by g ε := g * ε ρ ε satisfies

g ε ∈ W 1,∞ loc ( Ō), G ε := a • ∇g ε → a • ∇g in L 1 loc ( Ō),
as ε → 0, and

g ε → g in L p loc ( Ō), if p < ∞, g ε → g in L 1 loc ( Ō), (g ε ) bounded in L ∞ loc ( Ō), if p = ∞.
We skip the proof of Lemma 8. Proof of Theorem 8.3. Let us fix χ ∈ D( Ō) such that 0 ≤ χ ≤ 1 and denote R > 0 a real number such that supp χ ⊂ B R . We observe that χsign(a • n) ∈ L 1 (Σ). From Gagliardo trace theorem [170, Teor. 1.II], there exists ψ ∈ W 1,1 (O) such that γψ = χsign(a • n) and supp ψ ⊂ B R . Denoting T 1 : R → [-1, 1] the truncation function which is odd and is defined by T 1 (σ) = σ ∧ 1 for any σ ≥ 0, we see that γT 1 (ψ) = T 1 (γψ) = γψ, and thus we may assume ψ ∈ L ∞ (O) up to replacing ψ by T 1 (ψ). As a consequence, there exists a sequence (

ψ k ) of W 1,∞ (O) such that ψ k → ψ in W 1,1 (O), with (ψ k ) bounded in L ∞ (O), suppψ k ⊂ B R , and γψ k → χ sign(a • n) in L 1 (Σ), with (γψ k ) bounded in L ∞ (Σ).
Let us then consider the sequences (g ε ) and (G ε ) defined in Lemma 8.5. The classical Green formula for Lipschitz functions writes

Σ (g ε|Σ -g ε ′ |Σ ) 2 |a • n| χ dσ y = Σ (g ε|Σ -g ε ′ |Σ ) 2 a • n ψ k dσ y + Σ (g ε|Σ -g ε ′ |Σ ) 2 [|a • n| χ -a • n ψ k ] dσ y = O [2 (g ε -g ε ′ ) (G ε -G ε ′ ) ψ k dy + (g ε -g ε ′ ) 2 div(aψ k )] dy + Σ (g ε|Σ -g ε ′ |Σ ) 2 [|a • n| χ -a • n ψ k ] dσ y ≤ 4 ψ k L ∞ g L ∞ G ε -G ε ′ L 1 (BR) + ψ k W 1,∞ BR (|a| + |diva|)(g ε -g ε ′ ) 2 dy + 2 g 2 L ∞ (a • n)γψ k -χ |a • n| L 1 (Σ)
, for any ε > 0 and k ≥ 1. We deduce that (g ε|Σ ) is a Cauchy sequence in L 2 (|a • n| χ dσ). From the fact that (g ε ) is bounded in L ∞ (O), we deduce that the sequence (γg ε ) is also bounded in L ∞ (Σ). As a consequence, there exists a function γg ∈ L ∞ (Σ) such that γg ε → γg in L 2 (|a • n| χ dσ). Next, we may write the Green formula

O G ε ϕ + g ε div(aϕ) dy = Σ γ g ε ϕ a • n dσ y ,
for any test function ϕ ∈ C 1 c ( Ō), and we may pass to the limit as ε → 0. We deduce that the Green formula

O β ′ (g ε )G ε ϕ + β(g ε )div(aϕ) dy = Σ β(g ε|Σ ) ϕ a • n dσ y ,
for any renormalizing function β ∈ Lip(R) and any test function

ϕ ∈ C 1 c ( Ō). Using that β ′ (g ε )G ε → β ′ (g)G, β(g ε ) → β(g), β(g ε|Σ ) → β(γg)
respectively in L 1 loc ( Ō) and in L 1 loc (Σ), and that the two last sequences are bounded in L ∞ , we may pass to the limit ε → 0 in the last Green formula, and we thus get

O β ′ (g)G ϕ + β(g)div(aϕ) dy = Σ β(γ g) ϕ a • n dσ y .
Together with (8.23) and by uniqueness of the trace function, we conclude to γβ(g) = β(γg).

Let us state several variants of the preceding trace result. For the transport evolution equation (8.10) a first possible trace result writes as follows.

Theorem 8.6. Assume that g ∈ L ∞ ((0, T ) × O), a ∈ L 1 (0, T ; W 1,1 loc ( Ō)) and G ∈ L 1 loc ([0, T ] × Ō)
satisfy the evolution transport equation (8.10) in the distributional sense. Then,

g ∈ C([0, T ]; L 1 loc ( Ō)
) and there exists a unique function

γg ∈ L ∞ ((0, T ) × Σ\Σ 0 ; dt ⊗ dσ y ), γg L ∞ ≤ g L ∞ ,
which satisfies the renormalized Green formula

t1 t0 O β ′ (g) G ϕ + β(g) [∂ t ϕ + div(aϕ)] dydt (8.24) = O β(g(t, •))ϕdy t1 t0 + t1 t0 Σ β(γg) ϕ a • n dσ y dt, for any ϕ ∈ C 1 c ([0, T ] × Ō), any t 0 , t 1 ∈ [0, T ] and any β ∈ C 1 pw (R).
In particular renormalization and trace operations commute: (8.18) holds.

We skip the proof of Theorem 8.6 which is very similar to the proof Theorem 8.

(8.26) a • ∇ y β(g) + bβ ′ (g)g = β ′ (g)G,
in the distributional sense for any renormalizing function β ∈ C 1 * (R) the set of C 1 (R) functions such that β admits some finite limits in ±∞ and s → s β ′ (s) is bounded on R, in particular

C 1 * (R) ⊂ C 1 b R). We also denote by β ∈ C 1 pw, * (R) the C 1 piecewise variant of C 1 * (R).
We will repeatedly use the family of functions β δ ∈ C 1 * (R) defined by β δ (s) := s/(1 + δs 2 ) 1/2 for any δ ∈ (0, 1]. We observe that β ′ δ (s) = (1 + δs 2 ) -3/2 , so that sβ ′ (s) → 0 as s → ±∞. Let us start formulating some basic facts on renormalized solutions to equation (8.25). in the renormalized sense.

Proof of Lemma 8.7.

We briefly sketch the proof and for more details we refer to [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF], in particular to [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF]Lem. II.2]. It is worth mentioning that only the case b ∈ L ∞ (O) is considered in [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF], but it readily extends to our framework. Assertion ( 1) is just a consequence of the chain rule β ′ (s) = (β • α -1 ) ′ (α(s))α ′ (s) for smooth enough solutions and thus for any solution thanks to Lemma 8.5 (see the proof of [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF]Cor. II.1]) and to a standard approximation procedure in order to deal with piecewise C 1 functions. In order to establish (2), we consider two renormalized solutions g i , a renormalized function β ∈ C 1 * (R) and we write

a • ∇ y β(β δ (g 1 ) + β δ (g 2 )) = β ′ (β δ (g 1 ) + β δ (g 2 ))[(G 1 -bg 1 )β ′ δ (g 1 ) + (G 2 -bg 2 )β ′ δ (g 2 )]
, where we have added the two renormalized formulations (8.26) associated to β δ (g i ) and renormalized once more the resulting solution using [START_REF] Abdallah | Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system[END_REF]. Letting δ → 0, we immediately obtain

a • ∇ y β(g 1 + g 2 ) = β ′ (g 1 + g 2 )[G 1 + G 2 -b(g 1 + g 2 )]
in the distributional sense. For proving (3), we introduce the mollified sequence (g ε ) and (Φ ε ) defined as in the statement of Lemma 8.5 so that

a • ∇g ε + bg ε = G ε , a • ∇Φ ε = c ε with G ε → G and c ε → c in L 1 loc ( Ō) as ε → 0. The smooth function h ε := g ε e -Φε satisfies a • ∇ y h ε + (b + c ε )h ε = G ε e -Φε
and then

a • ∇ y β(h ε ) + β ′ (h ε )(b + c ε )h ε = β ′ (h ε )G ε e -Φε
for any β ∈ C 1 * (R). Passing to the limit ε → 0, we obtain the renormalized formulation of (8.27).

We now generalize the trace result to the framework of renormalized solutions. Theorem 8.8. Assume that a ∈ W 1,1 loc ( Ō), b, G ∈ L 1 loc ( Ō) and that g ∈ L 1 loc ( Ō) is a renormalized solution to the transport equation (8.25). Then there exists a unique function γg ∈ L(Σ\Σ 0 ; dσ y ) which satisfies the renormalized Green formula

(8.28) O β ′ (g) (G -bg) ϕ + β(g) div(aϕ) dy = Σ β(γg) ϕ a • n dσ y , for any ϕ ∈ C 1 c ( Ō) and any β ∈ C 1 pw, * (R).
Proof of Theorem 8.8. We fix

β 1 : R → R defined by β 1 (s) := s(1 + s 2 ) -1/2 , so that β 1 ∈ C 1 b (R) and β 1 : R → (-1, 1) is a bijection. Since then β 1 (g) ∈ L ∞ (O) and β ′ (g)(G -bg) ∈ L 1 loc ( Ō), we know from Theorem 8.3 that γβ 1 (g) is well defined in L ∞ (Σ\Σ 0 ) through the Green formula O β ′ 1 (g)(G -bg) ϕ + β 1 (g) div(aϕ) dy = Σ γβ 1 (g) ϕ a • n dσ y ,
for any test function ϕ ∈ C 1 c ( Ō). We set γg := β -1 1 (γβ 1 (g)) ∈ L(Σ\Σ 0 ; dσ y ), with the convention β -1 1 (±1) = ±∞. For any β ∈ C 1 pw, * (R), we then deduce

γβ(g) = γ[(β • β -1 1 )(β 1 (g))] = β • β -1 1 (γβ 1 (g)) = β(γg)
, where we have used the renormalization result stated in Theorem 8.3 and the chain rule (1) stated in Lemma 8.7 in the second equality and the very definition of γg in the third equality. In other words, the renormalized Green formula (8.28) holds. Remark 8.9. (1) We will see in Section 8.4 that under the same conditions as in Theorem 8.8 the information on γg can be slightly improved, in particular γg ∈ L 0 (Σ\Σ 0 ).

(2) Theorem 8.8 in particular holds when we assume a ∈ W 1,p ′ loc ( Ō), b ∈ L p ′ loc ( Ō), G ∈ L 1 loc ( Ō) and g ∈ L p loc ( Ō) satisfy the transport equation (8.26) in the distributional sense. Indeed, in that situation one knows from the DiPerna-Lions renormalizing theory [139, Cor. II.2] that g is also a renormalized solution to the transport equation (8.26) (in the above sense).

(3) Assuming more interior integrability on the functions g, b, G and a, we may deduce more accurate information on γg. A typical example, is that

Σ∩BR |γg| r (|a • n| ∧ 1) 2 dσ y < ∞,
for some r ∈ [1, ∞) and any R > 0, under the additional assumption

|g| r (|diva| + |a • ∇T 1 (a • n)| + |b|) ∈ L 1 loc ( Ō), |g| r-1 |G| ∈ L 1 loc ( Ō). The proof follows by choosing ϕ := T 1 (a • n) χ, χ ∈ C 1 c ( Ō), 0 ≤ χ ≤ 1, and β k (s) = (|s| ∧ k)
r in the associated Green formula (8.16), and then to pass to the limit k → ∞. (4) Even more integrability on γ ± g is available on one part of the boundary if additional integrability assumption is made on γ ∓ g on the other part of the boundary. A typically example, is that

Σ±∩BR |γ ± g| r |a • n|dσ y < ∞, under the additional assumption |g| r (|diva| + |a| + |b|) ∈ L 1 loc ( Ō), |g| r-1 |G| ∈ L 1 loc ( Ō), |γ ∓ g| r a • n ∈ L 1 loc ( Σ∓ ). The proof follows by choosing ϕ ∈ C 1 c ( Ō), 0 ≤ ϕ ≤ 1, and β k (s) = (|s| ∧ k) r
in the associated Green formula (8.16), and then to pass to the limit k → ∞.

(5) The results stated in Lemma 8.7, in Theorem 8.8 and in points ( 1), ( 2), ( 3) and ( 4) above may be straightforwardly adapted to the evolution transport equation (8.10). We refer to [START_REF] Mischler | On the trace problem for solutions of the Vlasov equation[END_REF][START_REF] Mischler | On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system[END_REF][START_REF] Mischler | Kinetic equations with Maxwell boundary conditions[END_REF][START_REF] Boyer | Trace theorems and spatial continuity properties for the solutions of the transport equation[END_REF] where such results are established in a slightly less general framework. Let us emphasize again that when a ∈ W 1,1 loc ([0, T ] × Ō) (what it is the case in the time independent case when a satisfies (8.5)) this extension is directly implied by Theorem 8.8 applied to the vector field (1, a(t, y)) in the open set (0, T ) × O.

Well-posedness for the transport equation with given inflow at the boundary.

We deduce from the previous trace theorems and standard tools the well-posedness for the transport equation with several boundary conditions. In this section, we deal with the transport equation with given inflow at the boundary. We are first concerned with the stationary transport equation

λg + a • ∇g + bg = G in O, γ -g = g on Σ -, (8.29)
for a real number λ ∈ R large enough, a given source term G : O → R and a boundary term g : Σ -→ R. As we will see, our analysis also apply to the associated dual equation

λϕ -a • ∇ϕ + (b -diva)ϕ = Φ in D ′ (O), γ + ϕ = ψ on Σ + . (8.30)
We will also consider the related evolution equation (8.31)

   ∂g ∂t + a • ∇g + bg = G on (0, T ) × O, γ -g = g on (0, T ) × Σ -, g(0, •) = g 0 on O,
with given source term G : (0, T ) × O → R, boundary term g : (0, T ) × Σ -→ R and initial datum g 0 : O → R.

A possible simple framework consists in imposing the following conditions

(8.32) a ∈ W 1,1 loc ( Ō), b ∈ L 1 loc ( Ō), and
(8.33) b -, diva ∈ L ∞ (O), a y b + ∈ L 1 (O) + L ∞ (O).
The first condition on a is useful for the renormalization trick and the definition of the trace, the second condition is needed for the existence results in a L p framework when p = ∞ and the last condition is used for proving the uniqueness result. In order to be able to apply our results to more general (and realistic) situations, we rather consider the following situation. We assume that a and b satisfy (8.32), and defining

(8.34) ̟ = ̟ p := b - 1 p diva -a • ∇m m ,
for some smooth enough weight function m : Ō → (0, ∞) and some exponent p ∈ [1, ∞], we assume

(8.35) ̟ -∈ L ∞ (O), b, diva ∈ L ∞ ̟+ -1 (O), a y ∈ L ∞ ̟+ -1 (O) + L 1 (O)
In the case p = 1 and p = ∞, we will additionnaly assume (̟ q ) -∈ L ∞ for any q ∈ (1, ∞). It is worth emphasizing that condition (8.35) automatically holds when m ≡ 1 and a, b satisfy (8.33).

We also define the critical real number (8.36)

λ * p = λ * p (a, b, m) := ̟ -L ∞
, and we may observe that (8.37)

λ * p ′ (-a, b -diva, m -1 ) = λ * p (a, b, m
), what links up the primal and the dual problems. In order to shorten notations, we introduce the three weight functions (8.38) m

O := m ̟ + 1/p , m O := m ̟ -1/p ′ , m Σ := m|a • n| 1/p .
We start with a general discussion about a priori bounds, formal representation formulas and general stability results.

A priori estimates. Consider a solution g to the stationary equation (8.29). For any renormalizing function β : R → R + and any function ϕ : Ō → (0, ∞), we (at least) formally have

O [(λ + b)gβ ′ (g)ϕ -β(g)(div(aϕ))] + Σ+ a • nβ(γ + g)ϕ = O β ′ (g)Gϕ + Σ- |a • n|β(g)ϕ.
Choosing β(s) := |s| p , 1 ≤ p < ∞, and ϕ := m p , we get in particular (8.39)

O |g| p m p λ + ̟ + 1 p Σ+ |γ + g| p m p a • n = O Gg|g| p-2 m p + 1 p Σ- |g| p m p |a • n|.
For p = 1 and any λ > λ * 1 , we get

O |g|m λ -λ * 1 + ̟ + + Σ+ |γ + g|m Σ ≤ O |G|m + Σ- |g|m Σ .
For p ∈ (1, ∞), we split G = G 1 + G 2 and using the Young inequality, we have

O Gg|g| p-2 m p ≤ ε 1 g p m p + ε 2 g p m p O + 1 p 1 (p ′ ε 1 ) p/p ′ G p 1 m p + 1 p 1 (p ′ ε 2 ) p/p ′ G p 2 m p O , for any ε i > 0. For λ > λ * p , we choose ε 1 := (λ -λ * p )/2 and ε 2 := 1/2, we get 1 2 O |g| p m p O + λ -λ * p 2 O |g| p m p + 1 p Σ+ |γ + g| p m p Σ ≤ 2 p-1 p(p ′ ) p/p ′ (λ -λ * p ) 1-p O |G 1 | p m p + 1 p(p ′ /2) p/p ′ O |G 2 | p m p O + 1 p Σ- |g| p m p Σ .
We thus deduce (8.40)

g L p m ≤ C p λ -λ * p G 1 L p m + C p (λ -λ * p ) 1/p ( g L p m Σ + G 2 L p m O
) and (8.41)

g L p m O + γ + g L p m Σ ≤ C p (λ -λ * p ) 1/p ′ G 1 L p m + C p ( g L p m Σ + G 2 L p m O ),
for some numerical constant C p ∈ (0, ∞) and any p ∈ (1, ∞) and also for p = 1 because of the previous estimate. Finally, for λ > λ * ∞ and α ∈ (0, λλ * ∞ ), we may proceed exactly as above, but throwing also away the contribution of ̟ + , and we may thus first write

(8.42) λ -̟ -L ∞ - α p ′ p ′ g p L p m (O) ≤ 1 p G/α p L p m (O) + g|a • n| 1/p p L p
m (Σ-) , for any p ∈ (1, ∞) large enough in such a way that the coefficient in front of g p L p m (O) is positive. Taking the power 1/p in both sides and passing first to the limit p ր ∞ and next to the limit α ր λλ * ∞ , we end with

(8.43) g L ∞ m (O) ≤ max 1 λ -λ * ∞ G L ∞ m (O) , g L ∞ m (Σ-)
. Consider now a solution to the evolution equation (8.31). For any renormalizing function β : R → R + and any test function ϕ : [0, ∞) × Ō → (0, ∞), we (at least) formally have ) ds (8.45)

t 0 O β ′ (g)Gϕ -β ′ (g)gϕ + β(g)[∂ t ϕ + div(
≤ e pκt g 0 p L p m + t 0 e pκ(t-s) ( G s p L p m O + g s p L p m Σ
) ds, ∀ t ≥ 0, with κ := ̟ -L ∞ . Passing to the limit p → ∞, we also have

max g(t) L ∞ m , γ + g(t) L ∞ m ≤ e κt max g 0 L ∞ m , sup [0,t] ( G s L ∞ m + g s L ∞ m ) , (8.46)
for any t ≥ 0.

Representation formulas. In a smooth functions framework or still formally, one classically knows that the solution g to the evolution transport equation (8.31) is given by

g(t, y) = g 0 (Y -t (y))e -t 0 b(Ys-t(y))ds 1 t<t b + g(t -t b , y b )e -t b 0 b(Ys-t b (y))ds 1 t>t b (8.47) + t ′ b 0 G(s, Y s-t ′ b (y))e -t ′ b -s 0 b(Y-τ (y))dτ du,
∂ t f + a • ∇f + bf = 0 in (0, ∞) × O, γ -f = 0 on (0, ∞) × Σ -.
For G, g : O → R, we next define (8.50)

g := g + ∞ 0 e -λt S b (t) G dt, with G := G -λg + a • ∇ x g -bg.
By construction, it is a solution to the stationary transport equation (8.29).

Stability. We present some stability and continuity results. Generalizing slightly [272, Definitions 2.6 and 3.1], we say that a sequence (g n ) of L(E) converges in the renormalized sense to g, we note g n r ⇀ g, if for any δ ∈ ∆, ∆ ⊂ (0, 1], 0 ∈ ∆, there exists βδ ∈ L ∞ (E) such that (8.51)

β δ (g n ) ⇀ βδ * σ(L ∞ , L 1 )
as n → ∞ and β 1 ( βδ ) → β 1 (g) L 1 loc ( Ō) as δ → 0. We may observe that in particular g n ⇀ g weakly L 1 (E) or g n → g a.e. in E implies g n r ⇀ g. We refer to [START_REF] Mischler | Kinetic equations with Maxwell boundary conditions[END_REF] and the references therein for more material about the subject. Proposition 8.10. Let us consider four sequences

(g k ) of L 1 loc ( Ō), (a k ) of W 1,1 loc ( Ō), (b k ) and (G k ) of L 1 loc ( Ō) such that a k • ∇β(g k ) + b k β ′ (g k )g k = β ′ (g k )G k in D ′ (O),
for any k ≥ 1 and any β ∈ C 1 * (R) and four functions g

∈ L 1 loc ( Ō), a ∈ W 1,1 loc ( Ō), b, G ∈ L 1 loc ( Ō) such that a k → a in W 1,1 loc ( Ō) and b k → b, G k → G in L 1 loc ( Ō).
Let us denote by Σ 0 the boundary singular subset associated to a. Proof of Proposition 8.10. We split the proof into two steps.

Step 1. We establish [START_REF] Abdallah | Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system[END_REF]. We fix β 1 as in the proof of Theorem 8.8 and we write the Green formula

O β ′ 1 (g k )G k ϕ -β ′ 1 (g k )g k b k ϕ + β 1 (g k ) div(a k ϕ) dy = Σ β 1 (γg k ) ϕ a k • n dσ y ,
for any test function ϕ ∈ C 1 c ( Ō). There exists β1 ∈ L ∞ (Σ\Σ 0 ) and a subsequence (g n k ) such that β 1 (γg n k ) ⇀ β1 weakly σ(L ∞ , L 1 ). Passing to the limit in the above equation, we get

O β ′ 1 (g)G ϕ -β ′ 1 (g)gb ϕ + β 1 (g) div(aϕ) dy = Σ β1 ϕ a • n dσ y .
From Lemma 8.7 and Theorem 8.8, we deduce that β1 = β 1 (γg), so that

β 1 (γg n ) ⇀ β 1 (γg) weakly σ(L ∞ , L 1 ). Fixing now β 2 := β 2 1 ∈ C 1 * (R)
and repeating the same argument, we get β 2 (γg n ) ⇀ β 2 (γg) weakly σ(L ∞ , L 1 ). We then immediately deduce that

(β 1 (γg n ) -β 1 (γg)) 2 ⇀ 0 weakly σ(L ∞ , L 1 ), so that β 1 (γg n ) → β 1 (γg) in L 1
loc (Σ\Σ 0 ). We conclude by using that β 1 is one-to-one.

Step 2. We establish [START_REF] Agoshkov | Problema sledov funktsiȋ iz prostranstv H 1 p (Ω × D) i issledovanie nekotorykh variatsionnykh kraevykh zadach perenosa[END_REF]. We fix β δ as defined just before the statement of Lemma 8.7 and we write the Green formula

O β ′ δ (g k )G k ϕ + β δ (g k ) div(aϕ) dy = Σ β δ (γg k ) ϕ a • n dσ y , for any test function ϕ ∈ C 1 c ( Ō). There exist β δ , β δ , β ′ δ ∈ L ∞ (O), γβ δ ∈ L ∞ (Σ\Σ 0 ) and a subsequence (g n k ) such that β δ (g n k ) ⇀ β δ , g n k β ′ δ (g n k ) ⇀ β δ , β ′ δ (g n k ) ⇀ β ′ δ and β δ (γg n k ) ⇀ γβ δ weakly σ(L ∞ , L 1
). Passing to the limit in the above equation, we get

a • ∇ y β δ + b β δ = β ′ δ G in D ′ (O), γβ δ = γβ δ on Σ\Σ 0 .
From the fact that (g k ) is locally uniformly integrable, we classically deduce that

β δ , β δ → g in L 1 loc (O), β ′ δ G → G in L 1 loc (O)
, as δ → 0. More precisely, the two first convergences come from the elementary inequalities [START_REF] Abdallah | Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system[END_REF], and the last convergence comes from the convexity inequality β ′ δ ≥ β ′ δ (g) and the elementary inequalities

∀ M > 0, ∃ c M > 0, |s -β δ (s)| ≤ |s -sβ ′ δ (s)| ≤ c M δ + |s|1 |s|>M , for any s ∈ R, δ ∈ (0,
∀ M > 0, ∃ c M > 0, 0 ≤ 1 -β ′ δ (s) ≤ c M δ + 1 |s|>M , ∀ s ∈ R, ∀δ ∈ (0, 1). From
Step 1, we deduce that g satisfies (8.26) and that, up to the extraction of a subsequence, γβ δ = γβ δ → γg a.e. on Σ\Σ 0 . Using the Cantor diagonal process, we obtain that there exist two sequences (δ m ) and (g n k ) such that δ m ց 0 and γg n k r ⇀ γg in the renormalized sense associated to (δ m ).

Existence. We establish two existence results of solutions to the transport equation ( 8

, b ∈ C 1 b ( Ō), G ∈ C 1 c (O), g := g|Σ-with g ∈ C 1 c ( Ō)
, both definitions (8.48) and (8.50) provide a classical (and thus also renormalized) solution g to (8.29). In such a situation, we may justify the computations made in the above a priori estimates paragraph and we conclude that g satisfies the L ∞ estimate (8.43). In the general case for a, b, g and G, we introduce some sequences (a ε ), (b ε ), (g ε ) and (G ε ) of regular and approximating functions so that we may apply the first step above. In that way, we build a sequence (g ε ) of renormalized solutions to the approximated problem which is uniformly bounded and thus converges (up to the extraction of a subsequence) in the weakly * σ(L ∞ , L 1 ) sense to a function g ∈ L ∞ (O) satisfying (8.46). We then immediately conclude by passing to the limit ε → 0 thanks to Proposition 8.10. We give a first version of an existence result in a L p framework with strong assumption on the boundary condition. Lemma 8.13 (Existence in L p ). We assume that a and b satisfy (8.35) for some p ∈ [1, ∞) and some weight function m : Ō → [1, ∞). For any λ > λ * p , G ∈ L p mO (O) and g ∈ L p mΣ (Σ -), there exists g ∈ L p mO (O) a renormalized solution to the transport equation (8.29). This one satisfies (8.40), (8.52) and γ + g ∈ L p mΣ (Σ + ). Proof of Lemma 8.13. We argue similarly as during the proof of Lemma 8.12. When g = g|Σ-with g, a, b, G smooth and with compact support, the classical solution built above satisfies (8.40), and thus

(8.53) g L p m O (O) G L p m O + gm L ∞ a 1/p L 1 (suppg)
. For p > 1, and under the general conditions (8.35) on a and b, but still assuming g = g|Σ-and G, g ∈ C 1 c ( Ō), we may introduce two sequences (a ε ) and (b ε ) of smooth functions approximating a and b. Since the resulting solution g ε satisfies (8.53), so that the sequence (g ε ) is bounded in L p mO , we may argue with the same (compactness) argument as in the proof of Lemma 8.12. We then conclude to the existence of a (renormalized) solution g ∈ L p mO to the transport equation (8.29) satisfying (8.40). Still for p > 1, but assuming G ∈ L p mO and g ∈ L p mΣ (Σ -), we may introduce two sequences (G ε ) and (g ε ) of smooth functions approximating G and g. Thanks to (8.40), the associated sequence of solutions (g ε ) is bounded (and better it is a Cauchy sequence) in L p mO (O) and we conclude again to the existence of a (renormalized) solution g ∈ L p mO (O) to the transport equation (8.29) satisfying (8.40). Finally, in the case p = 1 and λ > λ * 1 , we may find q > 1 small enough such that λ > λ * q . For G, g ∈ L 1 ∩ L q , the last step imply the existence of a renormalized solution g ∈ L q mO (O) to the transport equation (8.29). Renormalizing the equation, we deduce that g satisfies (8.40) for p = 1. When G, g ∈ L 1 , we introduce two sequences (G ε ) and (g ε ) of L 1 ∩ L q functions approximating G and g, and using (8.40) for p = 1, we deduce that the resulting sequence (g ε ) is a Cauchy sequence in L 1 mO (O). We easily conclude again. Finally γ + g ∈ L p mΣ (Σ + ) from (8.41) (see also Remark 8.9-( 4)).

Uniqueness. We present now a uniqueness result. λg

+ a • ∇g + bg = 0 in D ′ (O), γ -g = 0 on Σ -,
we have g ≡ 0.

Proof of Lemma 8.14. We main follow the proof of [139, Cor. II.1]. We fix

β ∈ W 1,∞ (R), β(0) = 0, in such a way that β(g) ∈ L p mO ∩ L ∞ is a solution to (λ + b)gβ ′ (g) + a • ∇β(g) = 0 in D ′ (O), γ -β(g) = 0 on Σ -.
For any ψ ∈ C c (O) and any λ > λ * m,p , we solve in

L p ′ m -1 ̟+ 1/p ′ ∩ L ∞ the dual problem (8.55) λϕ -a • ∇ϕ + (b -diva)ϕ = ψ in D ′ (O), γ + ϕ = 0 on Σ + ,
thanks to Lemma 8.12 and Lemma 8.13, where we observe that, because of (8.37), the necessary condition on λ in these results is precisely the one made here. For

χ ∈ C 1 c (R D ), 1 B1 ≤ χ ≤ 1 B2
, and R > 0, we define χ R (x) := χ(x/R). Using the Green formula (8.21), we have

0 = O ((λ + b)ϕ -ψ)β(g)χ R - O (λ + b)ϕgβ ′ (g)χ R + O ϕβ(g) a R • (∇χ) R .
Because on the one hand ϕβ(g) ∈ L 1 ̟+ ∩ L ∞ and on the other hand a/R • (∇χ) R → 0 a.e. and is bounded in L ∞ ̟+ -1 + L 1 we deduce that the last term vanishes when R → ∞. Using also that bϕg ∈ L 1 thanks to (8.35), we may pass to the limit R → ∞ in the above equation and we get

0 = O ((λ + b)ϕ -ψ)β(g) - O (λ + b)ϕgβ ′ (g).
We take β := β δ and we observe that b

|̟||β δ (g) -gβ ′ δ (g)| ≤ b |ϕg| ∈ L 1 (O)
. We may then pass to the limit δ → 0 in the last equation, and we get

0 = - O ψg, ∀ψ ∈ C c (O),
from which we conclude that g ≡ 0.

We come to the time dependent transport equation by formulating a general continuity result.

Proposition 8.15. Assume that a ∈ W 1,1 loc ( Ō), b ∈ L 1 loc ( Ō), G ∈ L 1 loc ([0, T ]× Ō). Any renormalized solution g ∈ L 1 loc ([0, T ] × Ō) to the first equation in (8.31), meaning ∂ ∂t β(g) + a • ∇β(g) + β ′ (g)bg = β ′ (g)G in D ′ ((0, T ) × O),
for any renormalizing function

β ∈ C 1 * (R), satisfies g ∈ C([0, T ]; L 0 (O)), meaning that β(g) ∈ C([0, T ]; L 1 loc ( Ō)) for any β ∈ C b (R). Proof of Proposition 8.
   ∂β(g) ∂t + a • ∇β(g) + β ′ (g)bg = β ′ (g)G on (0, T ) × O, γ -β(g) = β(g) on (0, T ) × Σ -, β(g)(0, •) = β(g 0 ) on O, for any β ∈ C 1 pw, * . Furthermore, g ∈ C([0, T ]; L p m ) when p ∈ [1, ∞) and g ≥ 0 if g 0 , G, g ≥ 0.
Remark 8.17.

(1) The above result extends some previous results due to Bardos (2) We immediately deduce from the above result and Lemma 8.7-(2) a weak maximum principle:

g 1 ≤ g 2 if g i is renormalized solution to the transport equation (8.31) associated to the data g 0i , G i , g i such that g 01 ≤ g 02 , G 01 ≤ G 02 , g 01 ≤ g 02 .
Proof of Proposition 8.16. We proceed similarly as during the proof of Lemma 8.12.

Step 1. Characteristics. We assume first a

∈ C 1 (R D ), g 0 ∈ C c (O), b ∈ C b ( Ō), g ∈ C c ((0, T ) × Σ 0 ), G ∈ C 1 c ((0, T ) × O).
We use the characteristics representation (8.47). We may verify that ḡ both satisfies the transport equation in the renormalized sense and the boundary conditions in (8.56) and we may justify the computations leading to the a priori estimates (8.45) and (8.46).

Step 2. Existence. In the general case, we define some regularized sequence (a ε ), (g 0,ε ), (b ε ) (g ε ), (G ε ) and thanks to the first step we deduce the existence of an associated function g ε ∈ C([0, T ]; L p m ) satisfies both the equation (8.56) in the renormalized sense and the a priori estimates (8.45) or (8.46). When p > 1, the sequence (g ε ) is bounded in L ∞ (0, T ; L p m ) and (up to the extraction of a subsequence) we may pass to the limit ε → 0 using Proposition 8.10-(2) and Remark 8.11. We have established the existence of a renormalized solution to the transport equation which satisfies the estimate (8.45) or (8.46). When p = 1, we may for instance proceed in the following way by first assuming 0

≤ g 0 ∈ L 1 m , 0 ≤ G ∈ L 1 m ((0, T )×O), 0 ≤ g ∈ L 1
mΣ ((0, T )×Σ). We may thus consider some nonnegative approximating sequences (g 0,ε ) in

L p m ∩ L 1 m , G ε ∈ L p mO ∩ L 1 m , g ε ∈ L p mΣ ∩ L p
mΣ such that g 0,ε ր g 0 , G ε ր G and g ε ր g. The same construction as above implies the existence of 0 ≤ g ε ∈ L ∞ (0, T ; L 1 m ∩ L p m ) renormalized solution to the transport equation associated to these data and such that (g ε ) is increasing and uniformly bounded in L ∞ (0, T ; L 1 m ) thanks to the a priori L 1 m estimate (8.45). There thus exists 0 ≤ g ∈ L ∞ (0, T ; L 1 m ) such that g ε ր g, and we get that g is a renormalized solution to the transport equation by using again Proposition 8.10-( 2) and Remark 8.11. We remove the nonnegative condition on g 0 , G and g by introducing the positive and negative parts of each function, using the preceding step in order to prove the existence of two solutions 0 ≤ g ± ∈ L ∞ (0, T ; L 1 m ) associated respectively to (g 0+ , G + , g + ) and (g 0-, G -, g -), and finally defining g := g + -g -which is a renormalized solution to the transport equation thanks to Lemma 8.7 and Remark 8.9-(5).

Step 3. Continuity. From Proposition 8.15, we already know that g ∈ C([0, T ]; L 0 (O)). Together with the a priori estimate (8.45) or (8.46), we also have g ∈ C([0, T ]; L 1 loc (O)) when p > 1. When p ∈ [1, ∞), we may improve the above continuity properties by arguing in the following way. We define g := gm and we observe that it is a solution to the transport equation

∂ t g + a • ∇ g + b g = G, γ g = g, g(0) = g 0 ,
with b := ba • ∇m/m, G := Gm, g := gm and g 0 := g 0 m. We write the associated renormalized equation (8.44) for the renormalizing function β M (s) := (|s| ∧ M ) p , M > 0, and the test function

ϕ := χ R , with χ ∈ C 1 c (R d ), 1 B1 ≤ χ ≤ 1 B2 and χ R (y) := χ(y/R). Observing in particular that t 0 O β M ( g)a • ∇χ R → 0 as R → ∞,
because of (8.45) and (8.35) by arguing as in the proof of Lemma 8.14, we may pass to the limit in the associated renormalized equation as R → ∞, and we obtain

O β M ( g)dy t 0 = t 0 O β ′ M ( g) Gdyds - t 0 Σ β M (γ g)a • n dσ y ds + t 0 O β M ( g) 1 g>M diva dyds - t 0 O pβ M ( g) 1 g≤M ̟dyds.
Using again (8.45) and (8.35), we may next pass to the limit as M → ∞ in the above equation, and we get

d dt O | g| p = -p O | g| p ̟ + O p G g| g| p-2 + Σ |γ g| p a • n ∈ L 1 (0, T ).
We deduce that t → g(t) L p m = g(t) L p is continuous. Consider then t ∈ [0, T ] and t k → t, so that in particular g t k L p m → g t L p m as k → ∞. On the other hand, we have yet established that β 0 (g t k )β 0 (g t ) L 1 (O∩BR) → 0 as k → ∞ for any R > 0. There exists thus a subsequence (g t k ′ ) such that g t k ′ → g t a.e. on O. Thanks to Brézis-Lieb theorem [START_REF] Brézis | A relation between pointwise convergence of functions and convergence of functionals[END_REF], we deduce that g t k ′ → g t in L p m and it is the whole sequence which converges by uniqueness of the limit. We have thus established

g ∈ C([0, T ]; L p m ) when p ∈ [1, ∞). Step 4. Uniqueness.
Because of Lemma 8.7 and Remark 8.9-( 5), we just have to prove that g ≡ 0 if g is a renormalized solution associated to vanishing data g 0 = 0, G = 0 and g = 0. When p ∈ [1, ∞), the previous step implies that

d dt O |g| p m p = O |g| p m p ̟ ∈ L 1 (0, T ), O |g(0)| p m p = 0,
and together with the Gronwall lemma, we deduce that g = 0. The case p = ∞ may be tackled thanks to a duality argument exactly as in the proof of Lemma 8.14.

Corollary 8.18. The semigroup S b defined by (8.49) extends to a positive semigroup of contractions in L p m . Proof of Corollary 8.18. We just apply Proposition 8.16 with G = g = 0. When p ∈ [1, ∞), we define in that way a mapping L p m → C(R + ; L p m ), g 0 → g, where g denotes the unique renormalized solution. Defining then S(t)g 0 := g(t) we have built a strongly continuous semigroup in L p m . The case p = ∞ is identical, except the fact that the semigroup is only weak * σ(L ∞ m , L 1 m -1 ) continuous. The positivity has been established in Proposition 8.16 and the contraction property comes from the estimates (8.45) and (8.46). Remark 8.19. It is worth emphasizing that in Bardos [START_REF] Bardos | Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d'approximation; application à l'équation de transport[END_REF] the semigroup is defined by its representation formula for smooth data and by Hille-Yosida theory for L 2 data. Here we proceed in another way, by rather following [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF][START_REF] Mischler | On the trace problem for solutions of the Vlasov equation[END_REF][START_REF] Mellet | Uniqueness and semigroup for the Vlasov equation with elastic-diffusive reflexion boundary conditions[END_REF].

8.3.

Optimal weighted trace theorem and transport equation with reflection at the boundary. We define the functions τ ± as the solutions to (8.57)

λ 0 τ ± ∓ a • ∇τ ± = 1 in D ′ (O), γ ± τ ± = 0 on Σ ± , with λ 0 := 1 + diva L ∞ .
Lemma 8.20. Each of the two equations (8.57) has a unique solution τ ± ∈ L ∞ (O) and

0 < τ ± ≤ 1 a.e. in O, 0 < γ ± τ ± ≤ 1 a.e. on Σ ∓ .
Proof of Lemma 8.20. We follow a similar proof as in [START_REF] Boyer | Trace theorems and spatial continuity properties for the solutions of the transport equation[END_REF]Prop. 5.1] (see also [START_REF] Mischler | On the trace problem for solutions of the Vlasov equation[END_REF]Sec. 5]). We only deals with τ -since the case of τ + can be handled in the same way. The existence of τ -∈ L ∞ , its non negativity and the upperbound are consequences of Lemma 8.12 while the uniqueness is ensured by Lemma 8.14. In order to prove the strict positivity we argue as follows. We first fix A ∈ O, |A| ∈ (0, ∞) and we solve

λ 0 ϕ -div(aϕ) = 1 A in D ′ (O), γ + ϕ = 0 on Σ + ,
for which there exists a unique solution ϕ ∈ L 1 (O) thanks to Lemma 8.12 and Lemma 8.14, which furthermore satisfies ϕ ≥ 0 and ϕ ≡ 0. We observe that τ

-ϕ ∈ L 1 (O) satisfies div(aτ -ϕ) = ϕ -τ 1 A in O, γ(τ -ϕ) = 0 on Σ\Σ 0 .
Thanks to the Green formula, first written for β δ (τ -ϕ) and next passing to the limit δ → 0, we deduce

0 = Σ γ(τ -ϕ) a • ndσ y = O div(aτ -ϕ)dy = O ϕ - A τ -dy,
so that the last integral does not vanish. This being true for any A ⊂ O, we get τ -> 0 a.e. on O.

For A ⊂ Σ + such that 0 < A (a • n) + dσ y < ∞, we solve λ 0 ϕ -div(aϕ) = 0 in D ′ (O), γ + ϕ = 1 A on Σ + ,
thanks to Lemma 8.12 and Lemma 8.14, and we get a unique solution 0 ≤ ϕ ∈ L 1 (O) such that ϕ ≡ 0. The Green formula again implies

A γτ -(a • n) + dσ y = O div(aτ -ϕ)dy = O ϕ,
so that the first integral does not vanish. This being true for any A ⊂ Σ + , we conclude that γ + τ -> 0 a.e. on Σ + .

Lemma 8.21 (Optimal weight). We assume that a satisfies (8.33) as well as a ∈ W 1,p ′ loc ( Ō) for some 1 ≤ p < ∞. For any g ∈ L p (O) satisfying (8.9) in the distributional sense with G ∈ L p (O), the associated trace function γg defined in Theorem 8.8 satisfies

γ g ∈ L p Σ, |n • a| τ dσ .
Proof of Lemma 8.21. One fixes β M (z) = (|z| ∧ M ) p . From the DiPerna-Lions renormalizing theory, we have

a • ∇(β M (g) τ + ) = β ′ M (g) G τ + + β M (g)(τ + -1) in D ′ (O). Because β M (g ε ) τ + ∈ L 1 (O) ∩ L ∞ (O)
and a/ y ∈ L 1 + L ∞ , we may use the Green formula (8.21) with φ ≡ 1, and we get

Σ- β M (γg) τ + |n • a|dσ = O (diva)β M (g) τ + -β ′ M (g) G τ + + β M (g)(τ + -1) g p-1 L p g L p + G L p .
Passing to the limit M → ∞, we obtain γ -g ∈ L p Σ -, |n • a| τ dσ . In a very same way, we prove γ + g ∈ L p Σ + , |n • a| τ dσ .

We give now a second version of an existence result in a L p framework with optimal assumption on the boundary condition in the sense that it is reverse with respect to Lemma 8.21. That also a posteriori justifies that Lemma 8.21 provides the optimal trace result in term of weight function on the boundary. Proof of Lemma 8.22. We only sketch the proof in the case of equation (8.29), arguing along the lines of Lemma 8.12. We start with an a priori estimate. Observing that div(aτ

+ |g| p ) = (diva)τ + g p + (τ + -1)g p + pτ + (Gg|g| p-2 -b|g| p -λ|g| p ),
we have

O |g| p 1 + pτ + (λ + b -1 p diva -1 p = Σ- |γ -g| p τ |a • n|dσ + O Gg|g| p-2 τ + .
Using the condition on λ, the property 0 ≤ τ + ≤ 1 and the Young inequality, we deduce

1 p O |g| p ≤ Σ- |γ -g| p τ |a • n|dσ + 1 p O |G| p .
We conclude in a similar way as in the proof of Lemma 8.13.

We consider now the time dependent transport equation with positive abstract kernels (8.58)

   ∂g ∂t + a • ∇g + bg = K [g] + G on (0, T ) × O, γ -g = R[g, γ + g] + g on (0, T ) × Σ -, g(0, •) = g 0 on O,
with notations introduced at the beginning of the Section. We will work in a weighted Lebesgue space L p m with the same conditions on p, m, a and b as introduced at the beginning of Section 8.2. On the other hand, we assume where we recall that the weight functions m O , m O and m Σ have been defined in (8.38). More precisely, recalling that R = R O + R Σ with R O and R Σ defined by (8.4), we assume

K : L p mO (O) → L p mO (O)
K [g] p L p m O ≤ α K g p L p m O + M K g p L p m , (8.61) R O [g] p L p m Σ ≤ α R g p L p m O + M R g p L p m , R Σ [h] p L p m Σ ≤ β R h p L p m Σ , (8.62) with α K , α R , β R ∈ [0, 1], M K , M R ≥ 0 and (8.63) ϑ O := (1 -α R -α K )/2 > 0, ϑ Σ := 1 -β R ≥ 0.
Let us emphasize that when p = 1, the assumption (8.61) is equivalent to the Lyapunov type condition We consider some data g 0 ∈ L p m (O), G ∈ L p m ((0, T ) × O) and g ∈ L p mΣ ((0, T ) × Σ + ) with either (1) β R ∈ [0, 1); or β R = 1. In the latter case, we assume that g = 0 and we make one of the following additional structural assumption

K * [m] ≤ α K ̟ + m + M K m.
(2) there exist an exponent p 0 ∈ [1, p] and a weight function m 0 such that K and R satisfy (8.59) in L p0 m0 and (8.60) in L p0 m0O (O) × L p0 m0Σ (Σ + ), with obvious definitions for the weight functions m 0O and m 0Σ , and with

L p m ⊂ L p0 m0 , L p mO ⊂ L p0 m0O , L p mΣ ⊂ L p0 m0Σ , where mΣ := m(τ + a • n) 2/p ; (3) p = 1 and R Σ is diffusive, namely R * Σ [τ + m Σ ] ≥ c Σ m Σ a.
e. on Σ + with c Σ > 0. In the above three cases, there exists a unique solution g ∈ L ∞ (0, T ; L p m (O)) ∩ C([0, T ]; L p0 m0 (O)) satisfying the transport equation (8.58) in the renormalized sense as well as g ∈ L p0 (0, T ; L p0 m0O (O)) and γg ∈ L p0 (0, T ; L p0 m0O (Σ)), with p 0 = p and m 0 = m in the first and the third cases. Remark 8.24. (1) The above result extends some previous results initiated by Bardos in [37, Chap. III] and Beals et al in [START_REF] Beals | Abstract time-dependent transport equations[END_REF]Thm. 1&7], where however only the kinetic case were considered. We refer to Section 10 for a discussion about that important model.

(2) When β R = 1, the existence part of the above result still holds (without any additional structural assumption).

(3) Similarly as observed in Remark 8.17, a weak maximum principle holds: g 1 ≤ g 2 if g 1 and g 2 are the renormalized solutions to two transport equations (8.58) such that (with obvious notations) Proof of Proposition 8.23. We split the proof into five steps.

b 1 ≥ b 2 , K 1 ≤ K 2 , R 1 ≤ R 2 , g 01 ≤ g 02 , G 01 ≤ G
Step 1. A priori estimates. For a positive solution, we formally compute

1 p d dt g p m p = 1 p Σ- (R[g, γ + g] + g) p m p Σ dσ y - 1 p Σ+ (γ + g) p m p Σ dσ y (8.64) + O g p-1 (K [g] + G) -g p ̟ m p .
Using the Young inequality and (8.61), we have

O g p-1 K [g]m p ≤ 1 p ′ O g p ̟ + m p + 1 p O K [g] p ̟ + -p/p ′ m p ≤ 1 p ′ + α K p O g p ̟ + m p + M K p O g p m p .
• When g = 0, using also (8.62) and once more the Young inequality, we then have

1 p d dt g p m p ≤ 1 p (β R -1) Σ+ (γ + g) p m p a • n + M R p + M K p + ̟ -L ∞ O g p m p + α R p + 1 p ′ + α K p + ε p ′ -1 O g p ̟ + m p + ε -p/p ′ p O G p m p ̟ + -p/p ′ for any ε > 0. Making the choice ε := ϑ O p ′ /p, we deduce d dt g L p m + ϑ O g p L p m O + ϑ Σ γ + g p L p m Σ ≤ pκ g p L p m + C O G p L p m , with κ := M R p + M K p + ̟ -L ∞ , C O := (ϑ O p ′ /p) -p ′ /p .
Using the Gronwall lemma, we then obtain

g(t) p L p m + t 0 e pκ(t-s) (ϑ O g s p L p m O + ϑ Σ γ + g s p L p m Σ ) ds (8.65) ≤ e pκt g 0 p L p m + C O t 0 e pκ(t-s) G s p L p m O ds, ∀ t ≥ 0.
• When g = 0 and thus ϑ Σ > 0, we control the ingoing boundary term by

Σ- (R[g, γ + g] + g) p m p Σ ≤ (1 + ε 1 ) Σ- R[g, γ + g] p m p Σ + C ε1 Σ- g p m p Σ , ∀ ε 1 > 0,
and a very similar computation as above leads to the a priori estimate

g(t) p L p m + t 0 e pκ(t-s) (ϑ ′ O g s p L p m O + ϑ ′ Σ γ + g s p L p m Σ ) ds (8.66) ≤ e pκt g 0 p L p m + t 0 e pκ(t-s) (C O G s p L p m O + C Σ g s p L p m Σ
) ds, for any t ≥ 0, with

ϑ ′ Σ := 1 -β R (1 + ε 1 ), ϑ ′ O := 1 -α R (1 + ε 1 ) -α K -ε p p ′ , κ := M R p (1 + ε 1 ) + M K p + ̟ -L ∞ , C O := ε -p ′ /p , C Σ := C ε1 ,
and where we have chosen ε, ε 1 > 0 small enough in such a way that ϑ ′ Σ > 0 and ϑ ′ O > 0.

• When ϑ Σ = 0 and thus g = 0, we further multiply the equation by m p τ ± , where τ ± is defined in (8.57), and integrating, we deduce

T 0 Σ∓ τ ± a • n(γg) p m p dσdt = O g p m p τ ± 0 T + T 0 O pg p-1 (K [g] + G)m p τ ± + T 0 O g p div(am p ) m p + 1 -λ 0 τ ± -pKm p τ ± .
Together with (8.65) and τ ± ∈ L ∞ (O), we obtain (8.67)

T 0 Σ+ (γ + g) p τ -m p Σ ≤ C T g 0 p L p m + G p L p (0,T ;L p m )
and (8.68)

T 0 Σ- [R Σ (γ + g)] p τ + m p Σ ≤ C T g 0 p L p m + G p L p (0,T ;L p m ) ,
for some constant C T ∈ (0, ∞). In particular, when p = 1 and R Σ is diffusive, we have

c Σ T 0 Σ+ (γ + g)m Σ ≤ T 0 Σ+ (γ + g)R * Σ (τ + m Σ ) = T 0 Σ- R Σ (γ + g)τ + m Σ ,
and together with (8.68), we deduce the additional estimate

(8.69) c Σ T 0 Σ+ (γ + g)m Σ ≤ C T g 0 L 1 m + G L 1 (0,T ;L 1 m ) .
Step 2. Existence. As a consequence of these a priori estimates, we may classically build a solution through an iterative scheme. For the sake of brevity, we only consider the (more interesting and more difficult) case b Σ = 1 (so that ϑ Σ = 0 and g = 0) and G = 0. For a given 0 ≤ g 0 ∈ L p m (O), we define a sequence of solution (h n ) starting from h 0 ≡ 0 thanks to the recursive definition

   ∂h n+1 ∂t + a • ∇h n+1 + Kh n+1 = K [h n ] on (0, T ) × O, γ -h n+1 = R[h n , γ + h n ] on (0, T ) × Σ -, h n+1 (0, •) = g 0 on O.
From Proposition 8.16, there exists a unique renormalized solution h n+1 ∈ C([0, T ); L p m (O)) to the above equation satisfying the estimate (8.45)

with g := h n+1 , G := K [h n ] and g := R[h n , γ + h n ] ∈ L p mΣ .
We observe that 0 ≤ h n ≤ h n+1 thanks to the weak maximum principle (see Remark 8.17) and that h n satisfies the estimates (8.65) and (8.68) where g is replaced by h n . Thanks to the monotonous convergence theorem of Beppo Levi, there exists g satisfying estimates (8.65) and h n → g in L p mO ((0, T ) × O). We may pass to the limit in the equation satisfied by (h n ) and we deduce that g is a renormalized solution to

∂g ∂t + a • ∇g + bg = K [g] on (0, T ) × O.
From Theorem 8.8 and Remark 8.9-( 5), the function g admits a trace γg and thanks to Proposition 8.10, we have γh n → γg a.e. on Σ\Σ 0 . Because of (8.4) and the Beppo Levi theorem again we deduce that

R[h n , γ + h n ] → R[g, γ + g] a.
e. on Σ -. Together with the fact that γ -h n → γ -g a.e. on Σ -, we have established that the boundary condition in (8.58) holds true.

It is worth emphasizing here that γ + g ∈ L 1 (Σ + ; dr Σ (y, •)) for a.e. y ∈ Σ -because of (8.68). For g 0 ∈ L p m (O), we separate the positive and the negative parts g 0 = g 0+g 0-and we obtain two renormalized solutions g ± ∈ L ∞ (0, T ; L p m ) associated to g 0± respectively. By linearity, the function g := g +g -∈ L ∞ (0, T ; L p m ) is a renormalized solution to the transport equation and the boundary condition is

γ -g = γ -g + -γ -g -= R O [g + ] -R O [g -] + R Σ [γ + g + ] -R Σ [γ + g -] = R O [g] + R Σ [γ + g],
where the last term is indeed well defined a.e. from the fact that γ + g ± ∈ L 1 (Σ + ; dr Σ (y, •)) for a.e. y ∈ Σ -and thus γ + g = γ + g + -γ + g -belongs to the same spaces. From Proposition 8.15, we already know that g ∈ C([0, T ]; L 0 (O)) and thus using an interpolation argument g ∈ C([0, T ]; L p1 m1 (O)) for any p 1 ∈ [1, p) and any weight function m 1 such that m 1 /m ∈ L pp1/(p-p1) when p > 1.

Step 3. When β R < 1 and p ∈ [1, ∞), we have (8.66), and we may just repeat the proof of Proposition 8.16 in order to get g ∈ C([0, T ]; L p m (O)) and the uniqueness of the solution.

Step 4. We assume β R = 1 and the structural assumption [START_REF] Agoshkov | Problema sledov funktsiȋ iz prostranstv H 1 p (Ω × D) i issledovanie nekotorykh variatsionnykh kraevykh zadach perenosa[END_REF]. From the estimate (8.64) on a solution g and the renormalized formulation of the equation, we deduce that

1 p 0 d dt |g| p0 m p0 = 1 p 0 Σ- |R[g, γ + g]| p0 m p0 0Σ - 1 p 0 Σ+ |γ + g| p0 m p0 0Σ + O g|g| p0-2 (K [g] + G)m p0 -|g| p0 m p0 0O ,
with a RHS term in L 1 (0, T ). As above, we thus deduce g ∈ C([0, T ]; L p0 m0 (O)) and next the uniqueness of the solution.

Step 5. We assume β R = 1 and the structural assumption [START_REF] Agoshkov | The existence of traces of functions in spaces used in problems of transport theory[END_REF]. In that case, we have p = 1, γ + g ∈ L 1 mΣ ((0, T ) × Σ + ) from (8.69) and then γ -g ∈ L 1 mΣ ((0, T ) × Σ -) from (8.62). We may thus justify the same computation as in Step 4 with p 0 = 1, and we deduce g ∈ C([0, T ]; L 1 m (O)) and next the uniqueness of the solution.

As an immediate consequence of the above analysis, we may associate to the transport equation (8.58) a semigroup. Corollary 8.25. Under the assumptions of Proposition 8.23, there exists a positive semigroup S on L p m such that for any

g 0 ∈ L p m (O), the function t → g(t) := S(t)g 0 ∈ C(R + ; L p0 m0 (O)) ∩ L ∞ loc (R + ; L p m (O))
is the unique renormalized solution to the transport equation (8.58) associated to the initial datum g 0 (and with G = g = 0). Furthermore the growth bound satisfies ω(S) ≤ κ.

We end this section by formulating the counterpart of the above result for the associated stationary problem (8.70) λg

+ a • ∇g + bg = K [g] + G on O, γ -g = R[g, γ + g] + g on Σ -.
Proposition 8.26. We make exactly the same assumptions as in Proposition 8.23 on a, b, K and R for some weight function m : Ō → [1, ∞) and some exponent p ∈ [1, ∞) as well as either β R < 1 holds or β R = 1 holds with g = 0 and one of the additional structure assumptions (1) or [START_REF] Agoshkov | Problema sledov funktsiȋ iz prostranstv H 1 p (Ω × D) i issledovanie nekotorykh variatsionnykh kraevykh zadach perenosa[END_REF]. There exists λ * * ∈ R such that for any λ > λ * * , G ∈ L p m (O) and g ∈ L p mΣ (Σ + ), there exists a unique solution g ∈ L p mO (O) satisfying the transport equation (8.70) in the renormalized sense and some additional a priori estimates listed during the proof.

Proof of Proposition 8.26. We just explain the main steps. We first establish an a priori estimate. We observe that any solution g to the stationary problem (8.70) (at least formally) satisfies

(8.71) O |g| p m p λ+̟ + 1 p Σ+ |γ + g| p m p Σ = O (K [g]+G)g|g| p-2 m p + 1 p Σ- |R[g, γ + g]+g| p m p Σ .
We then only consider the case g = 0. Repeating the same computations as in Step 1 of the proof of Proposition 8.23 and with the same notations, we get

(8.72) p(λ -κ) g p L p m + ϑ O g p L p m O + ϑ Σ γ + g p L p m Σ ≤ C O G p L p m . For λ > λ * * := max(κ, λ * p )
and G ≥ 0, we next consider the sequence (h k ) in L p mO defined iteratively as the solution given by Lemma 8.13 to

λh k + a • ∇h k + bh k = K [h k-1 ] + G on O, γ -h k = R[h k-1 , γ + h k-1 ] on Σ -,
for k ≥ 1 and starting from h 0 ≡ 0. We observe that (h k ) is increasing and satisfies the estimate (8.72) where g is replaced by h k . We may pass to the limit in the above equation and estimate and we obtain a renormalized solution g ∈ L p mO to the transport equation (8.70) and satisfying the estimate (8.72). By linearity, the same holds without sign condition on G. Finally, considering the three different cases as in Steps 3, 4 and 5 in Proposition 8.23, we similarly show that g ∈ L p0 m0O and γ + g ∈ L p0 m0Σ for suitable exponent p 0 ∈ [1, p] and weight function m 0 . For two such solutions g i to (8.70), the function g := g 2g 1 is also a renormalized solution to (8.70) for which we may justify the identity (8.71) with p = p 0 , m = m 0 , G = 0. We thus deduce that (8.72) holds with p = p 0 , m = m 0 , G = 0, and we conclude that g = 0, what ends the proof of the uniqueness.

Characteristics.

In this section we come back to the characteristics method for the evolution and the stationary transport equation. Our aim is in particular to discuss the representation formula (8.47). We consider a vector field a : O → R D which extends to R D and, denoting by the same letter a its extension, we at least assume

(8.73) a ∈ W 1,1 loc (R D ), diva ∈ L ∞ (R D ), a/ y ∈ L 1 (R D ) + L ∞ (R D ).
After DiPerna and Lions [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF][START_REF] Lions | Sur les équations différentielles ordinaires et les équations de transport[END_REF] (see also [START_REF] Hauray | Deux remarques sur les flots généralisés d'équations différentielles ordinaires[END_REF]Def. 1] or [203, Def. 1]), we introduce the following notion of flow.

Definition 8.27. We name almost everywhere flow associated to (8.6) a measurable function

Y : R × R D → R D , (t, y) → Y t (y), such that (i) for a.e. y ∈ R D , the map t → Y t (y) is continuous and Ẏt (y) = a(Y t (y)) in D ′ (R), Y 0 (y) = y;
(ii) for a.e. y ∈ R D and for any s, t ∈ R, there holds Y s+t (y) = Y s (Y t (y));

(iii) there exists C ≥ 0 such that

(8.74) ∀ t ∈ R, e -CT λ ≤ Y (t, •) ♯ λ ≤ e CT λ,
where for any t ∈ R and any ϕ ∈ L ∞ c (R D ), the space of L ∞ functions with compact support. In the compressible case (diva = 0) and when a only satisfies (8.73), it seems not clear that [139, Thm. III.2] or [9, Thm. 31 & Remark 32] provides an a.e. flow such that (8.75) holds. In that general case, the volume identity (8.75) must be replaced by the volume two sides estimate (or nearly-incompressible condition):

(Y (t, •) ♯ λ)(A) = λ(Y (-t, A)), A ⊂ R D ,
e -t diva ∞ R D ϕ(y)dy ≤ R D ϕ(Y t (y))dy ≤ e t diva ∞ R D ϕ(y)dy, for any 0 ≤ ϕ ∈ L ∞ c (R D ) and t ∈ R, what is nothing but (8.74) with C := diva ∞ .
It is however quite straightforward to prove from [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF][START_REF] Ambrosio | Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields[END_REF] that the a.e. flows Y satisfies (8.75) when a additionally satisfies diva ∈ C(R D ). One possible way to construct the a.e. flow Y is to define Y = Y t (y) as the unique renormalized solution in C(R; L) to the transport equation

∂ t Y -a • ∇ y Y = 0 on R × R D , Y 0 (y) = y on R D , or more explicitly ∂ ∂t β(Y ) -a(Y ) • ∇β(Y ) = 0 on R × R D , β(Y 0 ) = β(y) on R D ,
in the distributional sense for any β ∈ C 1 (R D , R) such that β and |∇β(z)|(1 + |z|) are uniformly bounded on R D . In particular, for any g 0 ∈ C 1 (R D ) and next for any g 0 ∈ L 0 (R D ) the function g ♯ (t, y) := g 0 (Y -t (y)) is the unique renormalized solution to the transport equation

∂ t g ♯ + a • ∇g ♯ = 0 on R × R D , g ♯ (0, •) = g 0 on R D .
We introduce some notations. We denote y ∈ Y if (i) and (ii) hold. In particular, Y is a measurable subset of R D and |R D \Y| = 0. Because of (i), for y ∈ V := O ∩ Y, we may define the backward exit time

t b (y) := sup τ > 0 | Y -t (y) ∈ O, ∀ s ∈ [0, τ ] ∈ (0, +∞],
the subset V b := {y ∈ V; t b (y) < +∞} and the associated entering position

y b (y) := Y -t b (y) (y) when y ∈ V b .
We observe that the function

t b : V → (0, +∞] is measurable, V b is a mesurable subset of O and t b (Y s (y)) = t b (y) + s, y b (Y s (y)) = y b (y), ∀ y ∈ V b , ∀ s ∈ [0, t b (y)). (8.76) meas({y ∈ O; t b (y) = t}) = 0, ∀ t > 0. (8.77)
The properties (8.76) are straightforward while (8.77) is a consequence of the fact that {y ∈ V; t b (y) = t} ⊂ Y t (Σ) and of the nearly-incompressible condition (8.74). We now introduce the following first regularity assumption on a at the boundary

(8.78) ∀ y 0 ∈ Σ, y → a(y) • n(y 0 ) is continuous on Ō.
Let us present two examples.

-It may happen that V b = ∅. For instance, choosing O := {y ∈ R 2 ; |y| < 1} the unit disk of the plane and a(y) := |y|y ⊥ ∈ C 0,1 (R 2 ; R 2 ), y ⊥ := (y 2 , -y 1 ), we have diva ≡ 0 and a(y)•n(y) = y ⊥ •y = 0 for any y ∈ R 2 , so that the flows do not encounter the boundary set Σ = {y ∈ R 2 ; |y| = 1}. In that situation

V = O and V b = ∅. -In the kinetic case, namely y = (x, v) ∈ O := Ω × R d , Ω ⊂ R d
an open set with smooth boundary with unit normal outward ν x , so that n(y) = (ν x , 0), and a(y) = (v, F (x, v)), we have diva = div v F and div v F = 0 when F = E(x) + v ∧ B(x), and we have a(y)

• n(y 0 ) = v • ν x0 which is a smooth function on Ō × Σ.
Lemma 8.28. Under the condition (8.78), the mapping

y b : V b → Σ -∪ Σ 0 is measurable.
Proof of Lemma 8.28. From the very definitions and composition rules, we have

y b : V b → Σ ∩ Y is mesurable. Take y ∈ V b , denote y 0 := y b (y) and consider a sequence t k ց 0 so that Y t k (y 0 ) → y 0 and Y t k (y 0 ) ∈ O for any k ≥ 1. From (8.78), we deduce 0 ≥ lim sup k→∞ Y t k (y 0 ) -y 0 t k • n(y 0 ) = lim k→0 1 t k t k 0 a(Y s (y 0 )) • n(y 0 ) ds = a(y 0 ) • n(y 0 ), which means that y 0 ∈ Σ -∪ Σ 0 .
For further references, we introduce the following second additional mild regularity assumption on a in the domain

(8.79) a ∈ L ∞ loc ( Ō), (a(y) • y) + y 2 .
This one may in fact replace the last boundedness condition on a in (8.73). We establish a technical result which will be useful in the sequel.

Lemma 8.29. Under assumptions (8.78) and (8.79), the following hold:

(1) For any R 0 , T > 0, there exists R T , L T > 0 such that for any y ∈ B R0 there hold

(8.80) sup [-T,T ] |Y t (y)| ≤ R T and |Y t2 (y) -Y t1 (y)| ≤ L T |t 2 -t 1 |, ∀ t 1 , t 2 ∈ [-T, T ].
(2) For a sequence

(y ε ) of V such that y ε → y 0 ∈ Σ -∩ Y, we have t b (y ε ) → 0 and y b (y ε ) → y 0 .
Proof of Lemma 8.29. Proof of (1). Take y ∈ V ∩ B R0 . On the one hand, from (8.79), we have

Y t (y) 2 -y 2 = 2 t 0 a(Y s (y)) • Y s (y))ds ≤ C t 0 Y s (y) 2 ds,
and we conclude to Y t (y) ∈ B RT thanks to the Gronwall lemma. As a consequence, we have

|Y t2 (y) -Y t1 (y)| ≤ |t 2 -t 1 | a L ∞ (BR T ) , ∀ t i ∈ [-T, T ].
Proof of (2). Assume by contradiction that lim sup t b (y ε ) ≥ τ > 0 and set T := τ + 1. By assumption, there exists R 0 > 0 such that y ε ∈ B R0 and thus by step 1, (8.80) holds uniformly in ε ∈ (0, 1]. Thanks to the Ascoli Theorem and the contradiction hypothesis, there exists ε k → 0 such that t b (y ε k ) ≥ τ and a there exists

Y ∈ C([-T, T ]) such that Y • (y ε k ) → Y • in C([-T, T ]).
Next, passing to the limit in the conditions

Y -t (y ε k ) ∈ O and Y -t (y ε k ) -y ε k = - t 0 a(Y -s (y ε k )) ds,
for any k ≥ 1 and any t ∈ [0, τ ], we get

Y -t ∈ Ō and Y -t -y 0 = - t 0 a(Y -s )ds.
We deduce

0 ≥ lim sup t→0 Y -t -y 0 t • n(y 0 ) = lim t→0 - 1 t t 0 a(Y -s ) • n(y 0 ) ds = -a(y 0 ) • n(y 0 ),
which is in contradiction with the hypothesis y 0 ∈ Σ -. Now, we may estimate

|y b (y ε ) -y 0 | ≤ |y ε -y 0 | + t b (yε) 0 |a(Y s (y ε ))|ds → 0,
as ε → 0, as a consequence of the convergence t b (y ε ) → 0, the first estimate in (8.80) and the first condition in (8.79).

We reformulate some "space continuity" of solution to the transport equation results picked up in [START_REF] Boyer | Trace theorems and spatial continuity properties for the solutions of the transport equation[END_REF]Sec. 7]. Defining

O α := {y ∈ O; δ(y) > α}, Σ α := {y ∈ O; δ(y) = α} = ∂O α ,
we know from [70, Sec. 2], that there exists α O > 0 such that for any α ∈ (0, α O ), the mapping

θ α : Σ → Σ α , θ α (z) := z -αn(z)
is an isomorphim with associated jacobian function J α and

Σα h(z ′ )dσ α (z ′ ) = Σ h • θ α (z)J α (z)dσ z , (8.81) O\Oα g(y)dy = α 0 Σ g • θ α ′ (z)J α ′ (z)dσ z dα ′ , (8.82)
for any h ∈ L 1 (Σ α ) and g ∈ L 1 (O\O α ), where dσ α denotes the Lebesgue measure on Σ α and where the jacobian function J α satisfies 1/2 ≤ J α ≤ 3/2 as well as J α → I as α → 0. where we denote by γ α g the trace of g on Σ α , and

(8.84) γ α g • θ α → γg as α → 0, a.e. on Σ\Σ 0 . Proof of Lemma 8.30. For ϕ ∈ C c ( Ō) ∩ W 1,1 (O) and β ∈ C 1 (R), the renormalized Green formula (8.24) writes Σ ϕβ(γg) a • n dσ = Σα ϕβ(γ α g) a • n dσ α + O\Oα [div(aϕ)β(g) + ϕβ ′ (g)a • ∇g] dy, and thus (8.85)
Σ ϕβ(γg) a • n dσ = lim α→0 Σα ϕβ(γ α g) a • n dσ α . Denoting ψ := (1 -(δ(x) -α)/s) + , α + s ∈ (α, α O )
, observing that ψ |Σα = 1, recalling that n = -∇δ and using ϕψ as a test function, we similarly have

Σα ϕβ(γ α g) a • n dσ α = 1 h Oα\Oα+s ϕβ(g) a • n dy + Oα\Oα+s [div(aϕ)β(g) + ϕβ ′ (g)a • ∇g]ψ dy, so that Σα ϕβ(γ α g) a • n dσ α = lim s→0 1 s Oα\Oα+s ϕβ(g) a • n dy.

We immediately deduce

Σα

ϕβ(γ α g) a • n dσ α = d dα Oα ϕβ(g) a • n dy and next α 0 Σ α ′ ϕβ(γ α ′ g) a • n dσ α ′ dα ′ = O\Oα ϕβ(g) a • n dy.
Together with (8.82) and (8.81), we have established (8.83) follows. On the other hand, using (8.81) and (8.85) together, we have

α 0 Σ [ϕβ(γ α ′ g) a • n] • θ α ′ (z)J α ′ (z)dσ z dα ′ = α 0 Σ [ϕβ(g) a • n] • θ α ′ (z)J α ′ (z)dσdα ′ , so that β(γ α ′ g) a • n = β(g) a • n a.e. on Σ α ′ for a.e. α ′ ∈ (0, α), from what
Σ ϕβ(γg) a • n dσ = lim α→0 Σ [ϕβ(γ α g) a • n] • θ α J α dσ α , which implies (8.86) [β(γ α g) a • n] • θ α J α ⇀ β(γg) a • n * σ(L ∞ loc (Σ), L 1 c (Σ)).
Repeating the same argument with g := a • n and using that J α → I uniformly, we get

[β(a • n) a • n] • θ α ⇀ β(a • n) a • n * σ(L ∞ loc (Σ), L 1 c (Σ)), for any β ∈ C 1 (R). Choosing β(s) = 1 and β(s) = s, we classically deduce that [a • n] • θ α → a • n
a.e. on Σ. We finally conclude to (8.84) by gathering that last information with (8.86) written for β(s) = s and β(s) = s 2 . Remark 8.31. During the proof, we have in fact established that

[0, α O ] → L 1 (Σ); α → [γ α g a • n] • θ α is continuous.
Lemma 8.32. We make the additional assumptions (8.78) and (8.79). If Y is an almost everywhere flow associated to (8.6), then the function t b ∈ L(O) is a renormalized solution to the equation

(8.87) a • ∇t b = 1 in O, γ -t b = 0 on Σ -∩ Y.
Proof of Lemma 8.32.

Step 1. We fix β ∈ C 1 * (R) and we recall that β(t b (Y s (y))) = β(t b (y) + s) for any s ∈ R for a.e. y ∈ O. For any ϕ ∈ C 1 c (O), we may compute

R d β ′ (t b (y) + s)ϕ(y) dy = d ds R d β(t b (y) + s)ϕ(y) dy = d ds R d β(t b (Y s (y)))ϕ(y) dy = d ds R d β(t b (y))ϕ(Y -s (y))e -0 -s (diva)(Yτ (y))dτ dy = R d β(t b (y))[-a • ∇ϕ -(diva)ϕ](Y -s (y))e -0 -s (diva)(Yτ (y))dτ dy.
Taking s = 0, we conclude to

R d β ′ (t b )ϕ dy = R d β(t b )[-a • ∇ϕ -(diva)ϕ] dy,
which is nothing but the distributional formulation of the equation

a • ∇β(t b ) = β ′ (t b ).
That last family of equations is the renormalized formulation of equation (8.87) in the domain.

Step 2. Using lemma 8.30 with g := β(t b ), we have

γ α β(t b ) • θ α → β(γt b ) a.e. on Σ.
Using Lemma 8.29, we also have

γ α β(t b ) • θ α → 0 a.e. on Σ -∩ Y.
Both together, we find γ -t b = 0 on Σ -∩ Y.

We establish the main result of this section.

Theorem 8.33 (characteristics method). Assume that Y is an almost everywhere flow associated to (8.6) with a satisfying (8.73). For any

g 0 ∈ L ∞ (O), g ∈ L ∞ ((0, T ) × Σ -), T > 0, b ∈ L ∞ (O), the function ḡ(t, y) := g 0 (Y -t (y))e -t 0 b(Yτ-t(y)) dτ 1 t<t b (y) (8.88) +g(t -t b (y), y b (y))e -t b (y) 0 b(Y τ -t b (y) (y)) dτ 1 t>t b (y) is the unique solution in C([0, T ]; L 1 loc (O)) ∩ L ∞ ((0, T ) × O) to the evolution transport equation (8.89)    ∂g ∂t + a • ∇g + bg = 0 on (0, T ) × O, γ -g = g on (0, T ) × Σ -, g(0, •) = g 0 on O.
First proof of Theorem 8.33. We additionally assume that (8.75), (8.78) and (8.79) hold, that Let us then fix ϕ ∈ D((0, T ) × O) and let us extend ḡ and ϕ by 0 outside of O. We compute

g ∈ C((0, T ) × Σ -), supp g ⊂ (0, T ) × (Σ -∩ Y)
0 = d ds T 0 R d ḡ(t, y)ϕ(t, y) dydt = d ds T 0 R d ḡ(t + s, Y s (y))e s 0 b(Yτ (y))dτ ϕ(t, y) dydt = d ds T 0 R d ḡ(t, y)ϕ(t -s, Y -s (y))e s 0 (b-diva)(Yτ (y))ds dydt = T 0 R d ḡ(t, y) d ds [ϕ(t -s, Y -s (y))e s 0 (b-diva)(Yτ (y))ds ] dydt = T 0 R d ḡ(t, y)[-∂ t ϕ -a • ∇ϕ + (b -diva)ϕ](t -s, Y -s (y)) dydt,
where we have used the relation (8.90) in the second line and the change of variables property (8.75) in the third line. Taking s = 0, we get

0 = T 0 R d ḡ(t, y)[-∂ t ϕ -a • ∇ϕ + (b -diva)ϕ](t, y) dydt,
which exactly means that ḡ is a solution to equation (8.89) in the distributional sense. Now, because t b (y) > 0 for any y ∈ Y, we have ḡ(0, y) = g 0 (y). Take t > 0, and for α ∈ (0, α O ), let us denote

A α := {y ∈ Σ -∩ Y; t b • θ α (y) < t}, so that ḡ(t, θ α (y)) = g(t -t b,α (y), y b,α (y))e - t b,α (y) 0 b(Y τ -t b,α (y) (θα(y)) dτ on A α ,
where we use the shorthands t b,α := t b • θ α and y b,α := y b • θ α . From θ α ′ (y) → y as α ′ → 0 when y ∈ Σ and Lemma 8.29, we deduce

ḡ(t, θ α ′ (•)) → g(t, •) as α ′ → 0, on A α for any fixed α > 0. Because ∪ α>0 A α = Σ -∩ Y, the same convergence holds on Σ -∩ Y.
On the other hand, we have γ α ḡ • θ α → γḡ as α → 0 a.e. on Σ\Σ 0 , from Lemma 8.30. We deduce that γḡ = g on Σ -∩ Y.

Second proof of Theorem 8.33. We do not make any additional assumption and we mainly repeat the proof presented in [START_REF] Hauray | Deux remarques sur les flots généralisés d'équations différentielles ordinaires[END_REF]Sec. 3]. Consider the unique solution g ∈ C([0, T ]; L 1 loc ( Ō)) ∩ L ∞ ((0, T ) × O) to the transport equation (8.89). Regularizing by convolution

g ε := g * t,x,ε ρ ε ,
for a time and space dependent mollifier sequence ρ ε similarly as in (8.20), we have

g ε ∈ C 1 ([0, T ]× Ō) and ∂g ε ∂t + a • ∇g ε + bg ε = R ε on (0, T ) × O,
g ε (t, y) := g ε (t, y) -g ε (0, Y -t (y))e B(-t,y) 1 t<t b (y) -g ε (t -t b (y), y b (y))e B(-t b(y) ,y) 1 t>t b (y) = t 0 (R ♯ ε e B )(s, y)ds1 t<t b (y) + t t-t b (y) (R ♯ ε e B )(s, y)ds1 t>t b (y) ,
for a.e. y ∈ O and any t > 0 and where we use (8.77) for getting rid of the set {y ∈ O; t b (y) = t}.

For T, ̺ > 0 and setting U T,̺ := (0, T ) × (O ∩ B ̺ ), we deduce

UT,̺ | g ε (t, y)| dydt ≤ T UT,̺ |(R ♯ ε e B )(s, y)|dyds ≤ T e (C+ b L ∞ )T UT,̺ |R ε (s, y)|dyds,
from the near-incompressibility condition (8.74) of the flow. From Proposition 8.10 and Remark 8.11, we know that

g ε (t, •) → g(t, •) in L 1 loc ( Ō) as ε → 0, for any t ∈ [0, T ]; g ε|Σ-→ γ -g = g in L 1
loc (Σ -) as ε → 0. Passing to the limit, we get with S b is defined by (8.49) in which formula Y and t b stand for the characteristics and backward exit time defined just as above.

Proof of Corollary 8.34. That is nothing but (2.13).

Adapting the second proof of Theorem 8.33, we obtain a more accurate characterization of the backward exit time t b with more general assumptions on the vector field.

(8.92) a • ∇τ = 1 in O, γ -τ = 0 on Σ -.
We also have

(8.93) t b = ∞ 0 S 0 (t)1dt,
where 1 stands for the unit function in O and S 0 is defined by (8.49) with b = 0 and in which formula Y and t b stand for the characteristics and backward exit time defined just as above.

Proof of Lemma 8.35.

Step 1. Existence. From Lemma 8.12, for any λ > λ * ∞ := 0, there exists a (unique) solution τ λ ∈ L ∞ (O) to the truncated backward exit time problem

λτ λ + a • ∇τ λ = 1 in O, γ -τ λ = 0 on Σ -.
From the weak maximum principle, we have τ λ ≥ 0. As a consequence, for 0 < λ < λ ′ , we have

λ(τ λ -τ λ ′ + a • ∇(τ λ -τ λ ′ = (λ ′ -λ)τ λ ′ ≥ 0 in O, γ -(τ λ -τ λ ′ ) = 0 on Σ -.
From the weak maximum principle again, we deduce that τ λτ λ ′ ≥ 0, and (τ λn ) is an increasing sequence when λ n ց 0. We set τ := lim n→∞ τ λn , so that τ ∈ L(O) is a nonnegative renormalized solution to the backward exit time problem.

Step 2. Characterization. By definition, for any

β ∈ C 1 * (R), β(0) = 0, the function τ satisfies a • ∇β(τ ) = β ′ (τ ) in O, γ -β(τ ) = 0 on Σ -.
With the notations of (8.20), we define b ε := β(τ ) * ε ρ ε , B ε := β ′ (τ ) * ε ρ ε , and thanks to Lemma 8.5, we have thus

a • ∇b ε = B ε + r ε in D ′ (O), with b ε → β(τ ), B ε → β ′ (τ ), r ε → 0, respectively in L p loc ( Ō), L p loc ( Ō) and L 1 loc ( Ō) for any p ∈ [1, ∞). Because then d ds b ε • Y s = (a • ∇b ε )(Y s ) = (B ε + r ε )(Y s ),
and defining

b t,ε (y) := b ε (Y -t (y)) + 0 -t B ε (Y s (y)) ds 1 t<t b (y) + (γ -b ε )(y b (y))) + 0 -t b (y) B ε (Y s (y)) ds 1 t>t b (y) ,
we have

b ε (y) -b t,ε (y) = R t,ε := 0 -t r ε (Y s (y)) ds1 t<t b (y) + 0 -t b (y) r ε (Y s (y)) ds1 t>t b (y) .
Arguing similarly as in the second proof of Theorem 8.33, we have b

t,ε → b t and R t,ε → 0 in L 1 (U ρ,T ) as ε → 0, with b t (y) := β(τ (Y -t (y))) + 0 -t β ′ (τ (Y s (y))) ds 1 t<t b (y) + 0 -t b (y) β ′ (τ (Y s (y))) ds 1 t>t b (y) .
We deduce that

β(τ (y)) = b t (y), for a.e. t > 0, y ∈ O.
Choosing a sequence (β n ) of renormalizing function in C 1 * (R) such that 0 ≤ β n (s) ր s and 0 ≤ β ′ n (s) ր 1 locally uniformly, writing the above equation for β = β n and passing to the limit n → ∞, we obtain

τ (y) = τ (Y -t (y)) + t 1 t<t b (y) + t b (y)1 t>t b (y) ,
and in particular τ = t b a.e. on O. That implies that t b is the unique renormalized solution to the backward exit time problem (8.92). From Corollary 8.34, for any λ > 0, we have

τ λ = ∞ 0 e -λt S 0 (t)1dt a.e on O,
and we deduce (8.93) by passing to the limit λ ց 0 in that identity. 8.5. On the Krein-Rutman theorem for the transport equation with kernel terms.

In this section we carry on our analysis of the transport equation with kernel term (8.1)-(8.3) for which we establish a Krein-Rutman result under strong positivity assumption on the kernel acting on the domain. As in section 8.3, we assume that a, b, K and R satisfy the conditions (8.35), (8.59), (8.60), (8.61), (8.62) and (8.63) for some weight function m : Ō → [1, ∞) and some exponent p ∈ [1, ∞). On the kernel K , we make the additional strong positivity hypothesis: for any x ∈ O, there exist r x , η x > 0 such that (8.94)

∀ f ≥ 0, ∀ y ∈ B(x, r x ), K [f ](y) ≥ η x B(x,rx) f * dy * and (8.95) ∃ x 0 , a, b ∈ L ∞ (B(x 0 , r 0 )), r 0 := r x0 ,
as well as one of the two following regularity assumptions

(8.96) K ∈ K(L p m (O)) or K : L p m (O) → L p1 (O) ∩ L p m1 (O), with p 1 > p and m 1 /m → ∞ when y → ∞.
We thus consider the operator

(8.97) Lf = -a • ∇f -bf + K [f ] = -div(af ) -Kf + K [f ]
with K := bdiva ≥ 0, which is complemented with the boundary condition

(8.98) γ -f = R O [f ] + R Σ [γ + f ] on Σ -.
More precisely, we define L in the Banach space L p m (O) with domain

D(L) ⊂ {f ∈ L 1 (O); a • ∇f ∈ L 1 loc ( Ō), γ -f = R[f, γ + f ]}.
Notice that because of Section 8.1 the trace function is well defined.

Example. The nonlocal operator with a drift (8.99)

∂ t f = -a ∂ x f -b + K [f ] in O, γ -f = 0 on Σ -, with O ⊂ R a bounded interval, a ∈ W 1,1 loc (O), a ′ ∈ L ∞ (O), b ∈ L ∞ (O)
, and thus the boundary kernel is R ≡ 0. Motivated by some non-local reaction-diffusion models, this problem was recently investigated in [START_REF] Cloez | On an irreducibility type condition for the ergodicity of nonconservative semigroups[END_REF][START_REF] Coville | On generalized principal eigenvalues of nonlocal operators with a drift[END_REF][START_REF] Li | On eigenvalue problems arising from nonlocal diffusion models[END_REF]. It is also used in the study of selection-mutation models in changing environment, see the even newer works [START_REF] Forien | Ancestral lineages in mutation selection equilibria with moving optimum[END_REF][START_REF] Henry | Time reversal of spinal processes for linear and non-linear branching processes near stationarity[END_REF].

We start by checking that with the above assumptions, the conditions (H1)-(H5) presented in the abstract part hold true.

Condition (H1). From Proposition 8.26, we know that for any λ > λ * * the stationary problem

(λ -L)g = G in O, γ -g = R[g, γ + g] on Σ -,
has a unique solution. More precisely, the associated inverse operator denoted by R L (without reference to the boundary operator

R) satisfies R L : L p m → L p m and R L G ≥ 0 if G ≥ 0. Condition (H2).
We first consider the case when R O ≡ 0 and we denote by L 0 the associated generator. We fix

f 0 ∈ C 2 c (O), such that B0 f 0 dy = 1, f 0 > 0 on B 0 , suppf 0 = B0 , as well as f -1 0 L ∞ (Bε) = ε -2 , ∇f 0 L ∞ (B0\Bε) = ε, ∀ ε ∈ (0, 1/2),
where we denote B ε := B(x 0 , (1ε)r 0 ). We also define C 0 := f 0 L ∞ (B0) and C 1 := ∇f 0 L ∞ (B0) , both may be bounded by a constant which only depends on r 0 and d. Because of (8.94), we have

K [f 0 ](y) ≥ η 0 1 B0 ≥ η 0 C 0 f 0 .
We observe that f 0 ∈ D(L 0 ) and we compute (8.100)

L 0 f 0 ≥ -a L ∞ (B0) C 1 1 B0 -b L ∞ (B0) C 0 1 B0 + η 0 1 B0 ≥ κ 0 f 0 , if κ 0 := η 0 /C 0 -a L ∞ (B0) C 1 /C 0 -b L ∞ (B0) ≥ 0.
More generally, we have

(8.101) L 0 f 0 ≥ -a L ∞ (B0) C 1 1 Bε -a L ∞ ε1 B0\Bε -b L ∞ (B0) f 0 + η 0 1 B0 ≥ κ 0 f 0 , with κ 0 := -a L ∞ (B0) C 1 ε -2 -b L ∞ (B0) ∈ R when a L ∞ ε ≤ η 0 .
Depending on how η 0 > 0 is large, we obtain in that way two constructive lower bounds of I thanks to Lemma 2.4-(ii) and we have thus established that L 0 satisfies (H2). Because f 0 ∈ D(L 0 ), we have S L0 (t)f 0 ≥ e κ0t f 0 for any t ≥ 0, from Remark 2.5-(2). On the other hand, we observe that S L (t) ≥ S L0 (t) for any t ≥ 0, from the weak maximum principle mentioned in Remark 8.24-(3). These two last observations together imply S L (t)f 0 ≥ e κ0t f 0 , for any t ≥ 0. We deduce from Lemma 2.4-(iv) that (H2) holds.

Condition (H3). We introduce the semigroup S B associated to the transport equation

∂g ∂t + a • ∇g + bg = 0, γ -g = R[γ + g],
which is well defined thanks to Corollary 8.25 and satisfies S B (t)g 0 L p m ≤ e κBt g 0 L p m for any t ≥ 0 and g 0 ∈ L p m with κ B := ̟ -L ∞ + M R /p because of the a priori estimate (8.65) particularized to the present case (in particular where we can take ε 1 = 0 because the influx function is g = 0 here). We formulate the first hypothesis (8.102)

η 0 > ̟ -L ∞ C 0 + M R C 0 /p + a L ∞ (B0) C 1 -b L ∞ (B0) C 0 ,
with the same definitions as above for B 0 , C 0 and C 1 , so that κ 0 > κ B because of (8.100). In a second case, we assume In that case, the semigroup S B is explicitly given by

(S B (t)f 0 )(y) =    f 0 (Y -t (y)) exp(- t 0 K(Y τ -t (y))dτ ), if t ∈ (0, t b (y))
0 otherwise, and in particular S B (t)f = 0 for any f and any t > T O . We immediately deduce κ B = -∞ and thus κ 0 > κ B because we have established that κ 0 ∈ R. We next define Af := K [f ]. Using Lemma 2.8 and Remark 2.9-(2) or Lemma 2.13 and Remark 2.14-(1) depending on the assumption (8.96) made on K , we deduce that the condition (H3) holds in both cases discussed above. Under the first condition in (8.96), we conclude to the existence of eigenvalue triplet (λ

1 , f 1 , φ 1 ) ∈ R × L p m × L p ′ m -1 .
Under the second condition in (8.96), we may also get the same conclusion by by using [337, Cor. 1 of Thm. II.9.9] when p = 1 or by observing that the dual problem is similar to the primal problem when p > 1 and thus we may apply the same arguments for the dual problem as those explained above for the primal problem.

Condition (H4). Let us consider

λ > λ * * and 0 ≤ f ∈ L p mO (O) a (renormalized) solution to λf + a • ∇f + bf -K [f ] = F in O, γ -f = R[f, γ + f ] on Σ -, with 0 ≤ F ∈ L p m (O). If f ≡ 0, there exists x 1 ∈ O such that B(x1,r 1 ) f (z) dz > 0. From (8.94), we deduce K [f ](y) ≥ B(x1,r1) κ(y, z)f (z) dz > 0, ∀ y ∈ B(x 1 , r 1 ).
Now, we argue similarly as during the proof of Lemma 8.14 and in particular we use the same notations. For A ⊂ B(x 1 , r 1 ), we define the solution 0

≤ ϕ ∈ L p ′ m -1 ∩ L ∞ to the equation λϕ -div(aϕ) + bϕ = 1 A in O, γ + ϕ = 0 on Σ + ,
thanks to Lemma 8.12 and Lemma 8.13, and we observe that ϕ ≡ 0 on B(x 1 , r 1 ) if |A| > 0. For the renormalizing function β δ and a truncation function χ R , we compute

0 ≥ Σ a • nβ δ (γf )γϕχ R = O [β ′ δ (f )(F + K [f ])ϕ -β δ (f )1 A ]χ R + O (β δ (f ) -f β ′ δ (f ))(λ + b)ϕχ R + O ϕβ(f ) a R • (∇χ) R .
Passing first to the limit R → ∞ and next to the limit δ → 0, we deduce

0 ≥ O [(F + K [f ])ϕ -f 1 A ],
so that in particular

A f dy ≥ B(x1,r1) ϕK [f ] > 0.
This being true for any A ⊂ B(x 1 , r 1 ), we deduce f > 0 a.e. on B(x 1 , r 1 ). By a classical continuity argument, we conclude that f > 0 a.e. on O. We have thus established (H4) for λ > λ * * from what we immediately and classically deduce the general case λ ∈ R. We summarize our analysis in the following result which is a straightforward consequence of the above checked conditions together with Theorem 2.21, Theorem 4.13, Theorem 5.16 and Theorem 5.23. We state the available result in that situation. We assume further that K satisfies the strong positivity conditions (8.94) together with (8.95) and the first compactness property formulated in (8.96). We finally assume that (8.102) holds or (8.103) holds. In both cases, the conclusion (C3) holds as well as the ergodicity (E2) in L 1 φ1 . We are not aware of any similar result for such a general transport equation, see however the next sections where more specific transport like equations are discussed. We do not try to improve the convergence result in the general case, but rather we aim to make one step further in the following particular situation where Doblin approach may be used.

Condition (H5). Assume that (λ, f ) ∈ C × D(L) satisfies L|f | = (ℜeλ)|f | in O, R[|f |, γ + |f |] = γ -|f | on Σ -, and
L|f | = ℜe(signf )Lf in O, R[|f |, γ + |f |] = ℜe(signγ -f )R[f, γ + f ] on Σ -. From (
Doblin condition. We suppose here that O is bounded, K ∈ L ∞ (O), R * Σ 1 = R Σ 1 = 1,
and k(y, y * ) ≥ k 0 > 0. We aim at establishing the Doblin condition

S L (T )f 0 ≥ κ f 0 , 1 ,
which is (6.2) with ψ 0 = 1 and g 0 = κ1. From (8.64) we have

d dt O f dy = O f Kdy ≥ -K ∞ O f dy and so O f (t, y) dy ≥ e -K ∞t O f 0 (y) dy.
Now we define, for ϕ 0 ∈ C 1 c (O), ϕ 0 ≥ 0, ϕ 0 = 1, the solution ϕ to the equation

∂ t ϕ + div(aϕ) = 0, γ + ϕ = R * Σ [γ -ϕ].
d dt O f ϕ = O K [f ]ϕ - O Kf ϕ ≥ k 0 O f -K ∞ O f ϕ.
We deduce from Grönwall's inequality that, for any fixed T > 0,

O f (T, y)ϕ 0 (y) dy ≥ e -K ∞t O f 0 (y)ϕ 0 (y) dy + k 0 T 0 e -(T -t) K ∞ O f (t, y) dydt ≥ k 0 T e -T K ∞ O f 0 (y) dy =: κ f 0 , 1 .
This is nothing but the Doblin condition (6.2) since ϕ 0 is any non-negative function in

C 1 c (O) with ϕ 0 = 1.
In order to verify (6.3) in a quantitative way, we suppose that the conditions (8.35), (8.59), (8.60), (8.61), (8.62) and (8.63) are verified with the weight function m = 1 and the exponent p = 1. Note that in this case we have ̟ = K ≥ 0. The first condition in (8.35) then imposes that K ∈ L ∞ (O), and (8.102) reads

η 0 > K L ∞ C 0 + M R C 0 + a L ∞ (B0) C 1 -b L ∞ (B0) C 0 .
We also assume that for some k 1 > k 0 > 0, Theorem 8.37. We assume that O is bounded and that the conditions (8.35), (8.59), (8.60), (8.61), (8.62) and (8.63) are satisfied by a, b, K and R for the weight function m = 1 and the exponent p = 1. We assume further that K satisfies the strong positivity conditions (8.94) together with (8.95) and the first compactness property formulated in (8.96). We finally assume that (8.102), (8.104) and (8.105) are satisfied. Then the exponential convergence in (E3 1 ) holds in L 1 with constructive constants C and ω.

(8.104) R * Σ 1 = R Σ 1 = 1,
Proof of Theorem 8.37. We work in X = L 1 (O) and we normalize φ 1 by φ 1 L ∞ = 1. We have proved above that (6.2) holds true with ψ 0 = 1 and g 0 = κ1 for some explicit κ > 0, recalling that the assumption that K ∈ L ∞ is nothing but the first condition in (8.35) when m = 1 and p = 1 since b = K +diva. Due to the normalization φ 1 L ∞ = 1, the condition (6.4) holds with R 0 = 1. It only remains to check the validity of (6.3) in order to be able to apply Theorem 6.2. Since we assume that the conditions (8.35), (8.59), (8.60), (8.61), (8.62) and (8.63) are satisfied for the weight function m = 1 and the exponent p = 1, we have that R B (λ 1 ) :

L 1 → L 1 with R B (λ 1 ) B(L 1 ) ≤ 1 κ0-κB . This yields by duality that R * B (λ 1 ) : L ∞ → L ∞ with R * B (λ 1 ) B(L ∞ ) ≤ 1 κ0-κB .
Since k is bounded by the constant k 1 , we have on the other hand that

A * = K * : L 1 → L ∞ with A * B(L 1 ,L ∞ ) ≤ k 1 . We thus get 1 = φ 1 L ∞ ≤ k 1 κ 0 -κ B φ 1 L 1 ,
which yields (6.3) with r 0 = κ(κ 0κ B )/k 1 , and the proof is complete.

8.6.

A word about the renewal equation. We look at the case O = (0, +∞) and a(y) = 1, which corresponds to the equation (8.106)

∂ t f + ∂ y f + Kf = 0
with the boundary condition

(8.107) (γ -f )(t, 0) = ∞ 0 r O (y * )f (t, y * )dy * .
This renewal age structured model is standard in structured population dynamics, and the Krein-Rutman theorem is well-known for it, see for instance [START_REF] Bansaye | Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions[END_REF][START_REF] Feller | On the integral equation of renewal theory[END_REF][START_REF] Greiner | A typical Perron-Frobenius theorem with applications to an age-dependent population equation[END_REF][START_REF] Gwiazda | Invariants and exponential rate of convergence to steady state in the renewal equation[END_REF][START_REF] Sharpe | A Problem in Age-Distribution[END_REF][START_REF] Webb | A semigroup proof of the Sharpe-Lotka theorem[END_REF]. The existence and uniqueness of (λ 1 , f 1 , φ 1 ) can even be obtained by explicit computations. However, it is not covered by the cases considered in Section 8.5 because K = 0 here.

The singularity of this transport equation lies in the fact that (H2) is only guaranteed by the boundary condition. To fall into our splitting framework, we may replace the boundary condition by a singular source term Af = (R O f )(0)δ 0 , where δ 0 is the Dirac mass at the origin, and write L = A + B with B the generator of the free transport equation with zero flux boundary condition. This forces working in a space of measures, as in [START_REF] Mischler | Weak and strong connectivity regimes for a general time elapsed neuron network model[END_REF][START_REF] Mischler | Relaxation in time elapsed neuron network models in the weak connectivity regime[END_REF]. We briefly present an alternative approach, which is more in the spirit of [START_REF] Bansaye | Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions[END_REF][START_REF] Gabriel | Measure solutions to the conservative renewal equation[END_REF] and which consists in working in the Lebesgue space L 1 , first to solve the dual problem in L ∞ = (L 1 ) ′ and next to use for instance Doblin's contraction to solve the primal problem.

We assume here that

0 ≤ K, r O ∈ L ∞ loc (0, ∞), (r O -αK) + ∈ L ∞ , (8.108)
lim y→∞ K(y) = +∞, lim inf y→+∞ r O (y) > 0, (8.109)
for some α ∈ (0, 1), and we verify the usual conditions for the direct or the dual problem. Condition (H1). Under assumption (8.108), the age structured equation (8.106)-(8.107) is wellposed in L 1 thanks to Proposition 8.23 and we may associate to it a positive semigroup S L in L 1 with growth bound ω(S L ) ≤ κ 1 := (r O -K) + L ∞ thanks to Corollary 8.25. We deduce that (H1) holds for the primal problem and thus also for the dual problem thanks to Lemma 2.2 and Lemma 2.3.

Condition (H2). The generator of the dual problem is

L * φ = ∂ y φ -K(y)φ + φ(0)r O (y) with domain D(L * ) ⊂ W 1,∞ loc (O).
From the second hypothesis in (8.109), there exist y 0 , η 0 ∈ (0, ∞) such that r O (y) ≥ η 0 for any y ≥ y 0 . We then define φ 0 (y) = 1 [0,y0) + η 0 (y 1y)1 [y0,y1) , y 1 := y 0 + 1/η 0 , and we compute

L * φ 0 = r O (y) -K ≥ -K -r O L ∞ (0,y0) on (0, y 0 ), L * φ 0 = r O (y) -η 0 -Kφ 0 ≥ -K L ∞ (y0,y1) φ 0 on (y 0 , y 1 ), L * φ 0 = 0 on (y 1 , ∞), so that in the three case L * φ 0 ≥ κ 0 φ 0 with κ 0 := -max( K -r O L ∞ (0,y0) , K L ∞ (y0,y1) ). Using Lemma 2.4-(i),
= {f ∈ L 1 (O); ∂ y f + Kf ∈ L 1 (O), f (0) = 0}, generates a contraction semigroup in L 1 m * (O)
, and thus a bounded semigroup in

L 1 (O) because m * , m -1 * ∈ L ∞ (O).
In other words, we have established that ω(S B ) = -∞. Now, we see that

R B * (λ) : L ∞ → D(B * ) ⊂ W 1,∞ loc ([0, ∞)) is bounded for any λ ∈ R and thus A * R B * (λ) : L ∞ → L ∞ is compact for any λ ∈ R.
We deduce from Lemma 2.8 and Remark 2.10, that L * satisfies (H3).

Using Lemma 2.8-(1), we conclude to the existence of (λ 1 , φ 1 ) solution to the dual eigenvalue problem. Now we turn to the existence, uniqueness, and exponential stability of f 1 ∈ L 1 , by verifying that Doblin's condition (6.2) is satisfied. Doblin condition. Denoting S t := S L (t), we have from the characteristics method

S t f (y) = f (y -t)e -t 0 K(y-s)ds 1 t<y + N (t -y)e -y 0 K(s)ds 1 t>y with N (t) = ∞ 0 r O (y * )S t f (y * )dy * .
Iterating this formula and using the positivity of S t we get that for any f ≥ 0

S t f (y) ≥ t-y 0 r O (y * )N (t -y -y * )e -y * 0 K(s)ds dy * e -y 0 K(s)ds 1 0<y<t ≥ t-y 0 r O (t -y -τ )N (τ )e -t-y-τ 0 K(s)ds dτ e -y 0 K(s)ds 1 0<y<t .
Choosing t 0 > 2y 0 so that r O (y) ≥ η 0 > 0 for all y ≥ t 0 /2, we obtain

S t0 f (y) ≥ η 0 e -t 0 0 K(s)ds t0/4 0 N (τ )dτ e -y 0 K(s)ds 1 0<y<t0/4 .
From the expression of N (t) we get by duality, using that r O ≥ η 0 1 (y0,∞) , that

S t0 f (y) ≥ η 2 0 e -2 t 0 0 K(s)ds ∞ 0 f (y * ) t0/4 0 S * τ 1 (y0,∞) (y * )dτ dy * 1 0<y<t0/4 . (8.110)
Applying S t1 to this inequality we deduce that for any t 1 > 0

S t0+t1 f (y) ≥ η 2 0 e -2 t 0 0 K(s)ds ∞ 0 f (y * ) t0/4 0 S * τ 1 (y0,∞) (y * )dτ dy * S t1 1 0<y<t0/4 ≥ η 2 0 e -2 t 0 0 K e -t 1 0 K ∞ 0 f (y * ) t0/4 0 S * τ 1 (y0,∞) (y * )dτ dy * 1 t1<y<t0/4+t1 .
On the other hand, replacing f by S t1 f in (8.110) we obtain

(8.111) S t0+t1 f (y) ≥ η 2 0 e -2 t 0 0 K(s)ds ∞ 0 f (y * ) t0/4 0 S * τ +t1 1 (y0,∞) (y * )dτ dy * 1 0<y<t0/4 .
The fact that S * t φ(y) ≥ φ(t + y)e -t 0 K(y+s)ds ensures that for t 1 > y 0

S * t1 1 (y0,∞) ≥ e -sup y∈[0,y 0 ] t 1 0 K(y+s)ds 1 [0,y0] .
All together, we have proved that for any t 0 > 4t 1 > 4y 0 we have

S t0+t1 f (y) ≥ c 0 ∞ 0 f (y * ) t0/4 0 S * τ 1(y * )dτ dy * 1 t1<y<t0/4
for some explicit constant c 0 and all f ≥ 0. This is Doeblin's condition (6.2) with T = t 0 + t 1 , and the functions ψ 0 = t0/4 0 S * τ 1 dτ and g 0 = c 0 1 (t1,t0/4) . We are now in position to prove the following result. Despite the numerous results about the renewal age-structured model, we are not aware of any previous result with a constructive rate of convergence under such general assumptions.

Proof of Theorem 8.38. The conditions (H1), (H2) and (H3) for L * ensure the existence of λ 1 ≥ κ 0 and φ 1 ∈ L ∞ , φ 1 > 0, that we normalize by φ 1 L ∞ = 1. If we can prove that the conditions (6.2), (6.3) and (6.4) are verified, then the conclusions (C1) and (E3 1 ) follow by applying Theorem 6.2. Indeed, the contraction argument in the proof of Theorem 6.2 does not require the existence of f 1 and it can even be used for deriving the existence and uniqueness of f 1 , see Remark 6.4. We have already proved (6.2) with the functions ψ 0 = t0/4 0 S * τ 1 dτ and g 0 = c 0 1 (t1,t0/4) . For proving (6.3), we start by recalling that φ 1 = R * B (λ 1 )A * φ 1 ∈ W 1,∞ loc due to the informations derived on R * B in (H3). Consequently, there exists y 1 > 0 such that φ 1 (y 1 ) > 1/2, and we deduce from φ

′ 1 ≤ (λ 1 + K)φ 1 that φ 1 (y) ≥ 1 2 e -y 1 y (λ1+K)
for all y ∈ (0, y 1 ). Choosing in the proof of the Doblin condition t 0 such that y 1 < t 0 /4, we obtain that

φ 1 , g 0 ≥ c 0 2 y1 y1/2
e -y 1 y (λ1+K) dy, which gives (6.3). For (6.4), we use that

φ 1 = e -λ1τ S * τ φ 1 ≤ e -λ1τ S * τ 1
for any τ > 0 to deduce that

φ 1 = 4 t 0 t0 0 e -λ1τ S * τ φ 1 dτ ≤ 4e |λ1|t0 t 0 ψ 0 .
Finally, we check that the condition (H5 ′ ) is verified, so that (C3) is valid by virtue of Theorem 5.18 and Remark 5.19. The condition (H5 ′ ) is actually a direct consequence of the fact that (8.111) is verified for any t 0 > 2y 0 and t 1 > 0 together with the estimate

S * τ +t1 1 (y0,∞) (y) ≥ e -t 1 0 K(y+s)ds > 0
for any t 1 > y 0 and τ > 0, and all y > 0.

The growth-fragmentation equation

In this section, we are interested in the growth-fragmentation equation with equal mitosis kernel (9.1)

∂ t f (t, x) + ∂ x a(x)f (t, x) + K(x)f (t, x) = 4K(2x)f (t, 2x)
and to its variant with an additional "growth speed" variable (9.2)

∂ t f (t, x, v) + v∂ x a(x)f (t, x, v) + K(x)f (t, x, v) = 4 2 1 K(2x)℘(v, v * )f (t, 2x, v * )dv * ,
with x > 0 and v ∈ [START_REF] Abdallah | Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system[END_REF][START_REF] Agoshkov | Problema sledov funktsiȋ iz prostranstv H 1 p (Ω × D) i issledovanie nekotorykh variatsionnykh kraevykh zadach perenosa[END_REF]. For both equations, we assume that the total fragmentation rate K is a continuous function defined on R + such that (9.3) ∃x 0 > 0, K = 0 on (0, 2x 0 ] and K > 0 on (2x 0 , ∞).

This condition ensures that no particle of size less than x 0 can be produced by division, and we thus consider the equations posed on the size space (x 0 , ∞) with zero flux boundary condition f (t, x 0 ) = 0 or f (t, x 0 , v) = 0. The growth rate a is supposed to be positive and globally Lipschitz on [x 0 , ∞), and we assume that (9.4) lim

x→∞ xK(x) a(x) = +∞.
For quantifying the positivity of the first eigenvalue, we also make the technical assumption that (9.5) ∃k > 0, lim x→∞ e x k K(x) = +∞. 9.1. The mitosis equation with mixing growth rate. We are interested here in the growthfragmentation equation (9.1) in the case where (9.6) ∃x 1 > x 0 , a(2x 1 ) = 2a(x 1 ).

As we will see below, this condition ensures some mixing property for the trajectories that guarantees the triviality of the boundary point spectrum.

We work in the space X = L 1 m with a weight m that can be

(9.7) either m(x) = x r , r > 1, or m(x) = exp η x x0 K a , 0 < η < 1.
Note that due to assumption (9.4), the weight exp η

x x0 K/a is always stronger than x r . Theorem 9.1. Suppose that (9.3), (9.4), (9.5) and (9.6) are satisfied. The first eigentriplet problem admits a unique solution (λ 1 , f 1 , φ 1 ) ∈ R×X + ×X ′ + with the normalization φ 1 = φ 1 , f 1 = 1, and this triplet additionally satisfies λ 1 > 0, f 1 > 0 and φ 1 > 0. Besides, there are some constructive constants C ≥ 1, ω > 0 such that

e -λ1t S L (t)f -φ 1 , f f 1 X ≤ Ce -ωt f -φ 1 , f f 1 X for any f ∈ X and t ≥ 0.
This result is contained in the recent paper [START_REF] Villemonais | A quasi-stationary approach to the long-term asymptotics of the growthfragmentation equation[END_REF]. The novelty here is that all the constants are obtained constructively, which is not clear in [START_REF] Villemonais | A quasi-stationary approach to the long-term asymptotics of the growthfragmentation equation[END_REF]. We also provide what seems to us to be a more direct and comprehensive proof. We also refer to [START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF][START_REF] Bernard | Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate[END_REF][START_REF] Cañizo | Spectral gap for the growth-fragmentation equation via Harris's theorem[END_REF][START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF] where the same result is obtained under stronger assumptions.

Before starting the proof of Theorem 9.1, let us briefly justify the relevance of the chosen weight functions m in (9.7). The dual operator associated to equation (9.1) is given by

L * φ(x) = a(x)φ ′ (x) -K(x)φ(x) + 2K(x)φ(x/2).
For m(x) = x r , r > 1, we can compute (9.8)

L * m(x) = r a(x) x -(1 -2 1-r )K(x) m(x),
and for m(x) = exp η x x0 K/a , 0 < η < 1, (9.9)

L * m(x) = 2 exp -η x x/2 K a -(1 -η) K(x)m(x).
Assumption (9.3) then ensures that L * m ∼ -ξKm as x → +∞, with ξ = 1 -2 1-r > 0 in the first case and ξ = 1η > 0 in the second case. In both cases, we deduce that (9.10)

L * m ≤ κ m + M 1 (x0,R) m
for any κ ≥ 0, by choosing M > 0 and R > x 0 large enough, and this type of Lyapunov inequality is pivotal in our analysis.

Condition (H1). Equation (9.1) is a particular case of equation (8.58) with

G = g = R = 0, b = K + diva and K [g](x) = 4K(2x)g(2x)
. We may then use Proposition 8.23-(1) to infer the well-posedness of equation (9.1) in X = L 1 m (x 0 , ∞), provided that the conditions (8.35) and (8.61) are met, with 0 < α K < 1, which is nothing but (8.63) when R ≡ 0. To do so, we define the function

̟ := K -a m ′ m ,
which corresponds to ̟ 1 in (8.34). When m(x) = x r with r > 1, we have

̟(x) = K(x) -r a(x) x ,
and for m(x) = exp η x x0 K/a with 0 < η < 1, we have

̟(x) = (1 -η)K(x).
In both cases, the fact that a ∈ Lip ensures that ̟ q := ̟ + (1 -1/q)a ′ enjoys (̟ q ) -∈ L ∞ for any q ∈ [1, ∞). On the other hand, (9.4) guarantees that K ̟ + and a/x ̟ + , and finally (8.35) is verified. The condition (8.61) is equivalent to the Lyapunov type condition (9.11)

K * [m] ≤ (α K ̟ + + M K )m,
where K * [m](x) = 2K(x)m(x/2). For m(x) = x r with r > 1, we compute

K * [m]/m = 2 1-r K,
and for m(x) = exp η

x x0 K/a with 0 < η < 1, K * [m](x) m(x) = 2 exp -η x x/2 K a K(x).
Using (9.4), we obtain that (9.11) is satisfied, for any α K ∈ (2 1-r , 1) in the first case, and for any α K ∈ (0, 1) in the second case, by choosing M K large enough. We can then apply Proposition 8.23-(1) for associating to equation (9.1) a strongly continuous semigroup S in X = L 1 m (x 0 , ∞), and (H1) then follows from Lemma 2.2-(i). Moreover, we readily have that κ 1 ≤ κ + M for any couple (κ, M ) such that (9.10) is verified.

Condition (H2). We aim at verifying (H2) for some κ 0 > 0. Recalling assumption (9.5), we pick up ℓ > k and we consider the function

φ 0 (x) = xe -x ℓ /n
with n large enough to be chosen later. We compute

L * φ 0 (x) φ 0 (x) = a(x) x 1 - ℓ n x ℓ + K(x) e 1-2 -ℓ n x ℓ -1 .
Choosing R > x 0 such that xK(x)/a(x) ≥ 2ℓ 1-2 -ℓ and K(x) ≥ e -x k for all x ≥ R, we get that

L * φ 0 (x) φ 0 (x) ≥ a(x) x + K(x) e 1-2 -ℓ n x ℓ -1 - 1 -2 -ℓ 2n x ℓ ≥ e -x k e 1-2 -ℓ n x ℓ -e 1-2 -ℓ 2n x ℓ ≥ e 1-2 -ℓ 2n x ℓ -x k e 1-2 -ℓ 2n R ℓ -1 on [R, ∞). Choosing then n ≥ ℓ 2 R ℓ , we have L * φ 0 (x) φ 0 (x) ≥ a(x) 2x
on (x 0 , R). Gathering the two above estimates, we deduce the existence of an explicit κ 0 > 0 such that L * φ 0 ≥ κ 0 φ 0 . We conclude by invoking Lemma 2.4-(i).

Condition (H3). We consider the weight function m(x) = x r for some r > 1 or m(x) = exp η

x x0 K/a with 0 < η < 1 and we define the stronger weight function m 1 (x) = exp η 1 x x0 K/a for some η 1 ∈ (η, 1). We fix κ B ∈ [0, κ 0 ), M > 0, and R > x 0 such that (9.

B(L 1 m 1 ) ≤ 1 κ -κ B .
The operator A maps L 1 m into L 1 m1 with

A B(L 1 m ,L 1 m 1 ) ≤ M m 1 (R) m(x 0 ) .
Besides, due to the derivative part ∂ x (a •) in the operator B, we also have that R B (κ) maps L 1 m1 into W 1,1 loc . Finally, we have R B (κ)A : L 1 m → L 1 m1 ∩ W 1,1 loc , and thus R B (κ)A ∈ K (L 1 m ), for any κ ≥ κ 0 > κ B . We deduce from Lemma 2.8-(2) that the condition (H3) holds for both the primal and the dual problems.

Proof of the existence part of Theorem 9.1. We deduce from Theorem 2.21 that the conclusion (C1) about the existence of a solution (λ 1 , f 1 , φ 1 ) ∈ R × X + × X ′ + to the first eigentriplet problem holds true. Moreover, we have λ 1 ≥ κ 0 > 0 and f 1 ∈ W 1,1 loc ∩L 1 m with m(x) = exp η x x0 K/a for any η ∈ (0, 1). For deriving similar additional estimates on φ 1 , we can check directly that the condition (H3) holds for the dual operator L * . Condition (H3) for L * . We consider the weight function m(x) = x r for some r > 1 or m(x) = exp η x x0 K/a with 0 < η < 1 and we define the weaker weight function m 0 (x) = x r0 for some r 0 ∈ (1, r). We fix κ B ∈ [0, κ 0 ), M > 0, and R > x 0 such that (9.10) is verified by m 0 . Using again the splitting L = A + B with Af = M 1 (x0,R) f , (9.10) means that B * m 0 ≤ κ B m 0 and this ensures that for any κ > κ B the operator κ -B * is invertible in L ∞ m0 , with positive inverse, and

(κ -B * ) -1 B(L ∞ m -1 0 ) ≤ 1 κ -κ B .
Because of the derivative part of B * , we also have that

R B * (κ) : L ∞ m -1 0 → W 1,∞ loc . Besides, the operator A * = A maps L ∞ m -1 into L ∞ m -1 0 with A * B(L ∞ m -1 ,L ∞ m -1 0 ) ≤ M m(R) m 0 (x 0 ) . Finally we have that R B * (κ)A : L ∞ m -1 → L ∞ m -1 0 ∩ W 1,∞ loc . Consequently φ 1 ∈ L ∞ m -1 0 ∩ W 1,∞ loc and (9.12) φ 1 L ∞ m -1 0 = (λ 1 -B * ) -1 A * φ 1 L ∞ m -1 0 ≤ m(R) m 0 (x 0 ) M κ 0 -κ B φ 1 L ∞ m -1 .
We also easily deduce quantitative estimates of φ 1 in W 1,∞ loc from the identity

φ ′ 1 (x) = 1 a(x) λ 1 φ 1 (x) + K(x)φ 1 (x) -2K(x)φ 1 (x/2) .
Condition (H4). The operator L satisfies the strong maximum principle. Let λ ∈ R and f ∈ X + ∩ D(L) \ {0} such that (λ -L)f ≥ 0, i.e.

λf (x) + (af ) ′ (x) + K(x)f (x) ≥ 4K(2x)f (2x) ∀x > x 0 .
Denoting by Λ λ a function such that Λ ′ λ (x) = λ+K(x) a(x) , we get that (9.13) a(x)f (x) ≥ 4

x x0 e Λ λ (y)-Λ λ (x) K(2y)f (2y) dy.

Since K(2y) > 0 for all y > x 0 , f ∈ X + \ {0}, and a(x) > 0 for all x > x 0 , we deduce from (9.13) that the set {x > x 0 , f (x) > 0} is an interval of the form (x, +∞). Using again (9.13) we remark that we must have x = max(x 0 , x/2), which enforces x = x 0 and finally f > 0.

Proof of the uniqueness and positivity part of Theorem 9.1. We deduce from Theorem 4.13 the validity of the conclusion (C2) about existence, uniqueness and positivity of a solution (λ 1 , f 1 , φ 1 ) to the first eigentriplet problem.

For deriving the exponential stability, we start by verifying a quantified irreducibility and aperiodicity condition on S, given in the next lemma, which then allows us to prove that the Harris condition (6.8) is met.

Lemma 9.2. Assume that (9.6) is satisfied. Then for all ε > 0, R 1 > x 0 , and R 2 > x 0 + ε, there exists T 1 > 0 such that for any T > T 1 , there exists c T > 0 such that

S * T φ ≥ c T 1 (x0,R1) R2 x0+ε φ dx, ∀ φ ≥ 0.
Proof of Lemma 9.2. Throughout the proof we denote by c t any positive constant that depend only on t. It is proved in [START_REF] Villemonais | A quasi-stationary approach to the long-term asymptotics of the growthfragmentation equation[END_REF]Prop. 5] the existence of (x 2 , x 3 ) ⊂ (x 0 , ∞) such that for all R 1 > x 0 there exists T 0 > 0 such that for any T > T 0 and any φ ≥ 0

S * t φ(x) = φ(X t (x))e -t 0 K(Xs(x))ds + 2 t 0 K(X t-s (x))S * s φ(X t-s (x)/2)e -t-s 0 K(X s ′ (x))ds ′ ds,
where X t (x) is the solution to the characteristic equation (9.16) Ẋt (x) = a(X t (x)) with X 0 (x) = x.

Applying (9.14) to S * t φ, that we bound from below by the first in Duhamel's formula (9.15), we obtain

S * T +t φ ≥ c T 1 (x0,R1) x3 x2 φ(X t (x))e -t 0 K(Xs(x))ds dx ≥ c T c t 1 (x0,R1) Xt(x3) Xt(x2)
φ(y)dy.

Choosing t 0 such that X t0 (x 2 ) = x 3 , we get that for all T > T 0 + t 0

S * T φ ≥ c T 1 (x0,R1) Xt 0 (x3) x2 φ(x)dx.
Iterating this argument and using the strict positivity of a we get for any R 2 > x 2 the existence of a time t 1 such that for all T > T 0 + t 1 (9.17)

S * T φ ≥ c T 1 (x0,R1) R2 x2 φ(x)dx.
For decreasing the lower bound of the integral from x 2 to x 0 + ε, we iterate once Duhamel's formula (9.15) to get

S * t φ(x) ≥ 2 t 0 K(X t-s (x))φ(X s (X t-s (x)/2
))e -t-s 0 K(X s ′ (x))ds ′ -s 0 K(X s ′ (x))ds ′ ds and then, using (9.17),

S * T +t φ ≥ c t c T 1 (x0,R1) t 0 R2 x2 K(X t-s (x))φ(X s (X t-s (x)/2))dx ds.
We can assume that x 2 > 2x 0 and R 2 > 2x 2 . The fact that x 2 > 2x 0 ensures, due to assumption (9.3), that K is bounded from below by a positive constant on [x 2 , X t (R 2 )]. We thus deduce, by using of a change of variables, that for any t > 0

S * T +t φ ≥ c t c T 1 (x0,R1) R2/2 Xt(Xt(x2)/2)
φ(y)dy.

Since X t (x) → x when t → 0, we deduce for all ζ > 0 the existence of t > 0 such that

S * T +t φ ≥ c t c T 1 (x0,R1) R2/2 x2/2+ζ
φ(y)dy.

Since R 2 > 2x 2 , we deduce by combining the above inequality with (9.17) that for all T > T 0 +t 1 +t

S * T φ ≥ c T 1 (x0,R1) R2 x2/2+ζ φ(x)dx.
Let us take ζ = x 0 . Since the sequence (u n ) defined by u 0 = x 2 and u n+1 = u n /2 + x 0 converges to 2x 0 , we obtain by an iteration argument the existence of a time t 2 such that for all T > T 0 + t 2

S * T φ ≥ c T 1 (x0,R1) R2 2x0+ε φ(x)dx.
Using a last time the argument with ζ = ε/2 yields the desired result.

We now prove another positivity result which allows making the time T independent of R 1 in Lemma 9.2.

Lemma 9.3. Let R 1 > 2x 0 . Then there exists t 0 > 0 such that for any R > R 1 we have

S * t0 1 (x0,R1) ≥ c R 1 (x0,R) for some c R > 0.
Proof of Lemma 9.3. Since a is Lipschitz continuous, we can find t 0 > 0 small enough so that X t0 (x) ≤ αx for all x > x 0 , with α > 1 to be determined later. Then for any t ∈ (0, t 0 ] and any x ∈ (x 0 , R1 α ), we have by using the first term in (9.15)

S * t 1 (x0,R1) (x) ≥ c t0 1 (x0,R1) (X t (x)) = c t0 > 0.
Iterating once (9.15) and keeping only the second term, we get that for any t ∈ (0, t 0 ] and any x ∈ (2x 0 , 2R1 α 2 )

S * t 1 (x0,R1) (x) ≥ c t0 t 0 1 (x0,R1) (X s (X t-s (x)/2)) ds = c t0 t.
Choosing α > 1 such that R1 α > 2x 0 and 2R1 α 2 > R 1 , we deduce that for any t ∈ (0, t 0 ] there exists c t > 0 such that

S * t 1 (x0,R1) ≥ c t 1 (x0,2α -2 R1) . Dividing [0, t 0 ] into n sub-intervals [ k n t 0 , k+1 n t 0 ], 0 ≤ k ≤ n -1
, and iterating the above inequality with t = t 0 /n, we deduce for all integer n ≥ 1 the existence of c n > 0 such that

S * t0 1 (x0,R1) ≥ c n 1 (x0,(2α -2 ) n R1
) and the proof is complete since 2α -2 > 1.

With Lemmas 9.2 and 9.3, we are now in position to prove the convergence result in Theorem 9.1.

Proof of the exponential stability part of Theorem 9.1. We apply Theorem 6.3. We start by proving that (6.9) is verified, in a quantitative way, for the function g 0 = 1 (x0+ε,R2) with a suitable choice of R 2 and ε. Choosing r 0 ∈ (1, r) if m(x) = x r or any r 0 > 1 is m(x) = exp η x x0 K/a and defining m 0 (x) = x r0 we have from (9.12), because of the normalization

φ 1 L ∞ m -1 = 1, φ 1 L ∞ m -1 0 ≤ C 0 for some explicit constant C 0 > 0. Defining R 2 := inf{R > 0; m 0 (x)/m(x) ≤ 1/2C 0 , ∀ x > R},
we have

1 = φ 1 L ∞ m -1 = sup (x0,∞) φ 1 m = sup (x0,R2) φ 1 m , because sup (R2,∞) φ 1 m ≤ sup (R2,∞) φ 1 m 0 m 0 m ≤ C 0 1 2C 0 < 1.
Together with the fact that φ ′ 1 ∈ L ∞ loc , with a quantitative estimate on φ ′ 1 L ∞ (x0,R2) , we see that φ 1 has some quantifiable mass on (x 0 + ε, R 2 ) for ε > 0 small enough, which exactly means that g 0 , φ 1 is quantified from below. Now we prove that the Harris condition (6.8) is verified. Choosing R 1 > 2x 0 and combining Lemma 9.2 and Lemma 9.3, we have for any ε > 0 and R 2 > x 0 + ε the existence of T > 0 such that for any R > R 1 (9.18)

S * T φ ≥ c R 1 (x0,R) R2 x0+ε
φ dx, ∀ φ ≥ 0.

Defining g 0 = 1 (x0+ε,R2) , we deduce by duality that for all f ≥ 0,

S T f ≥ c R f, 1 (x0,R) g 0 .
Let us now consider A > 0 and f ∈ X + such that f ≤ A[f ] φ1 . Since m 0 (x)/m(x) → 0 as x → +∞ and φ 1 /m 0 ∞ ≤ C 0 , we have

[f ] φ1 = R x0 f φ 1 m m + ∞ R f m φ 1 m 0 m 0 m ≤ f, 1 (x0,R) sup (x0,R) m + f C 0 sup (R,∞) m 0 m ≤ f, 1 (x0,R) m(R) + 1 2 [f ] φ1 ,
by choosing R large enough. We deduce that S T f ≥ cR 2m(R) [f ] φ1 g 0 , which is equivalent to (6.8) where we recall the definition S t := S t e -λ1t . Finally, we prove the Lyapunov condition (6.7). On the one hand, we get from (9.10) that

d dt S * t m = S * t (L * -λ 1 )m ≤ (κ B -λ 1 ) S * t m + M S * t (1 (x0,R) m).
On the other hand, arguing as in (6.6), we infer from (9.18) that

φ 1 = e -λ1T S * T φ 1 ≥ c R e -λ1T g 0 , φ 1 1 (x0,R) . Combining both we deduce that d dt S * t m ≤ (κ B -λ 1 ) S * t m + M φ 1 with M = m(R) cR e λ 1 T
g0,φ1 , and Grönwall's inequality then yields S * t m ≤ e (κB-λ1)t m + M te (κB-λ1)t φ 1 . This guarantees that (6.7) is verified with γ L = e (κB-λ1)T ∈ (0, 1) and K = M T . We have proved that the conditions (6.13), (6.7) and (6.9) are verified. The conclusion of the proof then follows from Theorem 6.3. 9.2. The mitosis equation with non-mixing growth rate. In this section, we investigate the case when the mixing condition (9.6) is not verified. In other words, we place ourselves under the singular condition that (9.19) ∀x > x 0 , a(2x) = 2a(x).

In this case, we still have the existence of a unique eigen-triplet (λ 1 , f 1 , φ 1 ) but the boundary point spectrum is not reduced to λ 1 . As a consequence, the long time behavior of the semigroup does not stabilizes along f 1 but it exhibits periodic oscillations.

Theorem 9.4. Suppose that (9.3), (9.4), (9.5) and (9.19) are satisfied. The first eigentriplet problem admits a unique solution (λ 1 , f 1 , φ 1 ) ∈ R × X + × X ′ + with the normalization φ 1 = φ 1 , f 1 = 1, and this triplet additionally satisfies λ 1 > 0, f 1 > 0 and φ 1 > 0. Besides, Σ + P (L) = {λ 1 + ikα, k ∈ Z} for some quantifiable α > 0, there exists a family (g k , ψ k ) k∈Z of corresponding primal and dual eigenvectors that verifies ψ k , g ℓ = δ kℓ , and for all f ∈ L 1 φ1 , we have the convergence

e -λ1t S L (t)(f -Πf ) L 1 φ 1 → 0 as t → +∞,
where Πf = lim This new result complements the scarce literature on the long time behavior of equation (9.1) in the singular case (9.19) which, to the best of our knowledge, is limited to the references [START_REF] Bernard | Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts[END_REF][START_REF] Gabriel | Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation[END_REF][START_REF] Greiner | Growth of cell populations via one-parameter semigroups of positive operators[END_REF].

We will actually prove that the convergence in Theorem 9.4 also holds in other spaces, and this will be the occasion to compare our method and results to the three above mentioned papers.

The proof of the conclusion (C2) in Section 9.1 does not use the mixing assumption (9.6). It thus also proves the existence, uniqueness and strict positivity of eigentriplet (λ 1 , f 1 , φ 1 ) under the assumptions of Theorem 9.4, as well as the fact that equation (9.1) is associated with a semigroup S in X. For proving the long time convergence result, we start by verifying that this semigroup extends to other relevant Banach spaces.

Well-posedness in entropic L p and M 1 spaces. The dual eigenfunction φ 1 satisfies by definition L * φ 1 = λ 1 φ 1 and the rescaled semigroup S t = S t e -λ1t is thus a contraction for the norm of L 1 φ1 . In particular S t is a bounded operator for this norm and, since L 1 m is dense in L 1 φ1 , we can uniquely extend the semigroup S into a strongly continuous semigroup in L 1 φ1 . Similarly, due to the weak- * density of L 1 φ1 into M 1 φ1 , this semigroup extends uniquely into a weakly- * continuous semigroup in M 1 φ1 . We still denote by S these extensions. The General Relative Entropy principle, see [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF][START_REF] Bernard | Loskot-Rudnicki's Inequality and General Relative Entropy Inequality for Cauchy Problems Preserving Positivity[END_REF], ensures that the weighted L p sub-spaces of L 1 φ1 defined by

X p := L p f 1-p 1 φ1 (x 0 , ∞) for p ∈ [1, ∞) and X ∞ := L ∞ f -1 1 (x 0 , ∞)
are invariant under the semigroup S and the restriction to these spaces is a contraction. Besides, Jensen's inequality yields that it is a decreasing sequence for the inclusion

p > q =⇒ X p ⊃ X q .
Since X ∞ ⊂ X p is dense, we can infer the strong continuity of S in X p from the strong continuity in X 1 by writing for any f ∈ X ∞

S t f -f p Xp ≤ S t f -f p-1 X∞ S t f -f X1 ≤ 2 p-1 f p-1 X∞ S t f -f X1 → 0, as t → 0.
Long-time convergence in M 1 φ1 . We start by giving some useful properties of the dual semigroup

S * in X ′ = L ∞ m -1 . Splitting L * as L * = A * 0 + B * 0 with A * 0 φ = K * [φ], so that B * 0 φ = aφ ′ -Kφ, the Duhamel formula S * L = S * B0 + S * B0 A 0 * S * L
ensures that φ(t, x) := S * t φ(x) is a fixed point of the operator Γ defined by Γϕ(t, x) := S * B0 (t)φ(x) + S * B0 A 0 * ϕ(•, x) (t) (9.20) = φ(X t (x))e -t 0 K(Xs(x))ds + 2 t 0 K(X t-s (x))ϕ(s, X t-s (x)/2)e -t-s 0 K(X s ′ (x))ds ′ ds,

where we recall that X t (x) is the solution to the characteristic equation (9.16). It turns out that Γ has a unique fixed point in L ∞ loc ([0, ∞) × (x 0 , ∞)), and that this fixed point also lies in any closed subset of L ∞ loc ([0, ∞) × (x 0 , ∞)) which is left invariant by Γ. This property is proved in [START_REF] Gabriel | Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation[END_REF] or in [START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF]Sec. 6.3], by building φ thanks to the Banach-Picard fixed point theorem. It has very useful consequences, as for instance the fact that if φ ∈ C(x 0 , ∞), then φ ∈ C([0, ∞) × (x 0 , ∞)). In particular, this implies that C(x 0 , ∞) ∩ L ∞ m -1 is invariant under the semigroup S * . Since C(x 0 , ∞) ∩ L ∞ m -1 is a dense subspace of C 0,φ1 (x 0 , ∞), this ensures that C 0,φ1 is invariant under S * and that the duality relation S t f, φ = f, S * t φ is valid for any f ∈ M 1 φ1 and φ ∈ C 0,φ1 . The proof of the next result crucially relies on another application of the fact that the fixed point of Γ belongs to any closed invariant subset. Proposition 9.5. Suppose that (9.3), (9.4), (9.5) and (9.19) are satisfied. Then Σ + P (L) = {λ 1 + ikα, k ∈ Z} for some α > 0, there exists a family (g k , ψ k ) k∈Z of corresponding primal and dual eigenvectors that verifies ψ k , g ℓ = δ kℓ , and for all f ∈ M 1 φ1 we have the convergence

(9.21) S t f -S t Πf → 0, as t → ∞, Πf := lim n→∞ 1 n n ℓ=0 ℓ k=-ℓ ψ k , f g k ,
both convergences having to be understood in the sense of the weak- * topology.

Note that we did not specify the space in which we define the boundary point spectrum Σ + P (L) in Proposition 9.5. It is because this set is the same in all the Banach lattices we consider. Indeed, any g ∈ M 1 φ1 such that Lg = λg for some λ ∈ C with ℜe(λ) = λ 1 satisfies |g| ∈ Span(f 1 ), so that g ∈ X = L 1 m for any weight m as in (9.7).

Proof of Proposition 9.5.

Step 1. The rescaled semigroup S is a contraction semigroup in M 1 φ1 = (C 0,φ1 ) ′ . This ensures in particular that for all f ∈ M 1 φ1 the trajectory ( S t f ) t≥0 is bounded in M 1 φ1 . We can thus use Theorem 5.23-(2) to infer the non-triviality of the boundary spectrum, by proving that the conclusion cannot hold, see . We start from the fact that for any φ ∈ C 0,φ1 (x 0 , ∞), the solution S * φ to the dual mitosis equation is the unique fixed point of Γ defined in (9.20), and that it belongs to any closed invariant subset of C([0, ∞) × (x 0 , ∞)). For y > x 0 we define the set E y = {x > x 0 , x = 2 k y for some k ∈ Z} and we consider a function φ such that φ(x) = 0 if x ∈ E y and φ(x) > 0 if x ∈ E y . Then, since (9.19) ensures that X t (2x) = 2X t (x) for all t ≥ 0, the set

ϕ ∈ C([0, ∞) × (x 0 , ∞)), ϕ(t, x) = 0 if X t (x) ∈ E y and ϕ(t, x) > 0 if X t (x) ∈ E y
is invariant under Γ. Consequently, the unique fixed point S * t φ belongs to this set, and we deduce that S * t φ(x) = 0 if and only if X t (x) ∈ E y . In other words, by duality, supp(S t δ x ) ⊂ E y if and only if X t (x) ∈ E y , and in particular supp(S t δ x ) ⊂ E Xt(x) for all x > x 0 and t ≥ 0. This prevents the convergence of S t δ x toward δ x , φ 1 f 1 and we infer from (the negation of) Theorem 5.23-(2) that the boundary point spectrum cannot be trivial.

Step 2. We next formulate the following simple but fundamental observation. If (λ, f ) is a solution to the eigenvalue problem in X = M 1 m , then f ∈ D(L) ⊂ BV loc ⊂ L 1 loc and it is a solution to the eigenvalue problem in

X = L 1 m . Symmetrically, if (λ, φ) is a solution to the eigenvalue problem in Y = L ∞ m -1 1 , then φ ∈ D(L * ) ⊂ Lip loc ∩ L ∞ m -1 2 , m 1 (x)/m 2 (x) → 0 as x → ∞,
and it is a solution to the eigenvalue problem in Y = C 0,m1 . In other words, the point spectrum and the associated eigenelements are the same in the two frameworks (L 1 m , L ∞ m -1 ) and (M 1 m , C 0,m ). Now, as a consequence of this observation and Step 1, we know that the boundary point spectrum Σ + P (L) is not trivial. Because we have proved that (κ -B) -1 A is compact in L 1 m , κ < λ 1 , we may apply Theorem 5.7 and we obtain that Σ + P (L) = {λ 1 } + iαZ for some α > 0, and each eigenvalue is algebraically simple. Using finally Theorem 5.25 in the situation (2), we get the weak- * convergence (9.21). This result is proved by means of entropy techniques in [START_REF] Gabriel | Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation[END_REF] for a linear growth rate a(x) = x, by taking advantage of the explicit formulation of the eigenvectors g k and ψ k in terms of f 1 and φ 1 in that case. Here we extend it to any a satisfying a(2x) = 2a(x). Note that arguing similarly as in [START_REF] Gabriel | Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation[END_REF], the convergence (9.21) may be strengthened into an exponential strong convergence in M 1 m for m(x) = x r , r > 1, or m(x) = exp η x x0 K/a , 0 < η < 1, meaning that there is a spectral gap between Σ + (L) and the rest of the spectrum in these spaces.

Long-time convergence in X p . We prove the following result, the case p = 1 of which corresponds to the convergence result of Theorem 9.4. Proposition 9.6. Under the same assumptions as in Proposition 9.5, the convergence (9.21) holds for the strong topology in X p , 1 ≤ p < ∞ for all f ∈ X p , and the convergence of the Fejér sum in the definition of the projector Π is also for this topology.

Proof. The case p = 1 is an immediate consequence of Theorem 5.25, case (4). The proof in the case p > 1 is a direct adaptation of the case p = 1. We aim at verifying that the trajectories ( S t f ) t≥0 are relatively compact in X p . We have already seen that X ∞ ⊂ X p is dense. Besides, the domain D(L) of the generator Lλ 1 of S in X p is also dense in X p , so that it suffices to check the relative compactness of (

S t f ) t≥0 for f ∈ X ∞ ∩ D(L). For f in X ∞ ∩ D(L) the bounds S t f Xp ≤ f Xp , L S t f Xp = S t Lf Xp ≤ Lf Xp and S t f X∞ ≤ f X∞
yield the relative compactness of ( S t f ) t≥0 , the second bound guaranteeing uniform W 1,1 loc estimates. We can thus apply the case (1) of Theorem 5.25 to deduce the convergence (9.21) in X p for the strong topology. Propositon 9.6 extends the result of [START_REF] Bernard | Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts[END_REF] where it is proved in the case p = 2 for a(x) = x by taking advantage of the Hilbert structure of X 2 and of the explicit formulation of the eigenvectors g k and ψ k in terms of f 1 and φ 1 . In this Hilbert setting it is proved that the Fourier series n k=-n f, ψ k g k converges as n goes to infinity, and Πf is then given by the limit.

About the value of α. For ensuring that the boundary spectrum is discrete, we have used a compactness argument. The period 2π/α of the periodic semigroup SΠ is thus not quantified. It is expected to be equal to the time needed for a particle to double its size by following the flow of a, namely

(9.22) 2π α = 2x x dt a(t) ,
which is independent of the choice of x ≥ x 0 due to the condition a(2t) = 2a(t). This is known to be true in the case of a linear growth rate a(x) = x, see [START_REF] Diekmann | On the stability of the cell size distribution[END_REF] or [START_REF] Bernard | Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts[END_REF], and also for a general when the size domain is (x 0 , 4x 0 ), see [START_REF] Greiner | Growth of cell populations via one-parameter semigroups of positive operators[END_REF], where explicit computations can be carried out. In the general case, we have not been able to prove (9.22). Yet the fact that for any x > x 0 the support of S t δ x is a subset of E Xt(x) guarantees that the period cannot be too small as shown now.

Proposition 9.7. We have the estimate

(9.23) 2π α ≥ ℓ a := 2x x dt a(t) .
Proof of Proposition 9.7. Let x > x 0 such that ψ 1 (x) = 0 (actually any x > x 0 is suitable). We have S t δ x -S t Πδ x → 0, as t → +∞, and supp S t δ x ⊂ E Xt(x) , so supp S t Πδ x ⊂ E Xt(x) since S t Πδ x is periodic. Besides,

S t Πδ x = lim n→∞ 1 n n ℓ=0 ℓ k=-ℓ ψ k (x)e iαkt g k
and, since ψ 1 (x) = 0, the period of this periodic function of time is 2π α . But since supp S t Πδ x ⊂ E Xt(x) and the period of the set E Xt(x) is ℓ a , we have proved that (9.23) holds.

On the other hand, we can use Theorem 6.5 for deriving a quantified lower bound on α, and thus an upper bound on the period. We work in the space X = L 1 m , recalling that Σ + P (L) is the same in this space and in M 1 φ1 . Proposition 9.8. There exists a constructive constant α 1 > 0 such that Σ(L) ∩ B(λ 1 , α 1 ) = {λ 1 }. In particular, α ≥ α 1 .

To prove this result, we check that the conditions (6.7), (6.9) and (6.13) are verified, and we invoke Theorem 6.5. We start with a lemma which, together with Lemma 9.3, will guarantee the validity of (6.13). Lemma 9.9. For all ε > 0, R 1 > x 0 , and R 2 > x 0 + ε, there exist T > 0 and c T > 0 such that

T 0 S * t φ dt ≥ c T 1 (x0,R1) R2 x0+ε φ dx, ∀ φ ≥ 0.
Proof of Lemma 9.9. Throughout the proof we denote by c t any positive constant that depend only on t. From the Duhamel formula (9.15), we get by positivity that for any φ ≥ 0

T 0 S * t φ(x)dt ≥ T 0 φ(X t (x))e -t 0 K(Xs(x))ds dt.
We deduce that for all x ∈ (x 0 , R 1 ) and for T 0 large enough so that X T0 (x 0 ) > R 2 , we have Iterating once Duhamel's formula and using that X t (x/2) = X t (x)/2 and (9.3) we get that for all t ≥ 0 and all y ∈ (2x 0 + ε, R 2 )

S * t φ(y) ≥ c t t 0 K(X t-s (y))φ(X s (X t-s (y)/2))ds ≥ c t φ(X t (y/2)).
This yields for all x ∈ (x 0 , R 1 )

T +t 0 S * s φ(x)ds ≥ T +t t S * s φ(x)ds = T 0 S * s S * t φ(x)ds ≥ c T 2R2 max(R1,2x0+ε) S * t φ(y)dy ≥ c T c t 2R2 max(R1,2x0+ε) φ(X t (y/2))dy = c T c t Xt(R2) max(Xt( R 1
2 ),Xt(x0+ ε 2 ))

φ(z)dz.

Choosing t > 0 small enough so that max(X t ( R1 2 ),

X t (x 0 + ε 2 )) ≤ max( R1+ε 2 , x 0 + ε), we get T1 0 S * s φ(x)ds ≥ c T1 R2 max( R 1 +ε 2 ,x0+ε)
φ(z)dz, for T 1 := T + t.

Iterating the argument we can build an increasing sequence of times T n such that

Tn 0 S * t φ(x)dt ≥ c Tn R2 max(un,x0+ε) φ(z)dz, ∀ n ≥ 0,
where (u n ) is defined by u 0 = R 1 and u n+1 = un+ε 2 . Since this sequence (u n ) converges to ε < x 0 + ε, we get the conclusion by taking n large enough.

We are now in position to prove Proposition 9.8.

Proof of Proposition 9.8. Arguing similarly as in the proof of Theorem 9.1 and using Lemma 9.9 instead of Lemma 9.2, we can prove that the conditions (6.7), (6.9) and (6.13) are verified. Applying Theorem 6.5 then gives the result. 9.3. The model with variability. In this last part, we consider the model with variability given by the equation (9.2). Compared to equation (9.1), the main consequence of introducing a variability in terms of the spectrum and the asymptotic behavior is that for equation (9.2) the boundary spectrum is trivial and the first eigenfunction is exponentially stable, no matter if a satisfies (9.6) or (9.19).

Additionally to the assumptions (9.3), (9.4) and (9.5), we ask that (9.24)

K(x) = O exp δ x/2 x0 K/a as x → +∞,
for some δ > 0. About the variability kernel ℘ we suppose that (9.25)

2 1 ℘(v, v * )dv = 1, ∀v * ∈ [1, 2], ℘ ∈ W 1,∞ ([1, 2] 2 ) and ℘ ≥ ℘ *
for some ℘ * > 0. We still work in the space X = L 1 m by considering the weight m = m(x) as function of (x, v) constant in v.

Theorem 9.10. Suppose that (9.3), (9.4), (9.5), (9.24) and (9.25) are satisfied. The first eigentriplet problem for equation (9.2) admits a unique solution (λ 1 , f 1 , φ 1 ) ∈ R × X + × X ′ + with the normalization φ 1 = φ 1 , f 1 = 1, and this triplet additionally satisfies λ 1 > 0, f 1 > 0 and φ 1 > 0. Besides, there are some constructive constants C ≥ 1, ω > 0 such that

e -λ1t S L (t)f -φ 1 , f f 1 X ≤ Ce -ωt f -φ 1 , f f 1 X
for any f ∈ X and t ≥ 0. Yet expected, this result was known only in the case of a discrete set of variabilities [START_REF] Cloez | Long-time behavior and Darwinian optimality for an asymmetric size-structured branching process[END_REF][START_REF] Rat | Growth-fragmentation model for a population presenting heterogeneityin growth rate: Malthus parameter and long-time behavior[END_REF]. Theorem 9.10 is thus new in the literature.

Because of the assumption (9.25), we easily see that the construction of the semigroup and the proof of the conditions (H1), (H2) and (H4) given in Section 9.1 for the model without variability readily extend to the model (9.2). We thus only have to verify (H3) and some Harris type condition.

Condition (H3). Let δ ∈ (0, 1) such that (9.24) is verified, and consider the weight function m(x) = x r with r > 1 or m(x) = exp η x x0 K/a with η ∈ (0, 1δ). We also use the two other weights

m 1 (x) = exp η 1 x x0 K/a , m 2 (x) = exp η 2 x x0 K/a
for some η 1 ∈ (η, 1δ) and η 2 = η 1 + δ. We combine the two different splittings L = A + B and L = A 0 + B 0 , where

Af (x, v) = M 1 (x0,R) (x)f (x, v), A 0 f (x, v) = 4 2 1 K(2x)℘(v, v * )f (x, v * ) dv * .
We prove that for any κ > κ B the operator

C := (κ -B 0 ) -1 A 0 (κ -B) -1 A is well defined and maps continuously L 1 m into L 1 m1 ∩ W 1,1 loc , in the sense that if (f n ) is bounded in L 1 m then the image is bounded in L 1 m1 ∩ W 1,1 ((x 0 , R) × [1, 2]
) for all R > 0. In particular, C ∈ K(L 1 m ). More precisely, for any κ > 0, we prove

L 1 m A -----→ L 1 m2 (κ-B) -1 ------→ L 1 m2 A0 -----→ L 1 m1 ∩ W 1,1 v,loc (κ-B0) -1 ------→ L 1 m1 ∩ W 1,1 loc where W 1,1 v,loc := {f ∈ L 1 loc ((x 0 , ∞) × [1, 2]), ∂ v f ∈ L 1 loc ((x 0 , ∞) × [1, 2
])}. The results for A and (κ -B) -1 are proved as in the case without variability. For the third one, the fact that A 0 maps L 1 m2 in L 1 m1 follows from assumption (9.24), and the fact that the range is in W 1,1 v,loc is a direct consequence of the assumption that ℘ ∈ W 1,∞ ([1, 2] 2 ). Finally we consider κ -B 0 and we first verify that it is invertible in L 1 m1 for any κ > 0. If (κ -B 0 )g = f , then necessarily (9.26) g(x, v) = 1 va(x)

x x0 e (Λκ(y)-Λκ(x))/v f (y, v) dy, where Λ κ (x) =

x x0 κ+K a , and consequently g(x, v)m 1 (x) = e (η1Λ(x)-Λκ(x))/v va(x)

x x0 e (Λκ(y)-η1Λ0(y))/v f (y, v)m 1 (y) dy.

Since Λ κ (x) -η 1 Λ 0 (x) = (1 -η 1 )Λ κ/(1-η1) (x)
we have for all

v ∈ [1, 2] ∞ x0 κ 1 -η 1 + K(x) g(x, v)m 1 (x) dx = ∞ x0 1 v Λ ′ κ/(1-η1) (x)e -1-η 1 v Λ κ/(1-η 1 ) (x) x x0 e (1-η1)Λ κ/(1-η 1 ) (y,v) f (y, v)m 1 (y) dydx = ∞ x0 e 1-η 1 v Λ κ/(1-η 1 ) (y) f (y, v)m 1 (y) ∞ y 1 v Λ ′ κ/(1-η1) (x)e -1-η 1 v Λ κ/(1-η 1 ) (x) dx dy = 1 1 -η 1 ∞ x0 f (y, v)m 1 (y) dy.
We deduce that the operator κ -B 0 is invertible in L 1 m1 with (κ -B) -1 ≤ 1/κ. We have also proved that (κ

-B 0 ) -1 maps L 1 m1 into L 1 Km1 with (κ -B 0 ) -1 B(L 1 m 1 ,L 1 Km 1
) ≤ 1 1-η1 . The fact that it maps W 1,1 v,loc into W 1,1 loc readily follows from the formula (9.26). We conclude to the compactness of C and then to the validity of (H3). Indeed, we can write (2.20)

as fn = R B (λ n )A fn + R B (λ n )ε n , but also as fn = R B0 (λ n )A 0 fn + R B0 (λ n )ε n .
Combining both, we get fn

= C fn + R B0 (λ n )A 0 R B (λ n ) + R B0 (λ n ) ε n .
Since C is compact, we conclude to (H3) with the same argument as in the proof of Lemma 2.8.

From (H1), (H2), (H3) and (H4) we infer the conclusion (C2) about existence and uniqueness of (λ 1 , f 1 , φ 1 ), which gives a part of Theorem 9.10. For the quantitative exponential stability, we start with a lemma.

Lemma 9.11. For all ε > 0, R 1 > x 0 , and R 2 > x 0 + ε, there exist T > 0 and c T > 0 such that

(9.27) S * T φ ≥ c T 1 (x0,R1)×[1,2] 2 1 R2 x0+ε φ dxdv, ∀ φ ≥ 0.
Proof of Lemma 9.11. Let us fix ε > 0, R 1 > x 0 , and R 2 > x 0 + ε. Throughout the proof we denote by c t any positive constant that depend on t, and also possibly on the ingredients of the model g, K, ℘ and on ε, R 1 , R 2 , but is independent of (x, v) ∈ (x 0 , R 1 ) × [START_REF] Abdallah | Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system[END_REF][START_REF] Agoshkov | Problema sledov funktsiȋ iz prostranstv H 1 p (Ω × D) i issledovanie nekotorykh variatsionnykh kraevykh zadach perenosa[END_REF]. First step. We prove first that there exists T 1 > 0 and

x 3 > x 2 > x 0 such that x 3 > max(R 2 , 2x 2 ) and (9.28)
S * T1 φ ≥ c T1 1 (x0,R1)×[1,2] 2 1 x3 x2 φ dxdv,
for all φ ≥ 0. We start from the Duhamel formula

S * t φ(x, v) = φ(X v t (x), v)e -t 0 K(X v s (x))ds (9.29) + 2 t 0 2 1 K(X v s (x))S * t-s φ(X v s (x)/2, v * )e -s 0 K(X v s ′ (x))ds ′ ℘(v * , v)dv * ds,
where X v t (x) is the solution to the characteristic equation Ẋv t (x) = v a(X v t (x)) with X v 0 (x) = x. Iterating twice (9.29), using positivity and the fact that K and a are locally bounded and ℘ is bounded from below, we deduce that

S * t φ(x, v) ≥ c t t 0 2 1 s 0 2 1 K(X v s ′ (x))K(X v * s-s ′ (X v s ′ (x)/2))φ X v * * t-s X v * s-s ′ X v s ′ (x)/2 /2 , v * * dv * * ds ′ dv * ds, on (x 0 , R 1 ) × [1, 2]
, for all φ ≥ 0. Let t 0 be such that X 1 t0 (x 0 ) = 2x 0 + 1. Then, for t > 2t 0 , we deduce, from the fact that K is locally bounded from below on (2x 0 , ∞), that

S * t φ(x, v) ≥ c t t 2t0 2 1 s t0 2 1 φ X v * * t-s X v * s-s ′ X v s ′ (x)/2 /2 , v * * dv * * ds ′ dv * ds.
For t > 2t 0 + 2, by using the Fubini-Tonelli theorem, we thus have

S * t φ(x, v) ≥ c t 2 1 t t-1 t0+1 t0 2 1 φ X v * * t-s X v * s-s ′ X v s ′ (x)/2 /2 , v * * dv * ds ′ dsdv * * .
Using now a change of variables, we get

S * t φ(x, v) ≥ c t 2 1 t t-1 t0+1 t0 X v * * t-s (X 2 s-s ′ (X v s ′ (x)/2)/2) X v * * t-s (X 1 s-s ′ (X v s ′ (x)/2)/2) φ(y, v * * )dy ds ′ dsdv * * ≥ c t 2 1 X 2 t-t 0 -2 (x0/2)/2 X 2 1 (X 1 t-t 0 (X 2 t 0 +1 (R1)/2)/2)
φ(y, v * * ) dydv * * .

Due to the strict positivity of a, we can choose t = T 1 large enough so that

X 2 T1-t0-2 (x 0 /2)/2 > max R 2 , 2X 2 1 (X 1 T1-t0 (X 2 t0+1 (R 1 )/2)/2
) , and we obtain (9.28) by setting x 3 = X 2 T1-t0-2 (x 0 /2)/2 and x 2 = X 2 1 (X 1 T1-t0 (X 2 t0+1 (R 1 )/2)/2), which concludes the first step of the proof. Second step. We deduce (9.27) from (9.28) as follows. On the one hand, applying (9.28) to the function S * t φ, we obtain

S * T1+t φ ≥ c T1 1 (x0,R1)×[1,2] 2 1 x3 x2 S * t φ dxdv.
On the other hand, iterating once the Duhamel formula (9.29), we get by positivity that

S * t φ(x, v) ≥ c t φ(X v t (x), v) + t 0 2 1 K(X v s (x))φ(X v * t-s (X v s (x)/2), v * )dv * ds .
We first assume x 2 > 2x 0 . In that case, the term K(X v s (x)) is bounded from below uniformly in s ∈ [0, t], v ∈ [1, 2] and x ∈ [x 2 , x 3 ], so that we infer from the two above inequalities that

S * T1+t φ ≥ c T1 c t 1 (x0,R1)×[1,2] 2 1 x3 x2 φ(X v t (x), v) + t 0 2 1 φ(X v * t-s (X v s (x)/2), v * )dv * ds dxdv.
By a change of variable, we have

x3 x2 φ(X v t (x), v)dx ≥ c t x3 X 2 t (x2)
φ(y, v)dy

and t 0 x3 x2 φ(X v * t-s (X v s (x)/2), v * )dxds ≥ c t x3/2 X 2 t (X 2 t (x)/2)
φ(y, v * )dy.

Since X 2 t (x) → x as t → 0, we deduce that we can find, for any ζ > 0, a time t > 0 such that Using a last time the argument with ζ ≤ ε/2 yields (9.27), since x 3 > R 2 .

S * T1+t φ ≥ c T1 c t 1 (x0,R1)×[1,2]
In the case where x 2 ≤ 2x 0 , (9.28) directly implies (9.30) with T 2 = T 1 , and only one iteration of the extension argument is enough for concluding.

We are now in position to finish the proof of Theorem 9.10.

Proof of Theorem 9.10. The proof is exactly the same as for Theorem 9.1, Lemma 9.11 replacing Lemma 9.2. The only missing information is a quantitative L ∞ loc estimate on the derivatives ∂ v φ 1 and ∂ x φ 1 , in order to use the same argument as in the proof of Proposition 9.8 for verifying (6.9). The estimate on ∂ x φ 1 follows directly from the equation L * φ 1 = λ 1 φ 1 , which also reads

∂ x φ 1 = 1 va(x) λ 1 φ 1 + Kφ 1 -2K(x) 2 1 φ 1 (x, v * )℘(v * , v)dv * .
For ∂ v φ 1 we argue by duality, using that

φ L ∞ = sup f L 1 =1
φ, f .

We start from

φ 1 = (λ 1 -B * 0 ) -1 A * 0 φ 1 , which yields ∂ v φ 1 L ∞ ((x0,R)×[1,2] = sup f L 1 =1 ∂ v (λ 1 -B * 0 ) -1 A * 0 φ 1 , f = sup f L 1 =1 φ 1 , A 0 (λ 1 -B 0 ) -1 ∂ v f
where the supremum can be taken over the functions f ∈ C 1 c ((x 0 , R) × (1, 2)). Using an integration by parts in v * , we have

A 0 (λ 1 -B 0 ) -1 ∂ v f (x, v) = 4 2 1 K(2x) v * a(x) x x0 e (Λ λ 1 (y)-Λ λ 1 (x))/v * ∂ v f (y, v * )dy ℘(v, v * )dv * = -4 2 1 x x0 ∂ v * K(2x) v * a(x) e (Λ λ 1 (y)-Λ λ 1 (x))/v * ℘(v, v * ) f (y, v * )dydv * . Since φ 1 L ∞ ((x0,R)×[1,2]) ≤ m(R), we deduce ∂ v φ 1 L ∞ ((x0,R)×[1,2]) ≤ 4m(R) sup (y,v * )∈(x0,R)×[1,2] R x0 2 1 ∂ v * K(2x) v * a(x) e (Λ λ 1 (y)-Λ λ 1 (x))/v * ℘(v, v * ) dvdx,
and this last quantity is finite due to the assumptions made on the functions a, K and ℘.

The kinetic linear Boltzmann equation

In this section, we consider the kinetic linear Boltzmann type equation (10.1)

∂ t f + v • ∇ x f -∇ x Φ(x) • ∇ v f = K [f ] -Kf, in (0, ∞) × O
on the function f = f (t, x, v), t ≥ 0, (x, v) ∈ O := Ω × R d . We assume that K = K(x, v) ≥ 0 and that the collision operator K is linear and defined by (10.2)

K = rK 1 , (K 1 g)(x, v) := R d k g * dv * ,
for a real number r > 0 and some collision kernel

k : Ω × R d × R d → R + .
Here and below, we use the common shorthands

g * := g(v * ), k := k(x, v, v * ), k * := k(x, v * , v).
The most classical example for the collisional operator C = K -K is the mass conservative operator

(10.3) (Cg)(v) := R d |v -v * | γ {M g * -M * g} dv * ,
for some function M ∈ L 1 + (R d ) and some exponent γ ∈ R, which includes the relaxation operator

(10.4) (Cg)(v) := σ(M ρ g -g), ρ g := R d g * dv * .
We make the follwing strong positivity and boundedness assumption on the collision kernel k and the function K. There exist γ ≥ 0 and K i > 0 such that (10.5)

∀ (x, v) ∈ Ω × R d , K 0 ≤ K(x, v) v -γ ≤ K 1 .
There exists a weight function m :

R d → [1, ∞) such that (10.6) ∀ p ∈ [1, ∞], k m -1 * m ∈ L ∞ x L p v L p ′ v * . For all R > 0, there exists k R > 0 such that (10.7) ∀ (x, v, v * ) ∈ Ω × B R × B R , k(x, v, v * ) ≥ k R .
It is worth emphasizing that for K and K defined in (10.3), the above assumptions are met when m := M -1/2 : R d → [1, ∞) (so that in particular M > 0 a.e.) and M 1/2 v γ ∈ L 1 ∩ L ∞ . We finally assume that for some weight function m 1 : R d → [1, ∞) such that m 1 /m → ∞ at infinity, we have

(10.8) k m -1 * m 1 ∈ L ∞ x L 2 vv * ,
what holds true for the relaxation operator when M m 1 ∈ L 2 (R d ) and m -1 ∈ L 2 (R d ), and that for some weight function m 0 : R d → [1, ∞) such that m 0 /m → 0 at infinity, we have

(10.9) k m -1 0 * m ∈ L ∞ x L 1 vv * ,
what holds true for the relaxation operator when M m ∈ L 1 (R d ) and m -1 0 ∈ L 1 (R d ). For the space domain Ω, we consider the two following cases:

(1) Ω := T d , the torus;

(2) Ω := R d , the whole space.

In case [START_REF] Abdallah | Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system[END_REF], and for the sake of simplicity, we will always assume that Φ = 0. In case (2), we will need a confinement mechanism which will be provided by the mean of the confinement force associated to the confinement potential Φ. We do not consider here the case of a bounded domain with zero influx boundary condition because (1) our approach applies exactly as for the torus case and (2) this case has already been considered in the pioneering work by Vidav [START_REF] Vidav | Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator[END_REF], where existence, uniqueness and exponential stability (with non constructive constants) have been established. We do not consider either the case of a bounded domain complemented with a reflection as we will consider in Section 11 for the kinetic Fokker-Planck evolution equation, because we have not been able to establish some crucial regularity estimates which seem to be necessary in our approach. We let this issue for a future work. 10.1. The torus. In this section, we are first concerned with the kinetic linear Boltzmann equation in the torus (10.10)

∂ t f + v • ∇ x f = K [f ] -Kf, in (0, ∞) × T d × R d .
We make the boundedness and strong positivity assumptions listed above together with the additional assumption Our result may be compared to [START_REF] Vidav | Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator[END_REF] which establishes the same result without constructive rate and to [START_REF] Champagnat | Exponential convergence to quasi-stationary distribution and Q-process[END_REF] which establishes the same result using a probabilistic approach, both in the case of a bounded domain with zero influx boundary condition. It also extends to a non mass conservative situation the many results devoted to the conservative framework, see for instance the recent papers [START_REF] Mokhtar-Kharroubi | On L 1 exponential trend to equilibrium for conservative linear kinetic equations on the torus[END_REF][START_REF] Han-Kwan | Geometric analysis of the linear Boltzmann equation I. Trend to equilibrium[END_REF][START_REF] Cañizo | Hypocoercivity of linear kinetic equations via Harris's theorem[END_REF] and the references therein. When γ > 0, we may probably establish the same above result under the sole condition r > 0 (no need for r to be large enough) by using some arguments developed in the next section.

(10.11) k m -1 * m 1 , k m -1/2 1 * m ∈ L ∞ xvv * , u∈Z d m -1/2 1 (u + •) L ∞ (T d ) < ∞, for some m 1 such that m/m 1 ∈ L 1 ∩ L 2 .
We present now the proof of Theorem 10.1 by establishing that the conditions presented in the abstract part are satisfied.

Condition (H1). For an exponent p ∈ [1, ∞) and a weight function m satisfying (10.6), we set

k ∞ := km -1 * m L ∞ x L p v L p ′ v * < ∞.
Considering then a solution f to the evolution equation (10.10), we compute

1 p d dt f p m p = K [f ]f p-1 m p -K(x, v)f p m p ≤ K [f ]m L p f p m p 1-1/p -K(x, v)f p m p ≤ rk ∞ f p m p -K 0 v γ f p m p ,
where we have used twice the Holder inequality. This differential inequality together with the Gronwall lemma provides an apriori estimates about the growth of the L p m norm. As a consequence, the same arguments as in section 8.3 imply that S L is a positive semigroup in L p m with growth bound ω(S L ) = rk ∞ -K 0 . In particular, condition (H1) holds thanks to Lemma 2.2.

Condition (H2). For f

0 := 1 T d ×B1 , where B 1 denotes the unit ball in R d v , we compute Lf 0 = K [f 0 ] -Kf 0 ≥ inf v∈B1 rK 1 [f 0 ] -K}f 0 .
Using (10.5) and the strong positivity condition (10.7), we get (10.12) inf v∈B1

K [f 0 ] -K} ≥ rk 1 -2 γ/2 K 0 =: κ 0 ,
which provides a constructive lower bound of the set I defined in (2.15) thanks to Lemma 2.4-(ii).

We have thus established that L satisfies (H2).

Condition (H3). We define the operator

Bf := -v • ∇ x f -K(x, v)f,
and we assume κ B :=inf K ≤ -K 0 < κ 0 , what holds whenever r ≥ r * , with r * > 0 large enough thanks to (10.12). In the sequel, we assume p = 2 and we work in X = L 2 m . We immediately deduce that Bκ is dissipative for any κ > κ B , and thus R

B (z) is bounded in B(L 2 m ), uniformly in z ∈ ∆ κ . For κ > κ B and g ∈ L 2 m , the function f = R B (κ)g satisfies v • ∇ x f + (κ + K)f = g in O,
from what we deduce

(κ -κ B ) O f 2 m 2 ≤ O (κ + K)f 2 m 2 = O f gm 2 ,
and finally

f 2 L 2 m ≤ 1 κ -κ B g 2 L 2
m . Because of assumption (10.8), and defining A := K , we immediately deduce that (10.13) AR B (κ) : L 2 m → L 2 m1 . On the other hand, from the classical averaging lemma [START_REF] Golse | Regularity of the moments of the solution of a transport equation[END_REF], we know that (10.14)

A ϕ R B (κ) : L 2 (O) → H 1/2 (O), where for ϕ = ϕ 1 ⊗ ϕ 2 ∈ C 1 c (O) ⊗ C 1 c (R d ), we have defined the mapping A ϕ : L 2 (O) → L 2 (O) by A ϕ (f )(x) := ϕ 1 (x, v) R d f (x, v * )ϕ 2 (v * ) dv * .
By classical approximation arguments, there exists a sequence (ϕ n ) such that ϕ n → k in the space

L ∞ (T d ; L 2 m1⊗m -1 (R d × R d )) and such that ϕ n is a linear combination of functions of C 1 c (O) ⊗ C 1 c (R d ).
As a consequence of (10.13) and (10.14), we deduce that AR B (κ) ∈ K (L 2 m ) and next (R B (κ)A) 2 ∈ K (L 2 m ) for any κ > κ B . We may use Lemma 2.8 (and Remark 2.9-( 2)) with N = 2 in X = L 2 m , and deduce that (H3) holds. Condition (H4). We start with a result of independent interest about strict positivity. Such an argument is reminiscent from [START_REF] Carleman | Sur la théorie de l'équation intégrodifférentielle de Boltzmann[END_REF][START_REF] Pulvirenti | A Maxwellian lower bound for solutions to the Boltzmann equation[END_REF] in the study of the Boltzmann equation and has been used for instance in [START_REF] Mouhot | Quantitative lower bounds for the full Boltzmann equation. I. Periodic boundary conditions[END_REF][START_REF] Cañizo | Hypocoercivity of linear kinetic equations via Harris's theorem[END_REF]. Lemma 10.2. For any ̺, ̺ * , t > 0, there exists c > 0 such that

(10.15) (S L (t)f 0 )(x, v) ≥ c 1 B̺ (v) T d ×B̺ * f 0 dv * dx * , for all f 0 ≥ 0 and (x, v) ∈ T d × R d .
Proof of Lemma 10.2. We observe that the semigroup S B has explicit representation

(S B (t)f 0 )(x, v) = f 0 (x -vt, v)e -t 0 K(x-τ v,v)dτ .
We next write the associated iterated Duhamel formula

S L = S B + S B K * S B + S B K * S B K * S B + S B K * S B K * S B K * S L .
Since all the terms are nonnegative, we may through away the first terms and the last one, and we get S L ≥ S B * K S B * K S B . On the one hand, using the explicit expression of S B and (10.7), we have

(K 1 S B (s)f 0 )(y, w) ≥ k ̺ ′ e -sK̺ * 1 B ̺ ′ (w) B̺ * f 0 (y -w * s, w * )dw * =: g(s, y, w),
for any s > 0 and any ̺ ′ ≥ ̺ * > 0, with K ̺ := sup z∈T d ,|v|≤̺ K(z, v). On the other hand, for the same reasons, we have

(K 1 S B * g(t))(x, v) = t 0 R d k(x, v, v * )g(s, x -v * (t -s), v * )e -t-s 0 K(x-τ v * ,v * )dτ dv * ds ≥ k ̺ ′ 1 B̺ (v) t 0 B ̺ ′ g(s, x -v * (t -s), v * )e -(t-s)K ̺ ′ dv * ds ≥ k 2 ̺ ′ e -tK ̺ ′ 1 B̺ (v) t 0 B ̺ ′ B̺ * f 0 (x -v * (t -s) -w * s, w * )dw * dv * ds ≥ k 2 ̺ ′ e -tK ̺ ′ 1 B̺ (v) t/2 0 B̺ * B(x+w * s,(t-s)̺ ′ ) f 0 (y * , w * ) dy * (t -s) d dw * ds ≥ k 2 ̺ ′ e -tK ̺ ′ (t/2) d 1 B̺ (v) t/2 0 B̺ * T d f 0 (y * , w * )dy * dw * ds ≥ k 2 ̺ ′ e -tK ̺ ′ (t/2) d 1 B̺ (v) t 2 B̺ * T d f 0 (y * , w * )dy * dw * ,
for any t > 0 and ̺ ′ ≥ max(̺, ̺ * ) such that t̺ ′ ≥ 2, in such a way that B(z, (t/2)̺ ′ ) ⊃ T d . We then have

(K 1 S B * K 1 S B (t))(x, v) ≥ k 2 ̺ ′ e -tK ̺ ′ (t/2) d-1 1 B̺ (v) T d B̺ * f 0 * dw * dx * for any t ≥ 2/̺ ′ . We finally conclude S L (t)f 0 (x, v) ≥ r 2 k 2 ̺ ′ e -tK ̺ ′ t 2/̺ ′ ds (s/2) d-1 1 B̺ (v) T d B̺ * f 0 * dw * dx * ,
from what we deduce (10.15) by choosing ̺ ′ = 8/t. We now consider λ ≥ λ 1 and 0

≤ f ∈ L 2 m , f ≡ 0, such that λf + v • ∇ x f + Kf -K [f ] ≥ 0 in T d × R d .
We fix ̺ * > 0 such that f ≡ 0 on B ̺ * . From (2.13), we have

f ≥ ∞ 0 e -(1+λ)t S L (t)f dt,
and we conclude that f > 0 a.e. on any set T d × B ̺ thanks to Lemma 10.2. We have established that the strong maximum holds true, and thus (H4).

Condition (H5

). Assume that (λ, f ) ∈ C × X\{0} satisfies Lf = λf, L|f | = (ℜeλ)|f | = ℜe(signf )Lf.
From (H4) and the first identity satisfied by |f |, we know that |f | > 0 a.e. on T d × R d . Using the second identity, we get

K [|f |] = ℜe(signf )K [f ]. Writing f = e iα |f |, we deduce R d k(x, v, v * )|f (x, v * )|(1 -cos(α -α * ))dv * = 0 a.e. on T d × R d ,
and thus α = α(x) thanks to (10.7). Next, coming back to the first equation, we have

λ|f |e iα = L(|f |e iα ) = e iα L|f | -|f |e iα iv • ∇ x α = e iα (ℜeλ)|f | -|f |e iα iv • ∇ x α.
The equation simplifies into v • ∇ x α = ℑmλ, so that α(x) = α is a constant and the reverse Kato's inequality holds.

Alternatively to (H5), we readily infer from Lemma 10.2 that the variant condition (H5 ′ ) is verified.

At this stage, because of Theorem 2.21, Theorem 4.13 and Theorem 5.16 (or Theorem 5.18), we deduce the conclusions (C1), (C2) and (C3) about the existence and uniqueness of the eigentriplet (λ 1 , f 1 , φ 1 ) which satisfies f 1 > 0, φ 1 > 0, λ 1 is algebraically simple and on the triviality of the boundary punctual spectrum. We now establish the exponential asymptotic stability with constructive constants.

We start with a gain of unifom boundness estimate.

Lemma 10.3. There exists

N ≥ 1 such that (AR B (κ)) N : L 1 m → L ∞ m , for any κ > κ B . As a consequence, φ 1 ∈ L ∞ m -1 .
Proof of Lemma 10.3. Step 1. We argue silmilarly as in [START_REF] Mischler | On a linear runs and tumbles equation[END_REF]Sec. 3.1]. On the one hand, denoting A 1 = K 1 , so that A = rA 1 , we have for any

f 0 ∈ L 1 m (A 1 S B (t)f 0 )(x, v) = R d k(x, v, v * )f 0 (x -v * t, v * ) e -t 0 K(x-v * τ,v * )dτ dv *
and, using estimate (10.11), we deduce that

m 1 A 1 S B (t)f 0 L 1 x L ∞ v ≤ km -1 * m 1 L ∞ xvv * O |f 0 (x -v * t, v * )|m * dv * dx e tκB f 0 L 1 m e tκB , for any t ≥ 0. Now, we consider f 0 ∈ L 1 x L ∞ vm , we write (A 1 S B (t)f 0 )(x, v) = u∈Z d T d k(x, v, v * )f 0 (x -ut -v * t, u + v * ) e -t 0 K(x-(u+v * )τ,u+v * )dτ dv *
and using estimate (10.11) again, we compute

(A 1 S B (t)f 0 )(x, v)m(v) ≤ km -1/2 1 * m L ∞ xvv * u∈Z d T d m 1/2 1 (u + v * )f 0 (x -ut -v * t, u + v * ) e tκB dv * e tκB u∈Z d m -1/2 1 (u + •) L ∞ (T d ) tT d f 0 (y, •) L ∞ m 1 dy t d e tκB 1 + 1 t d f 0 L 1 x L ∞ vm 1 . Defining u(t) := e -κt A 1 S B (t), κ > κ B , we have first established u : L 1 m → L 1 x L ∞ vm1 unifomly in time, and thus u : L 1 x L ∞ vm1 → L 1 x L ∞ vm1 unifomly in time because L 1 x L ∞ vm1 ⊂ L 1
m (we use here the fact that m/m 1 ∈ L 1 ). On the other hand, we have establised that

t d u : L 1 x L ∞ vm1 → L ∞ m uniformly in time. Using [276, Prop. 2.5] with X := L 1 x L ∞ vm1 and Y := L ∞ m , we deduce u * (d+1) : L 1 x L ∞ vm1 → L ∞ m
uniformly in time, and we thus conclude that u * N : L 1 m → L ∞ m uniformly in time, for N := d + 2. Using formula (2.13), we deduce that (AR B ) N (z) :

L 1 m → L ∞ m , uniformly for any z ∈ ∆ κ . Step 2. In particular, (AR B ) N (z) : L 1 m → L 2 m because L ∞ m1 ⊂ L 2
m (we use here the fact that m/m 1 ∈ L 2 ). By duality, we deduce that (R

B * A * ) N (z) : L 2 m -1 → L ∞ m -1 .
Coming back to the eigenvalue equation

A * φ 1 + B * φ 1 = λ 1 φ 1 ,
we deduce (10.16)

φ 1 = R B * (λ 1 )A * φ = (R B * (λ 1 )A * ) N φ 1 .
By construction φ 1 ∈ L 2 m -1 and we thus conclude that φ 1 ∈ L ∞ m -1 . From now on, we choose the normalization convention φ 1 L ∞ m -1 = 1 and f 1 , φ 1 = 1. Because of (10.9) and proceeding similarly as during the proof of condition (H3), we have

AR B (κ) : L 1 m0 → L 1 m , ∀ κ > κ B , so that R B * (κ)A * : L ∞ m -1 → L ∞ m -1 0 , ∀ κ > κ B .
From the first identity in (10.16), we deduce

φ 1 L ∞ m -1 0 ≤ C 01 φ 1 L ∞ m -1 ,
with constructive constant C 01 ∈ (0, ∞). We may here proceed along an already used argument. Consider 0

≤ f ∈ L 1 m and assume f L 1 m ≤ A[f ] φ1 . We then compute f φ 1 = O̺ f φ 1 m m + O c ̺ f m φ 1 m 0 m 0 m ≤ f, 1 O̺ sup O̺ m + f L 1 m C 01 sup O c ̺ m 0 m ≤ f, 1 O̺ sup O̺ m + 1 2 [f ] φ1
by choosing ̺ = ̺(A) large enough, where we denote O ̺ := T d × B ̺ . Together with Lemma 10.2, we deduce that there exists T > 0 and g A ≥ 0, g A ≡ 0, such that (10.17)

S L (T )f ≥ g A [f ] φ1 ,
what is nothing but the Harris condition (6.8). On the other hand, from the above regularization estimate, we have in the same time

1 = φ 1 L ∞ m -1 ≤ C 0 φ 1 L 1 m -1 , φ 1 L 1 m -1 0 ≤ C 1 φ 1 L 1 m -1 ,
for some constructive constants C i ∈ (0, ∞). We may thus compute

φ 1 m -1 ≤ B̺ φ 1 m -1 + sup B c ̺ m 0 m φ 1 m -1 0 ≤ B̺ φ 1 m -1 + sup B c ̺ m 0 m C 1 φ 1 m -1 ,
so that for ̺ > 0 large enough, we deduce (10.18)

C -1 0 ≤ φ 1 L 1 m -1 ≤ 2 B̺ φ 1 m -1 .
Together with the definition of g A , we deduce that the positivity condition (6.3) holds.

Finally, as during the proof of (H3) above, for 0 ≤ f 0 ∈ L 1 m and denoting f := S L (t)f 0 , we compute

d dt f mdvdx = K [f ]mdvdx -Kf mdxdv ≤ C 0 f m 0 dvdx + κ B f mdxdv, with C 0 := kmm -1 0 * L ∞ xv * L 1 v < ∞ and m 0 /m → 0 as v → ∞. Observing that f m 0 ≤ B̺ f φ 1 sup B̺ m 0 φ 1 + B c ̺ f m sup B c ̺ m 0 m ,
for any κ > κ B , we may choose ̺ > 0 large enough in such a way that sup

B c ̺ m0 m ≤ (κ -κ B )/C 0 and we deduce that d dt f mdvdx ≤ C 1 f φ 1 dvdx + κ f mdxdv with C 1 = sup B̺ m0 φ1 .
From the Gronwall lemma, we obtain

f t L 1 m ≤ e κt f 0 L 1 m + e λ1t -e κt λ 1 -κ C 1 f 0 φ 1 ,
from what we immediately deduce that S L satisfies the Lyapunov condition (6.7) for any t > 0.

It remains to quantify the constant C 1 . The dual formulation of (10.15) applied to the dual eigenfunction φ 1 with t = 1 and ̺ * = ̺ yields

φ 1 = e -λ1 S * L (1)φ 1 ≥ e -λ1 c 1 T d ×B̺ T d ×B̺ φ 1 dv * dx * .
Together with Equation (10.18), this provides the explicit bound C 1 ≤ 2C 0 e λ1 c -1 sup B̺ m0 infB ̺ m .

We have established that the three conditions (6.8), (6.7) and (6.9) hold, so that we conclude the proof of Theorem 10.1 by just applying Theorem 6.3.

10.2. The whole space case. In this section, we assume that Ω := R d and we consider the kinetic equation (10.1) with an additional force field confinement F = -∇ x Φ associated to a potential Φ. More precisely from now-on, we assume that

(10.19) Φ(x) = |x| β , β > 2, K(v) = v γ , γ > 0,
that (10.6) holds (for p = 2) and that there exist ζ, c ζ > 0 such that (10.20)

K [M ζ ] ≥ c ζ v γ M ζ , M := e -|v| 2 /2 .
Observe that condition (10.20) is satisfied when K is the positive part of the mass conservative operator (10.3). For further references, we write L := T + C with

T := -v • ∇ x f + ∇ x Φ • ∇ v , C f := K [f ] -Kf
and we define the Hamiltonian

H := Φ(x) + 1 2 |v| 2 .
In the sequel, we will only consider some weight functions m = ω(H) with ω(y) = y ̺ , ̺ ≥ 0, or ω(y) = e κy , κ ∈ (0, 1), so that ω(H) ∼ ω(|v| 2 )ω(Φ). For p ∈ [1, ∞), we further assume that v → ω -1 (|v| 2 ) ∈ L p ′ (which imposes ̺ > d/(2p ′ ) for a polynomial weight). We are not aware of any result on the first eigentriplet problem for such linear Boltzmann like equation in the whole space. We may however compare our result to [START_REF] Han-Kwan | Geometric analysis of the linear Boltzmann equation I. Trend to equilibrium[END_REF] where the corresponding mass conservative framework is considered. We present the proof of Theorem 10.4 in that situation by adapating the arguments presented in the previous section.

Condition (H1). Let us consider a weight function m = ω(H) as intruced before and let us fix p ∈ [1, ∞). For a solution to the evolution equation (10.1), we classically compute d dt

f p p m p dvdx = (Lf )f p-1 m p dvdx = (f p /p)T * m p + (K f )f p-1 m p -f p Km p ≤ 1 p (K f ) p m p + f p 1 p ′ -K m p
, by using an integration by parts and the Young inequality. For the first term, we have

(K f ) p m p dvdx ≤ c ω ω(Φ) f dv p dx = c ω ω(Φ) f p ω(|v| 2 )dvdx ω -1 (|v| 2 ) p L p ′ f p m p dvdx.
All together and thanks to the Gronwall lemma, we have established an apriori estimate on the evolution of the norm f L p m and we deduce as in section 8 that L generates a positive semigroup on L p m . In particular, the condition (H1) is satisfied thanks to Lemma 2.2. Condition (H2). We define f 0 := e -ζH and we compute

Lf 0 = C f 0 = re -ζΦ K [e -M ζ ] -Ke -ζΦ M ζ ≥ (rc ζ -K 0 ) v γ e -ζH ≥ 0,
for r > 0 large enough. That implies that I is lower bounded by κ 0 = 0 by using Lemma 2.4-(ii), and we have thus established that L satisfies (H2).

Condition (H3). We introduce the collisionless operator

Bf := T f -Kf and we define

B ♯ φ := 1 2 T * φ -Kφ.
Our analysis is mainly a consequence of the following moment estimate.

Lemma 10.5. There exist some weight functions w H and some real numbers α, c α , C α > 0 such that

(10.21) B ♯ w ≤ C α w -c α H 1+α .
Proof of Lemma 10.5. We split the proof into two steps.

Step 1. We first assume γ ≤ β -2 and we define

w := 1 + 1 2 [x] 1+γ/2 • v + H, with [x] δ := x|x| δ-1 . We observe that [x] 1+γ/2 • v ≤ 1 2 |x| 2+γ + 1 2 |v| 2 ≤ 1 2 H + 1 2 ,
so that w ∼ H . We now compute

T * w ≤ γ + 2 4 |x| γ/2 |v| 2 - β 2 |x| β+γ/2
and thus

B ♯ w ≤ C 1 |x| γ/2 |v| 2 - 1 2 |x| β+γ/2 - 1 2 |v| 2+γ .
Condition (H4) and (H5 ′ ). We recall that it has been proved in [START_REF] Cañizo | Hypocoercivity of linear kinetic equations via Harris's theorem[END_REF]Lem. 4.5] that the semigroup S t associated to the operator L satisfies the Harris condition: for any T > 0 and ̺ > 0, there exists α > 0 such that (10.23)

S T f ≥ α1 B̺ B ϑ̺ f dxdv, ∀ f ≥ 0,
for some constant ϑ ∈ (0, 1) and where B r := {(x, v) ∈ R 2d ; |x| < r, |v| < r}. Although the statement of [START_REF] Cañizo | Hypocoercivity of linear kinetic equations via Harris's theorem[END_REF]Lem. 4.5] is not written in that way, one may easily track the constants appearing in Lemmas 3.5, 3.6, 3.7 and 4.1 in [START_REF] Cañizo | Hypocoercivity of linear kinetic equations via Harris's theorem[END_REF] and one obtain (10.23) with ϑ := 1/2. Now, (10.23) immediately implies (H5 ′ ) which in turn implies (H4) thanks to Lemma 4.8-( 2)-(3).

Because L is the generator of a semigroup it also satisfies the weak maximum principle and Kato's inequalities (H1 ′ ). We are then in position to apply Theorem 2.21, Theorem 4.13, Theorem 5.18 and Theorem 5.23-(3) and thus complete the proof of Theorem 10.4.

The kinetic Fokker-Planck equation

In this part, we consider the kinetic Fokker-Planck evolution equation associated to the operator (11.1) We will consider the case when Ω is a bounded domain and the equation is complemented with a boundary condition. More precisely, we assume the classical balance between the values of the trace γf of f on the outgoing and incoming velocities subsets of the boundary

Lf := -v • ∇ x f + ∆ v f + b • ∇ v f + cf, on functions f : O → R, where O := Ω × R d , Ω ⊂ R d is a domain, b : O → R d is
(11.2) (γ -f )(x, v) = R x (γ + f (x, .))(v) on Σ -,
where in this context we define Σ x ± := {v ∈ R 3 ; ± v •ν x > 0} the sets of outgoing (Σ x + ) and incoming (Σ x -) velocities at point x ∈ ∂Ω, next the sets

Σ ± = {(x, v) ∈ Σ; ±ν x • v > 0} = {(x, v); x ∈ ∂Ω, v ∈ Σ x ± }
, and finally the outgoing and incoming trace functions γ ± f := 1 Σ± γf . Here and in the sequel, ν x denotes the unit normal outward vector field defined on the boundary set ∂Ω. We similarly define the grazing velocity set Σ

0 = {(x, v) ∈ Σ; ν x • v = 0}.
The reflection operator R x is local in position, but can be local or nonlocal in the velocity variable, so that it writes [START_REF] Cercignani | The Boltzmann equation and its applications[END_REF][START_REF] Cercignani | The mathematical theory of dilute gases[END_REF][START_REF] Cercignani | A new scattering kernel in kinetic theory of gases[END_REF]. The second (normalisation) condition corresponds to the fact that all the particles reaching the outgoing boundary are put back on the incoming boundary (no mass is lost) while the third (reciprocity) condition means (when M is a Gaussian function) that the wall is in a local equilibrium state and is not influenced by the incoming particles. The normalization condition implies the local mass conservation (11.4)

(R x g)(v) := Σ x + r(x, v, v * )g(v * )v * • ν x dv * , for a reflection kernel r : ∂Ω × R d × R d → R. Some classical general assumptions on r are (11.3) r ≥ 0, R * x 1 = 1, R x M = M , for some positive function M = M (v), see for instance
Σ x - R x g|ν • v|dv = Σ x + gν • vdv,
while the three assumptions (11.3) on r together also imply

Σ x - R x g 2 M -1 |ν • v|dv ≤ Σ x - R x (g 2 /M ) R x M M -1 |ν • v|dv = Σ x + g 2 * M -1 * (R * 1)ν • v * dv * ,
where we have used the Cauchy-Schwarz inequality (and the fact that r ≥ 0) in the first line and the reciprocity condition in the second line. As a consequence, we have (11.5)

Σ x - R x g 2 M -1 |ν • v|dv ≤ Σ x + g 2 M -1 ν • vdv,
where we have used the normalization condition in that last step. In the sequel, we will rather consider the possibly position dependent Maxwell boundary condition operator

(11.6) R x g = α(x)D x g + β(x)Γ x g,
where the accommodation coefficients α, β :

∂Ω → [0, 1] satisfy α(x) + β(x) =: ζ(x) ≤ 1, Γ x is the specular reflection operator (11.7) Γ x (g(x, •))(v) = g(x, V x v), V x v = v -2ν(x)(ν(x) • v),
and D x is the diffusive operator

(11.8) D x (g(x, •))(v) = c M M (v) g(x), g(x) = Σ x + g(x, w) ν(x) • w dw.
Here the constant c M := (2π) 1/2 is such that c M M = 1 and M stands for the standard Maxwellian (11.9)

M (v) := (2π) -d/2 exp(-|v| 2 /2),
or, more generally, M = M (|v|) ≥ 0 is such that (11.10)

D * x 1 = 1, D x M = M , v ϑ M ∈ L 1 (R d ),
with ϑ ≥ 1 (that last condition is necessary in order that the second relation above makes sense). The boundary condition (11.6) corresponds to the pure specular reflection boundary condition when β ≡ 1 and it corresponds to the pure diffusive boundary condition when α ≡ 1. When ζ ≡ 1, the Maxwell boundary condition operator (11.6) satisfies (11.3). On the contrary, when ζ ≡ 1, the L 2 estimate (11.5) holds but not anymore the mass conservation (11.4). However, the following L 1 estimate (11.11)

Σ x - |R x g||ν • v|dv ≤ ζ * Σ x + |g| ν • vdv holds, with 0 ≤ sup ζ ≤ ζ * ≤ 1.
Finally, the case ζ ≡ 0 corresonds to the zero inflow problem. Let us finally mention that similarly as in Part 8, the regularity needed on the domain Ω may be formulated in the following way: we assume that Ω is locally on one side of ∂Ω and there exists a function δ = δ Ω ∈ W 2,∞ (R d ) such that for all x in an interior neighborhood of ∂Ω one has δ(x) = dist(x, ∂Ω) and the vector field ν defined on R d by x → ν(x) = ν x := -∇ x δ(x) coincides with the previously defined unit outward normal vector field on ∂Ω and satisfies ν L ∞ = 1. We also assume that the Lebesgue measure on ∂Ω is well defined and it is denoted by dσ x .

11.1. The trace problem. We consider in this section the trace problem for a solution g = g(x, v) to the stationary Vlasov-Fokker-Planck equation (11.12)

Mg := v • ∇ x g -b • ∇ v g -∆ v g = G in O,
for a given a vector field b = b(x, v), a source term G = G(x, v) and for a solution g = g(t, x, v) to the evolution Vlasov-Fokker-Planck equation (11.13)

∂ t g + v • ∇ x g -b • ∇ v g -∆ v g = G in (0, T ) × O,
for a given a vector field b = b(t, x, v), a source term G = G(t, x, v). The second problem has been considered first in [START_REF] Carrillo | Global weak solutions for the initial-boundary-value problems to the Vlasov-Poisson-Fokker-Planck system[END_REF] and next in [START_REF] Mischler | Kinetic equations with Maxwell boundary conditions[END_REF]Sec. 4], where a strong (renormalized) trace function is proved to exist. In the sequel, we recall these results and slightly extending them by considering a possible L 2 H -1 source term. We introduce some notations. We denote dξ := |ν(x) • v|dvdσ x and dξ 2 := (ν(x) • v) 2 dvdσ x the measures on the boundary set Σ. We denote by B 1 the class of renormalized functions β ∈ W 2,∞ loc (R) such that β ′′ has a compact support, by B 2 the class of functions β ∈ W 2,∞ loc (R) such that β ′′ ∈ L ∞ (R) and by D 0 ( Ō) the space of test functions ϕ ∈ D( Ō) such that ϕ = 0 on Σ 0 . We finally define the dual operator

M * ϕ := -v • ∇ x ϕ + div v (bϕ) -∆ v ϕ. Theorem 11.1. We consider g, b ∈ L 2 loc,x H 1 loc,v , G ∈ L 2 loc,x H -1
loc,v and we assume that g is a solution to the stationary Vlasov-Fokker-Planck equation (11.12). Then there exists γg ∈ L 2 loc (Σ, dξ 2 ) such that the following Green renormalized formula

O β(g) M * ϕ -β ′′ (g) |∇ v g| 2 ϕ) dvdx + G, β ′ (g)ϕ = (11.14) = Σ β(γ g) ϕ ν(x) • v dvdσ x
holds for any renormalized function β ∈ B 1 and any test functions ϕ ∈ D( Ō), as well as for any renormalized function β ∈ B 2 and any test functions ϕ ∈ D 0 ( Ō). It is worth emphasizing that β ′ (g)ϕ ∈ L 2

x H 1 v so that the duality product G, β ′ (g)ϕ is well defined. If furthermore γ ∓ g ∈ L 2 loc (Σ; dξ) then γ ± g ∈ L 2 loc (Σ; dξ) and (11.14) holds for any renormalized function β ∈ B 2 and any test functions ϕ ∈ D( Ō).

Proof of Theorem 11.1. We only allude the proof which uses very similar arguments as those presented in Section 10 and that can also be partially found in [START_REF] Diperna | On the Fokker-Planck-Boltzmann equation[END_REF][START_REF] Mischler | Kinetic equations with Maxwell boundary conditions[END_REF]. In the one hand, considering the mollifier (ρ ε ) ε>0 defined in (8.19) with z := (x, y), we get that g ε is smooth and satisfies

g ε → g in L 2 loc,x H 1 loc,v , Mg ε = G ε → G in L 2 loc,x H -1 loc,v ,
) + β ′′ (g ε )|∇ v g ε | 2 = β ′ (g ε )G ε in O,
with. We may thus pass to the limit as ε → 0 and we obtain (11.14).

We also write without proof (since this one is similar to the proof of Proposition 8.10) a stability result that we will use several times in the sequel.

Proposition 11.2. Let us consider three sequences

(g k ), (b k ) of L 2 loc,x H 1 loc,v and (G k ) of L 2 loc,x H -1 loc,v such that v • ∇ x g k -b k • ∇ v g k -∆ v g k = G k in D ′ (O) for any k ≥ 1 and three functions g, b ∈ L 2 loc,x H 1 loc,v and G ∈ L 2 loc,x H -1 loc,v such that g k → g strongly in L 2 loc,x H 1 loc,v , b k ⇀ b weakly in L 2 loc ( Ō) and G k → G strongly in L 2 loc,x H -1 loc,v .
Then g satisfies (11.12) and, up to the extaction of a subsequence, γg k → γg a.e. on Σ\Σ 0 .

(2) If g k ⇀ g weakly in L 1 loc ( Ō) then g satisfies (11.14) and, up to the extaction of a subsequence, γg k r ⇀ γg on Σ\Σ 0 (we recall that the renormalized convergence has been defined in (8.51)).

L 2 H 1 m := f 2 L 2 m + ∇ v f 2 L 2 m , 2
and we assume that m satisfies the Poincaré type inequality (11.16)

f ∇m m L 2 m f L 2 H 1 m , ∀ f ∈ L 2 H 1 m .
Such a Poincaré inequality is classically known to be true when m := M -ϑ , M is the Maxwellian (11.9) and ϑ > 0. We also define

L 2 H -1 m := {F = g + div v G; g, G i ∈ L 2 m (O)}, so that when m = 1 the space L 2 H -1
m is nothing but the space of continuous and linear mappings on

L 2 H 1 . For F ∈ L 2 H -1 m and f ∈ L 2 H 1 m , we may thus write F, f m 2 ≤ F L 2 H -1 m f L 2 H 1 m . We finally define in this context W 2 := {f ∈ L 2 H 1 m ; v • ∇ x f ∈ L 2 H -1 m }, and W 2,Σ := {g ∈ W 2 ; γg ∈ L 2 (Σ; dξ m )}, with W 2,Σ = W 2 in general.
* := ess sup ̟ < ∞, ̟ := c + ∆m 2 2m 2 - 1 2 divb -b • ∇m m .
For any F ∈ L 2 H -1 m , g ∈ L 2 (Σ -; dξ m ) and λ > λ * , there exists a unique solution f ∈ W 2,Σ to the Dirichlet problem (11.15). We have furthermore f ≥ 0 if F ≥ 0 and g ≥ 0.

A similar result is proved in [129, Appendix A] in the case Ω = R d . Also observe that (11.17) holds with m := M -1/2 when M is the standard Maxwellian (11.9) and b(v) = ϑv, with ϑ > 1/2. Proof of Theorem 11.3. We split the proof into five steps.

Step 1. A priori estimates. We argue similarly as in [START_REF] Carrapatoso | The Kinetic Fokker-Planck equation in a domain: ultracontractivity and long time asymptotic[END_REF][START_REF] Carrapatoso | The Boltzmann equation in a domain[END_REF]. Multiplying the first equation in (11.15) by f m 2 , performing several integrations by part in the velocity variable and using the Green formula, we have

O (λ -̟)f 2 m 2 + 1 2 Σ (γf ) 2 m 2 ν • v + O |∇ v f | 2 m 2 = F, f m 2 .
Fixing λ > λ * , using the Young inequality

F L 2 H -1 m f L 2 H 1 m ≤ 1 2(λ -λ * ) + 1 2 F 2 L 2 H -1 m + λ -λ * 2 f 2 L 2 m + 1 2 ∇ v f 2 L 2 m
and the boundary condition on the incoming set Σ -in (11.15), we deduce

(11.18) (λ -λ * ) O f 2 m 2 + Σ+ (γ + f ) 2 dξ m + O |∇ v f | 2 m 2 ≤ 1 + λ -λ * λ -λ * F 2 L 2 H -1 m + Σ- g 2 dξ m .
• Because of the first equation in (11.15) and the above estimate, we find

(11.19) v • ∇ x f = 1 v F -λf + ∆ v f + b • ∇ v f + cf ∈ L 2 H -1 m , so that f ∈ W 2 .
• Multiplying the first equation in (11.15) by f ψ, ψ := ν(x) • ṽm 2 where here and below we use the notations v := v/ v , ṽ := v/ v 2 , v 2 := 1 + |v| 2 , and using the Green formula and one integration by part in the velocity variable, we get

1 2 Σ (γf ) 2 (ν • v) 2 m 2 = 1 2 O f 2 v • D x ν x v m 2 - O |∇ v f | 2 ψ + O f ∇ v f (bψ -∇ v ψ) + O f 2 ψ(c -λ) + F, f ψ . Observing that | F, f ψ | ≤ F L 2 H -1 m f ν(x) • ṽ L 2 H 1 m F L 2 H -1 m f L 2 H 1 m and f ∇ v ψ L 2 f L 2 H 1 m , recalling that b/ v ∈ L ∞ (
O) and using the Cauchy-Schwarz inequality, we deduce (11.20)

γf 2 L 2 (Σ;dξ 2 m ) ≤ C(1 + |λ|) f 2 L 2 H 1 m + C F L 2 H -1 m f L 2 H 1 m , for some constant C = C(b, c, m, ν), with dξ 2 m := (ν • v) 2 m 2 dvdσ x .
• For latter reference, we establish an estimate about the behaviour of the solution near the boundary. We now introduce the following Lions-Perthame [START_REF] Lions | Lemmes de moments, de moyenne et de dispersion[END_REF] type weight function (11.21) ψ := 2δ(x) 1/2 ν(x) • ṽ, and we observe that

ψ = 0 on Σ, v ψ ∈ L ∞ (O), ∇ v ψ ∈ L ∞ (O) and v • ∇ x ψ = 1 δ(x) 1/2 (v • ν(x)) 2 + 2δ(x) 1/2 v • D x ν(x)v.
Multiplying the first equation in (11.15) by f ψ, we have

1 2 O v • ∇ x f 2 ψ - O f b v • ∇ v f v ψ + O ∇ v (f ψ) • ∇ v f + O (λ -c)f 2 ψ = F, f ψ .
Using Cauchy-Schwarz and Young inequalities, we deduce

(11.22) O f 2 (v • ν(x)) 2 δ(x) 1/2 dvdx ≤ C(1 + |λ|)( f 2 L 2 H 1 + F 2 L 2 H -1 ),
for some constant C = C(b, c, n).

• We finally state a somehow classical regularity estimate when F ∈ L 2 m (O). Taking advantage of the fact that F ∈ L 2 m and f ∈ L 2 H 1 m and localizing the problem by introducing the function g

:= f χ ε ∈ L 2 x H 1 v (R d × R d ), χ ε ∈ C 2 c (O), 1 Oε ≤ χ ε ≤ 1, O ε := {(x, v) ∈ O; δ(x) > ε, |v| ≤ 1/ε}, we have v • ∇ x g -∆ v g + v 2 g = G in D ′ (R d × R d ), with G := (F -λf -cf -b • ∇ v f )χ ε -2∇ v f • ∇ v χ ε + v 2 f χ ε ∈ L 2 (R d × R d ).
From the quantitative Hormander's hypoellipticity estimate of Hérau & Pravda-Starov [210, Prop. 2.1], we then have

D 2/3 x g L 2 + D 2 v g L 2 G L 2 + g L 2 .
Coming back to the function f and using the previous estimates, we deduce (11.23)

D 2/3 x f L 2 (Oε) + D 2 v f L 2 (Oε) ≤ C( Lf L 2 (O) + f L 2 (O) ), for a constant C = C(λ, ε) > 0.
Step 2. Existence. We assume g = 0. A possible way for proving the existence is to use Lions' variant of the Lax-Milgram theorem [250, Chap III, §1] as in [START_REF] Baouendi | Sur une équation d'évolution changeant de type[END_REF][START_REF] Degond | Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in 1 and 2 space dimensions[END_REF] and as we proceed now. Defining the bilinear form E :

L 2 H 1 m (O) × C 1 c (O ∪ Σ -) → R, by E(f, ϕ) = O (λ -L)f ϕm 2 := O (λf -b • ∇ v f -cf )ϕm 2 + ∇ v f • ∇ v (ϕm 2 ) -f (v • ∇ x ϕ)m 2 , for any f ∈ L 2 H 1 m (O) and ϕ ∈ C 1 c (O ∪ Σ -), we observe that this one is coercive, namely E(ϕ, ϕ) = O (λ -̟)ϕ 2 m 2 + O |∇ v ϕ| 2 m 2 + 1 2 Σ- (γ -ϕ) 2 dξ m ≥ κ ϕ 2 L 2 H 1 m , for any ϕ ∈ C 1 c (O ∪ Σ -)
, with κ := min(λλ * , 1) > 0. From the above mentioned Lions's theorem, for any

F ∈ L 2 H -1 m , there exists f ∈ L 2 H 1 m such that (11.24) E(f, ϕ) = F, ϕm 2 , ∀ ϕ ∈ C 1 c (O ∪ Σ -).
In particular, f satisfies the first equation in (11.15)

f (L * -λ)ϕ + Fϕ) dvdx = Σ γf ϕ ν(x) • v dvdσ x ,
for any ϕ ∈ D( Ō). Particularizing to ϕ ∈ D(O ∪ Σ -) and comparing with (11.24), we deduce that γ -f = 0.

Step 3. Existence. The general case g ∈ L 2 (Σ -; dξ m ). When g ∈ C 2 c (Σ -), there exists a function h ∈ C 2 c (O ∪ Σ -) such that h |Σ-= g and we consider the source term

G := F + (L -λ)h ∈ L 2 H -1 m as well as the problem (λ -L)g = G in O, γ -g = 0 on Σ -.
From Step 2, there exists a solution g ∈ W 2,Σ to this problem and we set f := g + h, in such a way that f ∈ W 2,Σ and satisfies

O f (λ -L * )ϕ = O g(λ -L * )ϕ + O h(λ -L * )ϕ = O Gϕ + O (λ -L)hϕ - Σ h |Σ ϕ ν • v,
and thus

(11.26) O f (λ -L * )ϕ = O Fϕ - Σ- gϕ ν • v, for any ϕ ∈ C 2 c (O ∪ Σ -).
Together with (11.25), we get that γ -f = g on Σ -. In order to deal with the general case g ∈ L 2 (Σ -; dξ m ), we introduce a sequence (g n ) of C 2 c (Σ -) such that g n → g in L 2 (Σ -, dξ m ) and we next consider the associated sequence of solutions (f n ) of W 2,Σ just built above. Using the estimates exhibited in Step 1, we get that (f n ) is a Cauchy sequence in W 2 , so that it converges to a limit f ∈ W 2,Σ . We may pass to the limit in (11.26) written for the sequence (f n ) and deduce that the same equation holds at the limit for f .

Step 4. Uniqueness. Consider two weak solutions f i ∈ W 2 to the equation (11.15) in the sense that

E(f i , ϕ) = F, ϕm 2 , ∀ ϕ ∈ C 1 c (O ∪ Σ -). In particular, the difference f := f 2 -f 1 ∈ W 2 satisfies E(f, ϕ) = 0, ∀ ϕ ∈ C 1 c (O ∪ Σ -)
, and from the above discussion γ -f = 0 ∈ L 2 (Σ -; dξ m ). Thanks to the trace Theorem 11.1, we deduce that γf ∈ L 2 loc (Σ; dξ m ) and we may choose β(s) = s 2 in the Green formula (11.14): we get

O f 2 {v • ∇ x ϕ -div v (bϕ) + ∆ v ϕ + 2f (c -λ)ϕ} -2|∇ v f | 2 ϕ = Σ+ (γf ) 2 ν • vϕ, for any test function ϕ ∈ C 2 c ( Ō). Choosing ϕ = m 2 χ ̺ , with χ ̺ (v) := χ(v/̺), χ ∈ C 2 c (R d ), 1 B1 ≤ χ ≤ 1 B2 , we deduce O f 2 m 2 (λ -̟)χ ̺ + 1 2 b • ∇χ ̺ - ∇m m • ∇χ ̺ -∆χ ̺ ≤ 0.
Because f ∈ L 2 H 1 m , we may pass to the limit ̺ → ∞ thanks to the dominated convergence theorem and we obtain O f 2 m 2 (λ -̟) ≤ 0, and thus f = 0.

Step 5. Positivity. We assume now that F ≥ 0 and g ≥ 0. We proceed similarly as in the previous step by considering

β(s) = s 2 -, ϕ = m 2 χ M . Letting M → ∞, we deduce O f 2 -m 2 (λ -̟) ≤ 0,
and thus f -= 0.

Summing up, gathering the above estimates (11.18), (11.19), (11.20), (11.22), (11.23), we see that there exists a constant C > 0 such that any function f ∈ D(L) satisfies

f L 2 H 1 m + v • ∇ x f L 2 H -1 m + f v • ν δ 1/4 L 2 (11.27) + γf L 2 (Σ;dξ 2 m ) + γ + f L 2 (Σ;dξm) ≤ C( f L 2 + Lf L 2
) and for any ε > 0 there exists a constant C ε such that any function f ∈ D(L) satisfies

D 2/3 x f L 2 (Oε) + D 2 v f L 2 (Oε) ≤ C ε ( f L 2 + Lf L 2 ).
11.3. Well-posedness problem with reflection condition at the boundary. We consider now the well posedness problem associated to the stationary equation

(11.28) (λ -L)f = F in O, γ -f = Rγ + f on Σ -,
for a given datum F : O → R, where the kinetic Fokker-Planck operator L is still defined by (11.1) and the reflexion operator R is described in (11.6), (11.7), (11.8).

Theorem 11.4. Let us fix a vector field b ∈ H 1 loc ( Ō) and a function c ∈ L ∞ (O) which satisfy the assumptions of Theorem 11.3 with a given weight function m : R d → [1, ∞) for the pure specular reflection case α ≡ 0 and with the weight function m := M -1/2 when α ≡ 0, where M is the Gaussian function (11.9) or a more general equilibrium satsifying (11.10). In that last case, we furthermore assume one of the two following conditions (i) 1ζ + α 2 /2 ≥ δ * > 0, and we observe that L 2 (Σ; dξ m ) ⊂ L 1 (Σ; dξ), (ii) v 2 M ∈ L 1 , and we observe that L 2 (Σ; dξ 2 m ) ⊂ L 1 (Σ; dξ), where we recall that we have defined dξ

m := m 2 |ν(x) • v|dvdσ x and dξ 2 m := m 2 (ν(x) • v) 2 dvdσ x . For any F ∈ L 2 H -1
m and λ > λ * , there exists at least one solution f ∈ W 2 to the Dirichlet problem (11.28). Assuming furthermore that We also emphasize on the fact that the additional assumptions (i) or (ii) are made in order to prove the uniqueness of the solution during the proof. Proof of Theorem 11.4. We split the proof into four steps.

Step 1. A priori estimates. We multiply the first equation in (11.28)

(λ -̟)f 2 m 2 + 1 2 Σ (γf ) 2 m 2 ν • v + O |∇ v f | 2 m 2 = F, f m 2 .
Using for instance [55, Lem. 3.1], we have (11.30)

Σ (γf ) 2 m 2 ν • v ≥ Σ+ [(1 -ζ)(γ + f ) 2 + α(D ⊥ γ + f ) 2 ]dξ m =: E ζ,α (γ + f ) ≥ 0, with D ⊥ g := g -Dg.
Using that the contribution of the boundary is nonnegative in the first estimate, we first deduce

(λ -λ * ) f 2 L 2 m + ∇f 2 L 2 m ≤ F L 2 H -1 m f L 2 H 1 m , for λ > λ * , so that min(λ -λ * , 1) f L 2 H 1 m ≤ F L 2 H -1 m .
From the three above estimates together, for λ > λ * , we obtain (11.31)

O (λ -̟) + f 2 m 2 + O |∇ v f | 2 m 2 + 1 2 E ζ,α (γ + f ) ≤ 1 min(λ -λ * , 1) F 2 L 2 H -1 m .
There is no difficulty for also getting the pieces of information (11.19), (11.20), (11.22) and (11.23), so that in particular f ∈ W 2 . It is worth emphasizing here that when v 2 M ∈ L 1 , we have L 2 (dξ 2 m ) ⊂ L 1 (Σ; dξ) by using the Cauchy-Schwarz and (11.20), so that in particular the boundary condition is well defined.

Let us show now how the last conclusion also holds under condition (i) in the statement of the Theorem. We then assume ϑ = 1 in (11.10) and we show how to establish an additionnal a priori estimate. We indeed know from (11.20)

that Σ- (αD(γ + f )) 2 (ν • v) 2 m 2 dvdσ x ≤ Σ (γf ) 2 (ν • v) 2 m 2 dvdσ x ≤ C λ F 2 L 2 H -1 m ,
and similarly as in [START_REF] Arkeryd | On diffuse reflection at the boundary for the Boltzmann equation and related equations[END_REF] or [272,

proof of Lemma 2.2] that 1 = Σ x - |ν(x) • v|M dv = C Σ x - (ν(x) • v) 2 M dv, ∀ x ∈ ∂Ω, for some constant C ∈ (0, ∞), so that (11.32) Σ- (αD(γ + f )) 2 dξ m = C Σ- (αD(γ + f )) 2 (ν • v) 2 m 2 ≤ CC λ F 2 L 2 H -1 m .
Summing up (11.31) and (11.32), and using that

(γ + f ) 2 ≤ 2(D ⊥ γ + f ) 2 + 2(Dγ + f ) 2 ,
we deduce that (11.33)

Σ+ [1 -ζ + α 2 /2](γ + f ) 2 dξ m ≤ C λ F 2 L 2 H -1 m . Defining f ∈ W 2,R := {g ∈ W 2 ; γ -g = Rγ + g}, we see that W 2,R = W 2,Σ if 1 -ζ + α 2 /2 ≥ δ * > 0, but it is worth emphasizing that we may have W 2,R = W 2,Σ in the general case.
Step 2. Existence when F ≥ 0. With the help of Theorem 11.3, we define f 0 = 0 and, recursively for any n ≥ 1, we define f n ∈ W 2,Σ as the solution of

(11.34) (λ -L)f n = F in O, γ -f n = Rγ + f n-1 on Σ -.
It is worth emphasizing here that γ + f n-1 ∈ L 2 (Σ + ; dξ m ) implies R(γ + f n-1 ) ∈ L 2 (Σ -; dξ m ) because of (11.5). We also notice that f n ≥ 0 because F ≥ 0. By linearity

(λ -L)(f n+1 -f n ) = 0 in O, γ -(f n+1 -f n ) = Rγ + (f n -f n-1 ) on Σ -,
and we thus show recursevily that f n+1f n ≥ 0. In other words, (f n ) is an increasing sequence and thus also is (γf n ). From (11.30), we have

Σ (γf n ) 2 dξ m = Σ+ (γ + f n ) 2 dξ m - Σ- (Rγ -f n-1 ) 2 dξ m ≥ Σ+ (γ + f n ) 2 dξ m - Σ- (Rγ -f n ) 2 dξ m ≥ E ζ,α (γ + f n ),
f n → Rγ + f in L 2 (Σ -; dξ 2
m ) thanks to the monotonous convergence theorem. As a consequence, we may pass to the limit in the weak formulation of (11.34), and we get that f is a solution of (11.28). We may also pass to the liminf in the estimate (11.31) written for f n , and we thus deduce that the same estimate holds for f .

Step 3. Existence when F ∈ L 2 H -1 m . When F ∈ L 2 m , we may introduce the splitting F = F + -F -, just use the previous step for F ± and conclude by linearity of the equation. When F / ∈ L 2 m , we proceed similarly as in [START_REF] Mischler | Kinetic equations with Maxwell boundary conditions[END_REF] and in the following way. We first assume ζ ≤ ζ * ∈ [0, 1) and we consider the mapping Ψ : W 2,Σ → W 2,Σ , g → f = Ψ(g), where f is the solution to the stationary problem (11.35

) (λ -L)f = F in O γ -f = Rγ + g on Σ -.
The space W 2,Σ is endowed with the norm • W2,Σ defined by

g 2 W2,Σ = g 2 L 2 m + ∇ v g 2 L 2 m + γ + g 2 L 2
m (dξ1) . From (11.18) and the estimate Rg L 2 (Σ-;dξm) ≤ ζ * g L 2 (Σ+;dξm) what we obtain by repeating the proof of (11.5), we deduce 1

C λ f 2 L 2 H 1 m + γ + f 2 L 2 (Σ+;dξm) ≤ C λ F 2 L 2 H -1 m + Rγ + g L 2 (Σ-;dξm) ≤ C λ F 2 L 2 H -1 m + ζ * γ + g L 2 (Σ+;dξm) ,
for some cosntant C λ > 0. By linearity of (11.35), we deduce that for two functions g 1 , g 2 ∈ W 2,Σ , and denoting

f i := Ψ(g i ), we have 1 C λ f 2 -f 1 2 L 2 H 1 m + γ + f 2 -γ + f 1 2 L 2 (Σ+;dξm) ≤ ζ * γ + g 2 -γ + g 1 2 L 2 (Σ+;dξm) , so that Ψ is a contraction in W 2,
f n 2 L 2 H 1 m + γf n 2 L 2 (Σ;dξ 2 m ) + E 1,α (γ + f n ) ≤ C λ F 2 L 2 H -1 m .
When α ≡ 0, the above estimate or (11.33) also implies that (γ + f n ) belongs to a weakly compact set of L 1 (Σ + ; dξ). As a consequence, there exist f ∈ W 2 and γ± two functions defined on Σ ± such that, up to the extraction of a subsequence,

f n ⇀ f L 2 H 1 m , γ ± f n ⇀ γ± L 2 (Σ ± ; dξ 2 m ), γ + f n ⇀ γ+ L 1 (Σ + ; dξ), Rγ + f n ⇀ R γ+ L 1 (Σ -; dξ),
where we have used (11.11) for the last convergence. Using Proposition 11.2, we may thus pass to the limit in the equation (11.28) satisfied by f n with modified reflection kernel R n and we get that f is a solution of (11.28). In the pure specular reflection case α ≡ 0, only the first line of convergences holds, but that it is enough in order to pass to the limit in the equations (we refer to [START_REF] Mischler | On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system[END_REF][START_REF] Mischler | Kinetic equations with Maxwell boundary conditions[END_REF] for similar arguments).

Step 4. Other properties. We further assume λ > λ * * . We proceed similarly as in [START_REF] Mellet | Uniqueness and semigroup for the Vlasov equation with elastic-diffusive reflexion boundary conditions[END_REF]. Consider two weak solutions f i ∈ W 2 to the equation (11.28). In particular, the difference f :

= f 2 -f 1 ∈ W 2 satisfies (λ -L)f = 0 in O, γ -f = Rγ + f on Σ -.
Using the Green renormalized formula (11.14), we have

0 = O β ′ (f )(λ -c)f ϕ + β ′′ (f )|∇f | 2 ϕ + β(f )(div v (bϕ) -v • ∇ x ϕ -∆ v ϕ) + Σ β(γf )ν • vϕ. for any β ∈ C 2 (R), β ′ ∈ C 1 b (R) and any test function ϕ ∈ C 2 c ( Ō). We choose ϕ = ϕ(v) ≥ 0, β ≥ 0 and β ′′ ≥ 0, so that 0 ≥ O β ′ (f )(λ -c)f ϕ + β(f )(div v (bϕ) -∆ v ϕ) + Σ β(γf )ν • vϕ.
By an approximation argument, we may now take β(s) = |s|, and we get 0

≥ O |f | (λ -c)ϕ + (div v (bϕ) -∆ v ϕ) + Σ |γf |ν • vϕ.
We observe that in any cases we have f ∈ L 2 m (O) ⊂ L 1 (O) and γf ∈ L 1 (Σ; dξ). By an approximation argument, we may now take ϕ = 1 and using the L 1 estimate (11.11) on R (with ζ * = 1), we get

0 ≥ Σ- |Rγ + f ||ν • v| - Σ+ |γ + f ||ν • v| ≥ O |f | λ -c + div v b ≥ (λ -λ * * ) O |f |.
We deduce that f = 0. The proof of the positivity property follows the same arguments but choosing β(s) = s -.

For latter reference, we state the counterpart of the preceding result for the kinetic Fokker-Planck evolution equation.

∂ t f = Lf in (0, ∞) × O γ -f = Rγ + f on (0, ∞) × Σ -,
with L defined in (11.1) and R defined in (11.6).

The proof of Theorem 11.

+ v • ∇ x f -∆ v f -b • ∇ v f -cf = 0 in O γ -f = Rγ + f on Σ -,
and the associated dual problem. In this section, we assume that b and c satisfy the assumptions of Theorem 11.3 with the weight function m := M -1/2 when α ≡ 0 and for a given weight function m : R d → [1, ∞) when α ≡ 0 and R is given by (11.6). We additionnally assume that (11.38) lim inf

|(x,v)|→∞ ̟(x, v) = -∞,
where we recall that ̟ is defined in (11.17). When M is the Gaussian function, we find

̟ = c + |v| 2 + d 2 - 1 2 divb -b • v,
so that (11.38) holds when b is typically a bounded perturbation of the vector field b 0 (v) = ϑ 0 v, ϑ 0 > 1/2, and more precisely

div v b ∈ L ∞ (O) and inf x∈Ω lim inf |v|→∞ (b • v v -2 ) ≥ ϑ 0 > 1/2.
The above condition is quite technical but can be seen as a compatibility condition between the thermalization due to the boundary and to the Fokker-Planck collisional operator. We are then able to work in the functional space X := L 2 m (O). Theorem 11.6. Under the above conditions, the first eigentriplet problem associated to (11.1) has a unique solution (λ 1 , f 1 , φ 1 ) ∈ R × X × X ′ with f 1 > 0 and φ 1 > 0.

The proof of Theorem 11.6 follows from Theorem 2.21, Theorem 4.13 and Theorem 5.16 as a consequence of conditions (H1)-(H5). We prove now that each of these conditions is satisfied. Theorem 11.6 generalizes [START_REF] Lelièvre | Quasi-stationary distribution for the Langevin process in cylindrical domains, Part I: Existence, uniqueness and long-time convergence[END_REF]Thm. 2.12] where the same problem is tackled for the zero inflow condition (α = β = 0) with b = v -F (x) and c = 1 by using the classical Krein-Rutman theorem [START_REF] Kreȋn | Linear operators leaving invariant a cone in a Banach space[END_REF] in the space X = C b ( Ō). We also refer to [START_REF] Guillin | Quasi-stationary distribution for strongly feller markov processes by lyapunov functions and applications to hypoelliptic hamiltonian systems[END_REF]Thm. 6.8] for a variant and somehow generalisation of [START_REF] Lelièvre | Quasi-stationary distribution for the Langevin process in cylindrical domains, Part I: Existence, uniqueness and long-time convergence[END_REF].

Condition (H1). From Theorem 11.4, the operator L satisfies (H1) with κ 1 := max(λ * , λ * * ), with λ * defined by (11.17) and λ * * defined by (11.29). For later reference, let us state more precisely the available estimates for f . On the one hand, repeating the proof of Step 1 in the proof of Theorem 11.4, we establish that for any λ > κ 1 and F ∈ L 2 m , the solution f ∈ W 2 to the Dirichlet problem (11.28) satisfies (11.39)

O (λ -̟) + f 2 m 2 + O |∇ v f | 2 m 2 + 1 2 E ζ,α (γ + f ) ≤ 1 λ -λ * F 2 L 2 m .
On the other hand, adapting the proof of (11.22), we straightforwardly obtain

U c f 2 m 2 ≤ f 2 m 2 1 |v|≥̺ + f 2 m 2 1 Ax + f 2 m 2 1 B , with
A x := {v ∈ B ̺ , (v • ν(x)) 2 ≤ ε 2 v }, B := {(x, v); |v| ≤ ̺, (v • n) 2 ≥ ε 2 v , d(x, ∂Ω) ≤ ε x }.
For the second term, we have

f 2 m 2 1 Ax ≤ |A x | 2/r ′ f (x, •) 2 L r v dx (̺ d-1 ε v ) 2/r ′ f 2 L 2 H 1 m ,
where we have used the Holder inequality with r ∈ (1, 2 * /2) in the first line and the Sobolev inequality in the second line. For the third term, we have

f 2 m 2 1 B ≤ m 2 (̺) ε 1/2 x ε 2 v O f 2 n (v • ν(x)) 2 δ(x) 1/2 .
Gathering these last estimates with (11.39) and (11.40), we have established that the solution f to equation (11.28) furthermore satisfies (11.42)

U c f 2 m 2 ≤ C 1 ̺ 2 + ̺ d-1 ε v + m 2 (̺) ε 1/2 x ε 2 v F 2 m 2 ,
for a constant C = C(b, c, Ω, λ) and for any ε x , ε v , ̺ > 0.

The strong maximum principle. Let us now consider a function 0 ≤ f ∈ W 2 \{0} which satisfies the Dirichlet problem (11.28) associated to λ > κ 1 and a source term 0 ≤ F ∈ L 2 m ∩L ∞ . In order to simplify the discussion, we assume that the normalization f L 2 m = 1 holds. For proving the strong maximum principle, we briefly explain how we may adapt the arguments we have presented for the diffusive equation in Part 7 by taking in particular advantage of the above established estimates, the regularity results established in [START_REF] Golse | Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation[END_REF][START_REF] Guerand | Quantitative De Giorgi methods in kinetic theory[END_REF] and some spreading positivity results we learnt in [353,Cor. A.20]. We proceed in three steps.

Step 1. On the one hand, from (11.42), we may choose conveniently ̺ -1 , ε v , ε x > 0 small enough in such a way that

U c f 2 m 2 ≤ 1 2 f 2 L 2
m , where U is defined by (11.41). Because of the normalization condition, we have (11.43)

U f 2 m 2 ≥ 1 2 f 2 L 2 m and consequently f (x 0 , v 0 ) 2 ≥ δ 2 0 := f 2 L 2 m (2 1 U 2 L 2 m
) -1 for at least one point (x 0 , v 0 ) ∈ U.

Step 2. On the other hand, let us recall some integrability and regularity results established in [START_REF] Golse | Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation[END_REF] for a solution g to the kinetic Fokker-Planck evolution equation

∂ t g + v • ∇ x g = ∆ v g + B • ∇ v g + s in V,
or a sub-solution ∂ t g + v • ∇ x g ≤ ∆ v g + B • ∇ v g + s in V, for some bounded set V ⊂ (0, T ) × O, s ∈ L 2 (V) and B ∈ L ∞ (V). For that purpose, given some (t * , x * , v * ), we define Q r := {(t, x, v); t ∈ (t *r 2 , t * ], |xx * -(tt * )v * | < r 3 , |vv * | < r}.

We claim then that there exist 2 < p < q < ∞, α ∈ (0, 1) and for any 0 < r 1 < r 0 there exists C such that (11.44) g L p (Qr 1 ) ≤ C ( g L 2 (Qr 0 ) + s L 2 (Qr 0 ) )

for any nonnegative subsolution g on Q r0 from [181, Thm. 6],

(11.45) g L ∞ (Qr 1 ) ≤ C ( g L 2 (Qr 0 ) + s L q (Qr 0 ) )

for any nonnegative subsolution g on Q r0 from [181, Thm. 12] and

f L p (U ) ≤ C 0 ( f L 2 (O) + F + cf -λf L 2 (O) ) ≤ C 1 ( f L 2 (O) + F L 2 (O) ).
Observing that for ̺ = p/2 > 1, we have

v • ∇ x f ̺ -∆ v f ̺ -b • ∇ v f ̺ + ̺f ̺-1 (λf -cf -F) = -4 (̺ -1) ̺ |∇(f ̺/2 )| 2 ≤ 0,
so that f ̺ is a weak sub-solution to the kinetic Fokker-Planck equation, we may repeat the argument and obtain in that way that f ∈ L p k (U) for any k ≥ 1, with p k := ̺ k 2. Now, choosing k such that p k ≥ q and using (11.45) (as well as again a classical covering argument), we get

f L ∞ (U ) f L 2 (O) + F + cf -λf L q (O) f L 2 (O) + F L q (O) .
Using finally (11.46), we deduce that there exists a constant C = C(U, λ) such that

f C α (U ) f L 2 (O) + F L ∞ (O) .
Together with the conclusion of the first step, we deduce that there exists a constructive constante r 0 > 0 such that f ≥ δ 0 1 B((x0,v0),r0) .

Step Taking advantage of the fact that h 0 has compact support, we compute

1 = O h 2 0 m 2 = O (κ 1 -L)f 0 h 0 m 2 = O f 0 (κ 1 -L * )(h 0 m 2 ) ≤ C 1 f 0 L 2 m ,
with C 1 := m -1 (κ 1 -L * )(h 0 m 2 ) L 2 . On the other hand, from (11.27), we have (11.47)

f 0 L 2 H 1 m + f 0 v • ν δ 1/4 L 2 ≤ C 2 ,
for a constant C 2 only depending on h 0 L 2 m , κ 1 and the constant C which appears in (11.27). Arguing as in (11.43), we deduce that (11.48) U f 2 0 m 2 ≥ (2C 1 ) -1 , supp h 0 ⊂ U, with U = U ̺ defined in (11.41) and ̺ > 0 small enough (chosen constructively from C 2 and C 1 ).

From the above constructive strong maximum principle, we deduce that f 0 ≥ ε1 U ≥ 1/C 0 h 0 for some ε, C 0 > 0. We conclude as in the second constructive argument for (H2) in Section 7.1.

Coming back indeed to the equation, we have

Lf 0 = κ 1 f 0 -h 0 ≥ κ 1 f 0 -h 0 L ∞ 1 U ≥ (κ 1 -h 0 L ∞ C 0 )f 0 ,
so that (H2) holds with κ 0 := κ 1h 0 L ∞ C 0 from Lemma 2.4-(ii).

Condition (H3). Let us fix κ < κ 0 arbitrary. We define Bf := Lfnχ R (v)f for any f ∈ W 2,R , with χ R ∈ D(R d ) such that 1 BR ≤ χ R ≤ 1 B2R and for some given n, R ≥ 0 to be specified below. We observe that, at least formally,

f m 2 (B -κ)f = O (̟ -κ -nχ R )f 2 m 2 - 1 2 Σ (γf ) 2 m 2 ν • v - O |∇ v f | 2 m 2 .
Thanks to (11.38), there exists a constant R > 0 such that sup

v∈R d \BR ̟ ≤ κ.
Choosing n := sup ̟ +κ, we deduce that ̟κnχ R ≤ 0. On the other hand, because of (11.30), the contribution of the boundary term in the above identity is non positive. We thus deduce that (Bκ) is dissipative in L 2 m . We now establish that the associated operator B has compact resolvent. For F ∈ L 2 m , we consider f ∈ L 2 m the solution to (11.

f 2 ̟ -m 2 + 2 |∇ v f | 2 m 2 ≤ F 2 m 2 .
Together with the regularity estimate (11.23) and the compact imbedding H 2/3 (U) ⊂ L 2 (U), we conclude that B has compact resolvent. The operator A on L 2 m defined by Af := nχ R (v)f being bounbded, we may apply Lemma 2.8-(2) and we deduce that (H3) holds for both the primal and the dual problems.

Condition (H4) is nothing but the yet established strong maximum principle.

A variant of condition (H5). Consider (f, λ) a pair of eigenfunction and eigenvalue such that λ ∈ Σ P + (L). Arguing similarly as in the proof of condition (H5) in Section 7.1, we know that Lf = iϑf, ϑ ∈ R, L|f | = 0 and introducing the real and complex part decomposition f = g + ih, we have

O 1 |f | 2 |g∇ v h -h∇ v g| 2 = 0,
and finally g∇ v h -h∇ v g = 0 a.e. on O. Because of the regularity estimate presented during the above proof of the strong maximum principle, the functions f has Hölder regularity, and thus g and h are continuous on O. Because |f | ≡ 0, we may claim that there exists a point (x 0 , v 0 ) ∈ O such that h(x 0 , v 0 ) > 0 for instance. Denoting by ω the connected component of {(x, v) ∈ O; h(x, v) > 0} containing (x 0 , v 0 ), we have ∇(g/h) = 0 on ω, and thus g = α(x)h on ω for some continuous function α : Ω → R. Coming back to the eigenvalue equation that we may write in the following system form Lg = -ϑh, Lh = ϑg, we compute -ϑh = L(αh) = α Lhhv • ∇ x α = αϑghv • ∇ x α on ω, so that -ϑ = α 2 ϑv • ∇ x α on ω. We deduce that α is a constant on ω and finally ϑ = 0. We have thus established that λ = λ 1 .

At this stage, we may use Theorem 2.21, Theorem 4.13 and Theorem 5.16, in order to get the conclusions (C1), (C2) and (C3) about the existence and uniqueness of the eigentriplet (λ 1 , f 1 , φ 1 ) which satisfies f 1 > 0, φ 1 > 0, λ 1 is algebraically simple and on the triviality of the boundary punctual spectrum.

We briefly explain how we may deduce the stability of f 1 by adapting some arguments developped in [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF] and already mentioned. On the one hand, we know from [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF]Lem. 1.1] that any solution f to the rescaled evolution equation (11.36) with L replaced by L = Lλ 1 satisfies ∂ t (H(X)f 1 φ 1 ) + div x (vH(X)f 1 φ 1 )div v (φ 2 1 ∇ v (H(X)f 1 /φ 1 )) = -H ′′ (X)f

1 φ 1 |∇ v X| 2 ,
for any convex function H : R → R and with X := f /f 1 . After integration, we get

(11.51) d dt O H(X)f 1 φ 1 + Σ ν • vH(X)f 1 φ 1 = - O H ′′ (X)f 1 φ 1 |∇ v X| 2 ,
When H(s) := |s|, the boundary term is

Σ |γf |γφ 1 ν • v = Σ+ |γ + f |R * γ -φ 1 ν • v - Σ- |Rγ + f |γ -φ 1 |ν • v| ≥ Σ+ |γ + f |R * γ -φ 1 |ν • v| - Σ- R|γ + f |γ -φ 1 |ν • v| = 0,
dt dt O f 1 φ 1 (f /f 1 ) 2 + 2 O f 1 φ 1 |∇ v (f /f 1 )| 2 ≤ 0.
We next recall a classical compactness result.

Lemma 11.7. Let (g n ) be a sequence of functions such that (g n ) is bounded in L ∞ (0, T ; L 2 xv,loc ) ∩ L 2 (0, T ; L 2 x,loc H 1 v,loc ) and ∂ t g n + v • ∇ x g -∆ v g n = G n bounded in L 2 loc , then (g n ) belongs to a strong compact set of L 2 loc . Proof of Lemma 11.7. We just sketch it. Because

∂ t g n + v • ∇ x g = ∆ v g n + G n bounded in L 2
tx H -1 v , the usual averaging lemma in [START_REF] Golse | Regularity of the moments of the solution of a transport equation[END_REF][START_REF] Diperna | Global weak solutions of Vlasov-Maxwell systems[END_REF] implies that (g n * ρ) belongs to a strong compact set of L 2 loc , for any ρ ∈ D(R d ). On the other hand, introducing a mollifiers sequence (ρ ε ) and writing then g n = (g ng n * ρ ε ) + g n * ρ ε , we see that the first term is small uniformly in n as ε → 0 and the second term is relativelly compact thanks to the first step, from what we immediately conclude. Now, for 0 ≤ f 0 ∈ L 1 φ1 , we introduce the sequence f 0,k := (f 0 ∧ k)1 U k ∈ L 2 (f -1 1 φ 1 ) ∩ L 2 , with U k := {(x, v) ∈ O; δ(x) > 1/k, |v| ≤ k}, and the associated solution f k ∈ L ∞ (0, T ; L 2 ) ∩ L 2 (0, ∞; L 2

x H 1 v ). Because of (11.53), for any increasing sequence (t n ) which converges to ∞ and for any function ϕ m ∈ D(O), 1 Um ≤ ϕ m ≤ 1, the rescaled and truncated function g n (t) := f k (t + t n )f -1 1 e -λ1(t+tn) ϕ m meet the hypothesis of Lemma 11.7, from what we classically deduce that the sequence of fn (t) := f k (t + t n )f -1 1 e -λ1(t+tn) is relatively strongly compact in L 2 loc . Repeating the proof of Theorem 4.23 and Theorem 5.23 (see also [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF]Thm. 3.2]), we deduce that fn (t) → f 0,k , φ 1 f 1 as t → ∞. Together with the above non expansive property (11.53), we deduce that f t → f 0 , φ 1 f 1 in L 1 φ1 as t → ∞. We summarize our convergence result in the following theorem.

Theorem 11.8. For any f 0 ∈ L 2 m , the holds f t → f 0 , φ 1 f 1 in L 1 φ1 as t → ∞.

Theorem 11.8 generalizes [START_REF] Lelièvre | Quasi-stationary distribution for the Langevin process in cylindrical domains, Part I: Existence, uniqueness and long-time convergence[END_REF]Thm. 2.18] for the zero inflow condition and [5, Thms. 1.6 & 1.7] for the torus case. It is worth emphasizing that in these papers the longtime convergence is established with exponantial rate (with constructive estimate in [START_REF] Albritton | Variational methods for the kinetic fokker-planck equation[END_REF]). In [START_REF] Lelièvre | Quasi-stationary distribution for the Langevin process in cylindrical domains, Part I: Existence, uniqueness and long-time convergence[END_REF] the proof is based on a representation formula for the associated semigroup S which is proved to have a kernel p t ∈ (L 1 ∩ L ∞ ∩ C ∞ )(O) for any t > 0 (see [START_REF] Lelièvre | Quasi-stationary distribution for the Langevin process in cylindrical domains, Part I: Existence, uniqueness and long-time convergence[END_REF]Thms. 2.4 & 2.6] as well as [START_REF] Rey-Bellet | Ergodic properties of Markov processes[END_REF][START_REF] Herzog | A practical criterion for positivity of transition densities[END_REF][START_REF] Lelièvre | A probabilistic study of the kinetic Fokker-Planck equation in cylindrical domains[END_REF]). One then classically deduces that S t ∈ K (X) for any t > 0 and X = L p , p ∈ [1, ∞], or X = C 0 (see [START_REF] Lelièvre | Quasi-stationary distribution for the Langevin process in cylindrical domains, Part I: Existence, uniqueness and long-time convergence[END_REF]Thm. 2.18]), and next one may apply Theorem 5.28. We also refer to [START_REF] Guillin | Quasi-stationary distribution for strongly feller markov processes by lyapunov functions and applications to hypoelliptic hamiltonian systems[END_REF]Thm. 6.8], [START_REF] Nier | Boundary conditions and subelliptic estimates for geometric Kramers-Fokker-Planck operators on manifolds with boundaries[END_REF] and [START_REF] Hwang | The Fokker-Planck equation with absorbing boundary conditions[END_REF][START_REF] Hwang | Nonuniqueness for the kinetic Fokker-Planck equation with inelastic boundary conditions[END_REF][START_REF] Hwang | On the structure of the singular set for the kinetic Fokker-Planck equations in domains with boundaries[END_REF] for related results.

We follow now a similar approach as in [START_REF] Lelièvre | Quasi-stationary distribution for the Langevin process in cylindrical domains, Part I: Existence, uniqueness and long-time convergence[END_REF][START_REF] Guillin | Quasi-stationary distribution for strongly feller markov processes by lyapunov functions and applications to hypoelliptic hamiltonian systems[END_REF]. We start with a series of technical results. Here, we make the additional assumption Proof of Lemma 11.9. We first recall from Step 1 of the proof of Theorem 11.3 and (11.30) that

(Lf, f ) L 2 m = -|∇f | 2 m 2 - 1 2 Σ (γf ) 2 m 2 ν • v + f 2 ̟m 2 ≤ -|∇f | 2 m 2 + f 2 ̟m 2
and, with ψ defined in (11.21),

(-Lf, f ) L 2 ψ = - 1 2 f 2 (v • ∇ x ψ) -f b v • ∇ v f v ψ + ∇ v (f ψ) • ∇ v f -cf 2 ψ ≤ -f 2 (v • ν(x)) 2 δ(x) 1/2 dvdx + C (f 2 + |∇f | 2 ).
Defining then m := m-βψ, with β > 0 small enough, and summing up the two previous estimates, we get

(Lf, f ) L 2 m ≤ -β f 2 (v • ν(x)) 2 δ(x) 1/2 - 1 2 |∇f | 2 m 2 + f 2 (̟m 2 + 1).
Similarly as in (11.41), we define

U := {(x, v) ∈ O; δ(x) > ̺ x , |v| < ̺ v },
and we observe that

U c ⊂ A ∪ B ∪ C, with A := {v ∈ B ̺v , |v • ν(x)| ≤ ε v }, B := {v ∈ B ̺v , |v • n| ≥ ε v , δ(x) ≤ ̺ x },
for some ε x > 0, and C := B c ρv . We next repeat the proof of (11.42), and we get

U c f 2 m 2 (̺ d-1 v ε v ) 2/r ′ |∇ v f | 2 + m(̺ v ) 2 ̺ 1/2 x ε 2 v f 2 (v • ν(x)) 2 δ(x) 1/2 + 1 ̟ -(ρ v ) f 2 ̟ -m 2 .
Observing that

f 2 (̟m 2 + 1) ≤ κ f 2 m2 + C κ U f 2 m 2 + C κ U c f 2 m 2
with C κ := sup(̟ + 2κ) + < ∞, and A ≥ C κ 1 U for n := C κ , altogether, we conclude with (Bf, f ) L 2 m ≤ κ f L 2 m . We then classically deduce that (11.55) holds. Similarly as for the first estimate and in the proof of [START_REF] Mischler | Exponential stability of slowly decaying solutions to the kinetic-Fokker-Planck equation[END_REF]Lem. 3.8], for any smooth, rapidly decaying and positive function f , we have From Darozès-Guiraud (or Jensen) inequality, we know that the first (boundary) term is nonpositive (see [START_REF] Darrozès | Généralisation formelle du théor me h en présence de parois[END_REF] or [START_REF] Mischler | Kinetic equations with Maxwell boundary conditions[END_REF]Rem. 6.4]) and we then classically conclude to (11.56).

Lemma 11.10. There exists a finite family 2 = p 0 < p 1 < • • • < p k < ∞ and α ∈ (0, 1) such that such that both C = B, L, for any T > τ > 0 and V ⊂⊂ O,

f L p 1 (U0) ≤ C0 f L 2 (U1) + s L 2 (U1) .
The estimate (11.57) for j = 1 then follows from Theorem 11.5 (and the classical underlying energy estimate). On the other hand, [START_REF] Golse | Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation[END_REF]Thm. 12] similarly implies that there exists a constant Ck > 0 and p k ∈ (p 1 , ∞) such that f L ∞ (U0) ≤ Ck f L 2 (U1) + s L p k (U1) , and interpolating with the previous estimate, we get f L p j (U0) ≤ Cj-1 f L 2 (U1) + s L p j-1 (U1) , ∀ j, 2 ≤ j ≤ k -1.

The growth bound (11.56) and the two last estimates imply (11.58) and (11.57) for any 2 ≤ j ≤ k -1. Finally, [START_REF] Golse | Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation[END_REF]Thm. 3] similarly implies that there exists a constant Ck+1 > 0 and α ∈ (0, 1) such that f C α (U0) ≤ Ck+1 f L 2 (U1) + s L ∞ (U1) , from what we deduce (11.59) in the same way. ≤ C T f 0 L 2 m , where we have used (11.56) in the first line, the Fubini theorem in the second line, (11.57) with j = 1 in the third line and several times (11.55) in the last line. For κ < κ 0 , we may choose ε > 0 small enough such that (11.55) holds. From the very definition of A and S B , we may thus fix κ B ∈ (κ, κ 0 ) arbitrary and next T > 0 large enough such that V (T ) B(L 2 m ) ≤ 1 3 e κBT . We may next use (11.55) and fix τ > 0 small enough such that K c 1 (T ) B(L 2 m ) ≤ τ C T ≤ 1 3 e κBT . Last, because of (11.60), we may fix ν > 0 small enough, in such a way that

AΞ k+1 f 0 L ∞ m ,
thanks to (11.59). The three last estimates together and the compact support property suppχ ν ⊂⊂ O imply

Kf 0 C α ∩L 2 mp 1 f 0 L 2 m , ∀ f 0 ∈ L 2 m
, from what we deduce that K ∈ K (L 2 m ). We may apply Theorem 5.28 in order to conclude.

A mutation-selection model

In this section, we consider the mutation-selection evolution equation associated to the mutationselection operator (12.1) Lf := J * f -W (x)f defined on functions f : R d → R, where J is a the mutation kernel, * stands for the convolution operator and W is a confining potential.

12.1. Almost regular mutation kernel. We assume that the mutation kernel J is a positive finite measure of R d which is lower bounded on a neighborhood of the origin, or in other words (12.2) 0 ≤ J ∈ M 1 (R d ), J ≥ J * 1 Br , for some constants J * , r > 0. We also assume that the selection potential W : R d → R is continuous and satisfies (12.3) W (x) > W (0) = 0, ∀ x = 0, W (x) → +∞ as |x| → ∞.

We finally assume the following compatibility condition between mutation and selection: there exist β > 0 and a bounded Borel set A ⊂ R d such that a := ess inf 2) admits a unique solution (λ 1 , f 1 , φ 1 ) ∈ R × X + × X ′ + with the normalization φ 1 = φ 1 , f 1 = 1, and this triplet additionally satisfies

λ 1 ≥ κ 0 , 0 < f 1 ∈ L 1 W (R d ) ∩ L ∞ W (R d ) and 0 < φ 1 ∈ L 1 W (R d ) ∩ L ∞ W (R d ).
(2) Moreover, L generates a semigroup S L on X and for any f 0 ∈ X, there holds

(12.6) e -λ1t S L (t)f 0 -φ 1 , f 0 f 1 L 1 ≤ Ce -αt f 0 -φ 1 , f 0 f 1 L 1 ,
for any t ≥ 0 and for some constructive constants C ≥ 1, α > 0.

Let us comment on the above result.

Remark 12.2.

(1) Assumption (12.4) is satisfies when W is small enough in a neighborhood of the origin. It is for instance satisfied if W -1 / ∈ L 1 (B 1 ). That is in particular the case in dimension d = 1 when W is Lipschitz, because of the condition W (0) = 0.

(2) Assume J(x) = ε -d ρ(ε -1 x) with ρ ∈ C 1 c (R d ) ∩ P(R d ) and ρ ≥ ρ * 1 B1 , ρ * > 0, so that J = J 1 and J 2 = 0, and W = W (|x|). We may observe that for β > 0 and ε > 0 small enough inf β≤W (x)<2β β≤W (y)<2β ]. On the other hand, the conditions on J are relaxed here since J may have singular part in (12.5).

(4) Optimal conditions linking J and W for the existence of a spectral gap are still unknown. In the recent paper [START_REF] Alfaro | Confining integro-differential equations originating from evolutionary biology: ground states and long time dynamics[END_REF], using variational methods in a L 2 framework, the authors obtain a quantified spectral gap and the associated exponential stability when the mutation kernel J is additionally assumed to be symmetric. Up to our knowledge, Theorem 12.1 is the very first result providing a quantified spectral gap for a non-symmetric mutation kernel J.

(5) Condition (12.4) can be compared to the condition ā := ess sup

x∈R d R d J(x -y) W (y) dy < 1,
under which no first eigenfunction may exist in X. First, we claim that λ 1 ≥ 0. Indeed, considering ε > 0 and f ε = 1 Bε , we have

Lf ε ≥ -inf Bε W f ε ,
so that the condition (H2) holds for κ 0 =inf Bε W for any ε > 0. Since W is continuous and W (0) = 0, we deduce that λ 1 ≥ 0 by passing to the limit ε → 0. Assume now by contradiction that there exists f 1 ∈ X + \ {0} such that (12.7)

λ 1 f 1 = Lf 1 = J * f 1 -W f 1
and define, for any ε > 0, the function ϕ ε (x) = 1 ε+W (x) ∈ L ∞ (R d ). Testing (12.7) against ϕ ε we get for any ε ∈ (0, 1)

0 ≤ λ 1 f 1 , ϕ 1 ≤ λ 1 f 1 , ϕ ε = J(x -y) ε + W (x) f 1 (y) dx dy - W (x) ε + W (x) f 1 (x) dx ≤ ā f 1 - W (x) ε + W (x)
f 1 (x) dx, and passing to the limit ε → 0 we obtain the contradiction 0 ≤ λ 1 f 1 , ϕ 1 ≤ (ā -1) f 1 < 0. However, there always exists a principal eigenvector f 1 in M 1 (R d ), which might have an atom at the origin when ā < 1, see for instance [START_REF] Bürger | Stationary distributions under mutation-selection balance: Structure and properties[END_REF].

The proof of Theorem 12.1 follows from Theorem 2.21, Theorem 4.13 and Theorem 5.16 as a consequence of conditions (H1)-(H5) that we establish now. Setting D(L) := L 1 W (R d ), we observe that L is an unbounded closed operator with dense domain D(L). Condition (H1) and (H1 ′ ). We define the semigroup S W (t)f (x) := e -W (x)t f (x), ∀ f ∈ L p , p ∈ [1, ∞], which is clearly a positive semigroup of contractions. We next define S L as a bounded perturbation of S W . It is also positive and it satisfies the growth estimate S L (t) B(L p ) ≤ e J 1t , where we recall that J 1 stands for the L 1 norm or the total variation norm of J. We deduce that (H1) holds true with κ 1 := J 1 thanks to Lemma 2.2-(i). Multiplying Lf by sign f , for f ∈ D(L), we immediately get Kato's inequality

(sign f )Lf = (sign f )J * f -W |f | ≤ J * |f | -W |f | = L|f |.
Condition (H2). Let us define f 0 := 1 W (x) 1 A β , where A β is introduced in condition (12.4). We compute

Lf 0 = J * 1 A β 1 W -1 A β ≥ J * 1 A β 1 W -1 1 A β ≥ ess inf x∈A β J * (1 A β 1 W ) -1 1 A β = (a -1)1 A β ≥ (a -1) β W 1 A β = κ 0 f 0 ,
where in the second equality we have used the very definition of a in assumption (12.4). We conclude that (H2) holds thanks to Lemma 2.4-(ii). Condition (H3). We introduce the splitting Next, observing that (W + α)h = (α -B)h + J 2 * h, for any h ∈ D(L) and α ≥ κ 0 , we deduce that (12.10) (W + α)(α -B) -1 g = g + J 2 * ((α -B) -1 g), for any g ∈ X and α ≥ κ 0 . Together with (12.9), we deduce

(12.11) (α -B) -1 g L p W ≤ g L p + J 2 * ((α -B) -1 g) L p ≤ α α -κ * g L p ,
for any g ∈ L p and α ≥ κ 0 . Defining W(α) := (α -B) -1 A, we finally deduce from (12.10) the identity

W(α)f = 1 W + α Af + 1 W + α J 2 * ((α -B) -1 Af ),
for any f ∈ X and α ≥ κ 0 . We may then compute

W(α)f L ∞ ≤ 1 α Af L ∞ + 1 α J 2 1 (α -B) -1 Af L ∞ ,
and together with (12.9) for p = ∞ and (12.11), we deduce (12.12)

W(α)f L ∞ ≤ J 1 ∞ 1 α -κ * f L 1 ,
for any f ∈ X and α ≥ κ 0 . Starting from the same identity, we prove in a similar way (12.13)

W(α)f L ∞ W ≤ J 1 ∞ α α -κ * f L 1 ,
for any f ∈ X and α ≥ κ 0 . As a conclusion and gathering (12.9), (12.11), (12.12) and (12.13), we have established that (12.14) W(α) : L 1 → L 1 W ∩ L ∞ W , with uniform bound for any α ≥ κ 0 . Observing that L 1 W ∩L ∞ W ⊂ L 1 is weakly compact and using Lemma 2.13 with p = 1, we deduce that (H3) holds. We can actually strengthen the compactness by noticing that A : L 1 → L 1 W ∩ W 1,1 is bounded because of assumption (12.5). This ensures that A : L 1 → L 1 is compact, from what we deduce that W(α) : L 1 → L 1 is strongly compact for all α ≥ κ 0 . We may thus apply Lemma 2.8-(2) to infer that condition (H3) holds for both the primal and the dual problems.

Condition (H4). Assume that λ ≥ λ 1 and f ∈ D(L) = L 1 W satisfy (12.15)

f L 1 = 1, f ≥ 0, (λ -L)f ≥ 0. Denoting W R := inf B c R W , we compute BR f ≥ R d f - 1 W R B c R f W ≥ 1 - 1 W R f L 1 W ≥ 1/2,
for R > 0 large enough by taking advantage of the fact that W (x) tend to infinity when |x| → ∞.

In particular, there exists x f 0 ∈ B R such that

B r/2 (x f 0 ) f ≥ δ := 1 2 r 2R d > 0,
where we recall that r is defined in (12.2). We deduce that

(J * f )(x) ≥ J * B r/2 (x f 0 )
f (y)dy1 B r/2 (x f 0 ) (x) ≥ J * δ1 B r/2 (x f 0 ) (x).

Using the equation (12.15), we obtain That means that the (H4) holds, with constructive lower bound.

f (x) ≥ (J * f )(x) W (x) + λ ≥ J * δ W [R] + λ 1 B r/2 (x f 0 ) (x
Condition (H5). Let us consider f ∈ L 1 W \{0} and λ ∈ C such that (5.16) holds, in particular (12.17 Proof of theorem 12.1 part (1). We may use Theorem 2.21 in order to establish the existence of a solution (λ 1 , f 1 , φ 1 ) ∈ (0, +∞) × L 1 × L ∞ to the first eigentriplet problem (1.1)-(1.2). From Theorem 4.13 and Theorem 5.16, this solution is unique, f 1 > 0, φ 1 > 0, λ 1 is algebraically simple for both L and L * and it is the unique eigenvalue in Σ + (L). Due to (12.14), we actually have

f 1 ∈ L 1 W ∩ L ∞ W .
Observing that L * is of the same type as L, L * φ = J * φ -W φ, J(x) := J(-x), and considering the dual problem as a primal problem in L 1 , Theorem 2.21 also provides the existence of λ * 1 > 0 and 0 < φ

* 1 ∈ L 1 W ∩ L ∞ W such that L * φ * 1 = λ * 1 φ * 1 .
Because of Remark 4.16, we have in fact λ * 1 = λ 1 and the simplicity of λ 1 then yields that Span φ * 1 = Span φ 1 . This ensures that φ 1 ∈ L 1 W ∩ L ∞ W and also that φ 1 enjoys the explicit lower bound (12.16). Besides, we can prove

φ 1 L ∞ W ≤ J 1 L 1 λ 1 λ 1 -κ * φ 1 L ∞ ≤ J 1 L 1 κ 1 κ 0 -κ * φ 1 L ∞
by arguing similarly as for (12.13).

In order to prove Theorem 12.1 part (2) with constructive constants we use a Doeblin-Harris type argument Lemma 12.3 (Lyapunov Condition). Under the above assumptions, for any T > 0, there are γ L ∈ (0, 1) and K > 0 such that

S T f L 1 ≤ γ L f L 1 + K f φ1 .
Proof of Lemma 12.3. Writing f t = S t f = e -λ1t S L (t)f , we have, since λ 1 ≥ 0,

d dt R d |f t | ≤ J 1 R d |f t | - R d W |f t | ≤ B c R ( J 1 -W )|f t | + J 1 α R BR |f t |φ 1 ,
for any R > 0 and some α R the bound by below of φ 1 in B R . Choosing R large enough so that W (x) ≥ J 1 + 1 for |x| ≥ R, we get

d dt R d |f t | ≤ - R d |f t | + J 1 + 1 α R R d |f t |φ 1 . Since R d |f t |φ 1 ≤ R d S t |f 0 |φ 1 = R d |f 0 |φ 1 ,
we infer

S t f ≤ e -t f + J 1 + 1 α R (1 -e -t ) f φ1 ,
by Grönwall's lemma.

Lemma 12.4 (Harris's condition). Under the assumption above, there exist ψ 0 ∈ X ′ ++ , g 0 ∈ X + and T > 0 such that (12.18) S T f ≥ g 0 f, ψ 0 , ∀ f ∈ X + .

Proof of Lemma 12.4.

Step 1. proof of (12.18). From Duhamel's formula (3.9) we have Proof of Theorem 12.1 part (2). Let us consider A > 0 and f ∈ X + such that f ≤ A[f ] φ1 . For any integer n ≥ 1, we have

[f ] φ1 = Bn f φ 1 + B c n f φ 1 ≤ α n f, ψ 0 + β n f ≤ α n f, ψ 0 + β n A[f ] φ1 , with α n = φ 1 L ∞ / inf Bn ψ 0 and β n = φ 1 L ∞ W / inf B c n W .
Choosing n A such that β nA A ≤ 1/2, we deduce the constructive estimate [f ] φ1 ≤ 2α nA f, ψ 0 , and thus that (6.8) holds with g A := (2α nA ) -1 g 0 . Because of the constructive lower bound (12.16) on φ 1 , we have φ 1 , g R ≥ (2α nA ) -1 h 0 , g 0 =: r A , which provides (6.9) in a quantified way. The two above estimates and the Lyapunov condition established in Lemma 12.3 ensure that we may apply the Harris-Doblin Theorem 6.3 and thus conclude to (12.6) with constructive rate. 12.2. A singular mutation kernel. Here we consider a mutation kernel supported by a set of zero Lebesgue measure, which thus does not satisfy (12.2). The kernel J ∈ M 1 + (R d ) is defined for any test function ϕ ∈ C 0 (R d ) by

J, ϕ = ε -1 d i=1 R ϕ(0, • • •, 0, x i , 0, • • •, 0)J i (ε -1 x i )dx i ,
where (J i ) 1≤i≤d is a family of L 1 positive kernels on R and ε > 0 is a variance parameter. The operator L then reads

Lf (x) = ε -1 d i=1 R f (x -ze i )J i (ε -1 z)dz -W (x)f (x)
, where e i is the i-th unit vector of the canonical basis of R d . This model was recently considered and studied by [START_REF] Velleret | Exponential quasi-ergodicity for processes with discontinuous trajectories[END_REF] through a probabilistic approach. It shares similarities with a model of telomere shortening which is under study in [START_REF] Doumic | Ergodic behaviour of a multidimensional Markov jump branching process modelling telomere shortening in a non-compact setting[END_REF]. We show that the method developed in the first sections of the present paper allows us to handle this model, under similar yet slightly different assumptions on the J i and W than in [START_REF] Velleret | Exponential quasi-ergodicity for processes with discontinuous trajectories[END_REF]. In particular we consider more general fitness functions W than quadratic ones. More precisely, we assume that W is a continuous function that satisfies (12.3) and (12.19) log W (x) = O(|x| 2 ) as |x| 2 := d i=1

x 2 i → ∞.

The kernels J i are supposed to be centered Gaussian distributions

J i (z) = M i G σi (z) := M i σ i √ 2π e -z 2 2σ 2
i , for given masses (M i ) 1≤i≤d ∈ (0, +∞) d and variances (σ i ) 1≤i≤d ∈ (0, +∞) d . Similarly as in Section 12.1, we work in the Banach lattice X = L 1 (R d ) and we may prove the following result.

Theorem 12.5. Under the above assumptions, there exists a constructive ε 0 > 0 small enough, such that for any ε ∈ (0, ε 0 ) the following conclusions hold

(1) The first eigentriplet problem (1.1)-(1.2) admits a unique solution (λ 1 , f 1 , φ 1 ) ∈ R × X + × X ′ + with the normalization φ 1 = φ 1 , f 1 = 1, and this triplet additionally satisfies λ 1 > 0, f 1 > 0 and φ 1 > 0.

(2) Moreover, L generates a semigroup S L on X and for any f 0 ∈ X, there holds

(12.20) e -λ1t S L (t)f 0 -φ 1 , f 0 f 1 L 1 ≤ Ce -αt f 0 -φ 1 , f 0 f 1 L 1 ,
for any t ≥ 0 and for some constructive constants C, α > 0.

Remark 12.6. The assumption of small variance ε in Theorem 12.5 replaces (12.4)-(12.5) as a condition which guarantees the strict positivity of κ 0 in the condition (H2), and so the strict positivity of λ 1 . This property is fundamental for ensuring the existence of f 1 in L 1 and for the existence of a spectral gap. On the contrary, for large values of ε, there cannot exist f 1 ∈ L 1 , as it is proved in Remark 12.2-(5). The reason is a concentration phenomenon which creates an atom at the origin for the principal eigenvector when the dispersion due to the mutations is too big. This is already noticed in [350, Rk. 5.3.1], and we refer to [START_REF] Bonnefon | Concentration phenomenon in some non-local equation[END_REF][START_REF] Bürger | Stationary distributions under mutation-selection balance: Structure and properties[END_REF][START_REF] Cloez | Fast, slow convergence, and concentration in the house of cards replicator-mutator model[END_REF] for more details about the singularity of f 1 and the concentration phenomenon.

For proving Theorem 12.5, we first show that the conditions (H1), (H2) and (H3) are verified for the dual problem in L ∞ = X ′ = (L 1 ) ′ . Then we check that the Harris conditions are satisfied, thus ensuring the existence, uniqueness and exponential stability for the primal problem.

It is worth noticing that since the J i are symmetric, we have L * = L and the only difference between the primal and dual problems is the Banach lattice in which it is posed. Condition (H1) and (H1 ′ ). With the same proof as in Section 12.1, L generates a positive semigroup S in L 1 with ω(S) ≤ J 1 and satisfies Kato's inequality. We deduce that (H1) and (H1 ′ ) are verified for both L in X and L * in X ′ with

κ 1 = J 1 = d i=1 M i .
Condition (H2). In view of condition (H3), we aim at verifying (H2) with κ 0 close enough to κ 1 . More precisely, we define ρ ∈ (0, 1] the ratio between the geometric and arithmetic means of the masses M i , namely

ρ := d i=1 M i 1/d 1 d d i=1 M i , we set ζ := d d i=1 M i 2 κ d 1 = d 1-d ρ d 2 ∈ (0, 1/2],
and we prove that there exists ε 0 such that if ε ∈ (0, ε 0 ), then (H2) is verified with

κ 0 =
M i G ε/η * G εσi (x i ) G ε/η (x i ) -W (x) = d i=1 M i G ε √ η -2 +σ 2 i (x i ) G ε/η (x i ) -W (x) = d i=1 M i 1 + (ησ i ) 2 exp η 2 (ησ i ) 2 1 + (ησ i ) 2
x 2 i 2ε 2 -W (x)

≥ 1 + θ 2 d i=1
M i exp η 2 (ησ i ) 2 1 + (ησ i ) 2

x 2 i 2ε 2 -W (x).

Due to Assumptions (12.3) and (12.19) on W and using Jensen's inequality, we have

W (x) ≤ 1 -θ 2 min 1≤i≤d M i d e C|x| 2 /d ≤ 1 -θ 2 d i=1 M i e Cx 2 i
for some C > 0 large enough. Choosing ε 0 > 0 small enough so that M i .

We deduce that for any α ≥ κ 0 ,

(R B (α)A) d φ L ∞ ≤ κ -d 0 A s φ L ∞ + κ 1-d 0 R B (α)A r φ L ∞ ≤ κ d 1 -d d i=1 M i κ d 0 φ L ∞ + κ 1-d 0 A r φ κ 0 + W L ∞ .
For any R > 0 we have

A r φ(x) κ 0 + W (x) = d ε -d 1 BR (x) κ 0 + W (x) BR φ(x -y)J ⊗ (y/ε) dy + d ε -d 1 BR (x) κ 0 + W (x) B c R φ(x -y)J ⊗ (y/ε) dy + d ε -d 1 B c R (x) κ 0 + W (x) R d φ(x -y)J ⊗ (y/ε) dy ≤ d ε -d κ 0 d i=1 M i σ i √ 2π B2R φ(y) dy + d κ 0 B c R/ε 0 J ⊗ (y)dy φ L ∞ + d d i=1 M i κ 0 + W R φ L ∞ ≤ κ d-1 0 R d φ(x)ϕ R (y) dy + η R κ 0 φ L ∞ , where ϕ R = d d i=1 M i /σ i √ 2π(εκ 0 ) d 1 B2R and η R = d B c R/ε 0 J ⊗ (y)dy + dκ 0 W R d i=1 M i ,
and with W R = inf B c R W . We may therefore infer that

(R B (α)A) d φ L ∞ ≤ κ d 1 -d d i=1 M i + η R κ d 0 φ L ∞ + φ, ϕ R .
Since W (x) → +∞ and J ⊗ (x) → 0 as |x| → ∞, we can find R large enough so that

η R ≤ d 2 d i=1 M i = ζκ d 1 .
Recalling that κ d 0 = (1ζ)κ d 1 , we then obtain (12.21) with

γ = κ d 1 -d 2 d i=1 M i κ d 0 = 1 -ζ 1 -ζ 2 = 1 1 + ζ < 1 and ϕ = ϕ R ∈ L 1 .
Invoking Lemma 2.19, we deduce that (H3) holds true for L * = L in X ′ = L ∞ .

From conditions (H1)-(H2)-(H3), we infer the existence of a solution to the dual problem.

Lemma 12.7. If ε < ε 0 , where ε 0 is defined in the paragraph about Condition (H2) above, then there exist λ 1 ≥ κ 0 and φ 1 ∈ X ′ + , φ 1 L ∞ = 1, such that L * φ 1 = λ 1 φ 1 . Moreover, φ 1 ∈ L ∞ W and φ 1 , ϕ ≥ 1γ.

Proof. The existence of (λ 1 , φ 1 ) follows from applying Theorem 2.21. The equation Lφ 1 = λ 1 φ readily gives that φ 1 L ∞ W ≤ J * φ 1 L ∞ + λ 1 φ 1 L ∞ ≤ J 1 + λ 1 , and the estimate φ 1 , ϕ ≥ 1γ comes from Lemma 2. 19 We now aim at verifying (6.8), (6.7) and (6.9) in order to apply Theorem 6.3. Lemma 12.8 (Lyapunov Condition). Under the above assumptions, for any T > 0, there are γ L ∈ (0, 1) and K > 0 such that

S T f L 1 ≤ γ L f L 1 + K f φ1 .
Proof. The proof is exactly the same as for Lemma 12.3 in Section 12.1. Lemma 12.9 (Harris's condition). Under the above assumptions, there exists ψ 0 ∈ X ′ ++ , g 0 ∈ X + and T > 0 such that (12.22) S T f ≥ f, ψ 0 g 0 , ∀ f ∈ X + .

Proof. We prove the dual version of (12. Iterating and using the successive J i 's parts of J we finally get Corollary 12.10. For ε ∈ (0, ε 0 ), there exists f 1 ∈ X + such that Lf 1 = λ 1 f 1 with f 1 , φ 1 = 1. Moreover, the exponential convergence (12.20) holds for some constructive constants C ≥ 1 and α > 0.

Proof. Similarly as in the proof of Theorem 12.1 part (2), we can infer from Lemma 12.9 that (6.8) holds with g R = C R g 0 where C R > 0 is an explicit constant. The Lyapunov condition (6.7) is established in Lemma 12.8, and the positivity condition (6.9) readily follows from the estimate φ 1 , ϕ ≥ 1γ established in Lemma 12.7. We can thus apply Theorem 6.3 which, together with its attached Remark 6.4, gives the conclusion.

Proof of Theorem 12.5. It only remains to prove the uniqueness and strict positivity properties. Combining (12.22) and (12.22) with φ = g 0 , we get that

S 2T f = S T (S T f ) ≥ f, ψ 0 S T g 0 ≥ g 2 0 f, ψ 0 ψ 0 = 2 d f, ψ 0 ψ 0 .
for all f ∈ X + . Since ψ 0 > 0, this ensures that (4.12) is verified, and then (H4) because of point (4) in Lemma 4.8. This gives the result of uniqueness and strict positivity by using Theorem 4.13.

1. 3 . 2 .

 32 Transport equation. In Part 8, we are interested in the general transport equation (1.15)∂ t f + a • ∇ y f = K [f ] -Kf in (0, ∞) × O, on the function f = f (t, y), t ≥ 0, y ∈ O, with O ⊂ R D , D ≥ 1, a smooth open connected set. We assume that a : O → R D , K : O → R + ,and that the collision operator K is linear and defined by (K g)(y) := O k(y, y * ) g(y * ) dy * , for some kernel k : O × O → R + . When O = R D , we complement the equation with a boundary condition on the incoming boundary Σ -which writes(γ -f )(t, y) = R O [f (t, •)] + R Σ [γ + f (t, .)](y) on (0, ∞) × Σ -,whereγ ± f are the trace functions on the incoming and out going set Σ ± and (R O g)(y) := O r O (y, y * ) g(y * ) dy * , (R Σ h)(y) := Σ+ r Σ (y, y * ) h(y * ) dσ y * ,

1. 3 . 4 .

 34 Kinetic linear Boltzmann equation. In Part 10, we are interested in another important subclass of transport equations, namely in the kinetic linear Boltzmann equation (1.16)

Proof of Lemma 2 . 7 . 0 e 0 e

 2700 The condition (H1) holds because of Lemma 2.2-(i). Let us then consider three sequences (λ n ), ( fn ) and (ε n ) satisfying(2.20). Integrating along the rescaled flow, this yieldse -λnT S T fnfn = T -λnt S t (Lλ n ) fn dt = -T -λnt S t ε n dt =: εn , which also reads V fn + K fne λnT fn = e λnT εn .

  and β(M ) := sup g∈C g1 g≥M L p → 0, as M → ∞.

  264, Definition 2.4.1], as a consequence of [264, Thm. 2.4.2 (iii)] along with the fact that weakly convergent increasing sequences in Banach lattices are automatically norm convergent, see e.g. [264, Prop. 1.4.1].

  ) L = A + B with A and B satisfying (3.23), (3.25), (3.26) and (3.27).

Remark 3 . 5 .S

 35 [START_REF] Abdallah | Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system[END_REF] When S L satisfies (HS3) then (3.39) holds with B * (AS B ) ( * ℓ) , w := (S B A) ( * N ) .

Lemma 4 . 8 . 1 )

 481 For a positive semigroup S, the following hold: (The pointwise strong positivity condition (4.14) implies the condition (4.15);

( 2 )

 2 Theorem 4.13 has to be compared with the seminal Krein and Rutman Theorem 1.2 ([238]), to the many results gathered in [15, Part C-III] (see in particular [15, Prop. C.3.5], [15, Thm. C.3.8] and the original paper [185]) and to the more recent contributions [278, Thm. 5.3], [41, Thm. 14.15] and [231, Thm. 5.1]. Probably many of the conclusions of Theorem 4.13 are very similar (or even included) in the material of [15, Part C-III]. However, our assumptions slightly different since we do no make explicit reference to a positive semigroup but rather refer to the weak and strong maximum principles. (3) Our proof is quite direct and elementary and uses similar arguments as those used during the proof of [278, Thm. 4.3] and [231, Thm. 5.1]. We learnt this kind of technique in the (less abstract and general) proof of the uniqueness part of [313, Lem. 2.1].

Remark 4 . 19 .

 419 Under the same hypotheses as in Lemma 4.18, we have ψ

E×E 1 -

 1 and we observe that from the Radon-Nikodym theorem, there exist two measurable functions α, β : E → [0, 2π) such that f = e iα |f | and Af = e iβ |Af |. We next compute A|f | -|Af |, φ = ℜe A|f |, φ -Af, e -iβ φ = E×E ℜe 1e i(α(x)-β(y)) φ(y)Q(x, dy)|f |(dx) = cos(α(x)β(y)) φ(y)Q(x, dy)|f |(dx). In the case of equality A|f | = |Af |, we must have 1cos(α(y)β(x)) = 0 for µ-a.e. x ∈ E and |f |-a.e. y ∈ supp f = E. We deduce that β is a constant function, so that Af = e iβ |Af | = uA|f |, for the constant u = e iβ ∈ S 1 .

  5.15. (2) During the proof we use similar arguments as in [231, Thm. 5.1].

( 3 )

 3 Condition (H5) is reminiscent of PDE arguments as we may find for instance in [231, Proof of Thm. 5.1] or in the discussion in [252, 4th course] about an uniqueness argument due to L. Tartar. Proof of Theorem 5.16. Consider an eigenvalue λ ∈ C with normalized eigenvector f ∈ X\{0}, and more precisely |f |, φ 1 = 1 and Lf = λf . Thanks to the complex Kato's inequality (4.17), we have (ℜeλ)|f | = ℜe sign(f )(λf ) = ℜe sign(f )(Lf ) ≤ L|f |.

  ) and L|f | = ℜe(signf )Lf, so that multiplying both term of the equation by |f | and integrating, we have ℜe Lf, f = L|f |, |f | .

  a smooth open connected set. We assume that a = a(y), a : O → R D , K = K(y), K : O → R + and that the collision operator K is linear and defined by (8.2) (K g)(y) := O k g * dy * , for some kernel k : O × O → R + and for any (conveniently) bounded function g : O → R. Here and below, we use the common shorthands g * := g(y * ), k := k(y, y * ), k * := k(y * , y).

Remark 8 . 4 .

 84 [START_REF] Abdallah | Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system[END_REF] Because of the very general assumption (8.5) made on the vector field a : O → R D which is exactly the one made in the DiPerna-Lions theory for transport equation in the whole space developed in[START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF], the above trace result slightly improves the similar trace result established by Boyer in[START_REF] Boyer | Trace theorems and spatial continuity properties for the solutions of the transport equation[END_REF] Thm. 3.1], where an additional assumption a • n ∈ L ζ (∂O), ζ > 1, is made.

  5 since it follows by just repeating the proofs of [139, Lem. II.1], [271, Lem. 1] or [70, Lem. 3.1].

  g div(aϕ) dy = Σ γg ϕ a(y) • n(y) dσ y , holds for any ϕ ∈ C 1 c ( Ō). That clearly uniquely defines the trace function γg on Σ\Σ 0 . Now, on the one hand, from the DiPerna-Lions renormalizing theory [139, proof of Corollary II.1], we know that β(g) ∈ L ∞ (O) satisfies the transport equation (8.22) a(y) • ∇ y β(g) = β ′ (g)G in D ′ (O), for any renormalizing function β ∈ Lip(R) and any test function ϕ ∈ C 1 c (O). Using the already established trace result, we know that there exists γβ(g) ∈ L ∞ (Σ\Σ 0 ) such that (8.23) O β ′ (g)G ϕ + β(g)div(aϕ) dy = Σ γβ(g) ϕ a • n dσ y , for any test function ϕ ∈ C 1 c ( Ō). On the other hand, from the classical Green formula for Lipschitz functions and because β(g ε ) |Σ = β(g ε|Σ ), we have

  3 using the slight modifications that one can find in [271, Thm. 2] or [70, Thm. 3.1]. Under the slightly more regularity assumption a ∈ W 1,1 loc ([0, T ] × Ō), Theorem 8.6 is a direct corollary of Theorem 8.3 applied to the for field (1, a(t, y)) on the open set (0, T ) × O. For some additional function b : O → R, another possible variant is the following trace result for the stationary transport equation (8.25) a • ∇ y g + bg = G in O in the renormalized framework as introduced by DiPerna and Lions in [139]. Assuming a ∈ W 1,1 loc ( Ō), b, G ∈ L 1 loc ( Ō), we say that g ∈ L 1 loc ( Ō) is a renormalized solution to the transport equation (8.25) if

Lemma 8 . 7 .

 87 Assume a ∈ W 1,1 loc ( Ō), b, G ∈ L 1 loc ( Ō). (1) If g ∈ L 1 loc ( Ō) and α(g) satisfies equation (8.26) for one renormalizing function α : R → (-1, 1) which is bijective and belongs to C 1 pw, * (R) then β(g) satisfies equation (8.26) for any renormalizing function β ∈ C 1 pw, * (R). (2) If g 1 , g 2 ∈ L 1 loc ( Ō) are two renormalized solutions to the transport equations a • ∇ y g i + bg i = G i ∈ L 1 loc ( Ō), then g := g 1 + g 2 is a renormalized solution to the transport equation (8.25) with G := G 1 + G 2 . (3) If g is a renormalized solution to the transport equation (8.25) and Φ, c ∈ L ∞ (O) satisfy a • ∇ y Φ = c in the distributional sense, then h := ge -Φ satisfies (8.27) a • ∇ y h + (b + c)h = Ge -Φ

  where we recall that the characteristics Y and the backward exit time t b are defined in (8.6)-(8.7) and we denote t ′ b := min(t, t b ). Similarly, the solution g to the stationary transport equation (8.29) is given by (8.48) g(y) = g(y i (y)))e -t b 0 b(Y-s(y))ds + t b 0 G(Y -s (y))e -s 0 b(Y-τ (y))dτ du. Alternatively, we may define a semigroup S b (say on L ∞ (O)) by (8.49) (S b (t)f 0 )(y) :=    f 0 (Y -t (y)) exp(-t 0 b(Y τ -t (y))dτ ), if t ∈ (0, t b (y)), 0 otherwise. Given f 0 : O → R, the function f (t, y) := (S b (t)f 0 )(y) is thus a solution to the evolution equation

( 1 )

 1 If g k → g a.e. in O then g satisfies (8.26) for any β ∈ C 1 * (R) and, up to the extraction of a subsequence, γg k → γg a.e. on Σ\Σ 0 . (2) If g k ⇀ g weakly in L 1 loc ( Ō) then g satisfies (8.26) and, up to the extraction of a subsequence, γg k r ⇀ γg on Σ\Σ 0 . Remark 8.11. Because of Remark 8.9-(5) and the time independence made on a and b in (8.35), exactly the same stability result holds for the evolution equation (8.31) as a consequence of Proposition 8.10.

Lemma 8 . 14 (

 814 Uniqueness). We assume that a and b satisfy (8.35) for some exponent p ∈ [1, ∞] and some weight function m : Ō → [1, ∞) as well as for p = 1 and m ≡ 1. We additionally assume diva ∈ L ∞ loc ( Ō) (what is automatically true under assumption (8.33)). With obvious notations, for any λ > max(λ * p (m), λ * 1 (1)), and any solution g ∈ L p mO (O) to the transport equation (8.54)

15 .

 15 The proof is a variant of the proof of[START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF] Thm. II.3] and we just allude it. Because β(g) ∈ L ∞ ((0, T ) × O) is a solution to the transport equation with source termβ ′ (g)Gβ ′ (g)bg ∈ L 1 loc ([0, T ] × Ō), we have β(g) ∈ C([0, T ]; D ′ (O)) for any β ∈ C 1 * (R). Fixing β 0 ∈ C 1 * (R)strictly increasing, we deduce that β 0 (g), β 0 (g) 2 ∈ C([0, T ]; D ′ (O)), so that β 0 (g) ∈ C([0, T ]; L 2 loc ( Ō)), and the conclusion. We consider now the time dependent transport equation(8.31).

Proposition 8 . 16 (

 816 Renormalized solutions). We assume(8.35) for some p ∈ [1, ∞] and some weight function m. For anyg 0 ∈ L p m (O), G ∈ L p mO ((0, T ) × O), g ∈ L p mΣ ((0, T ) × Σ), there exists a unique g ∈ C([0, T ]; L 1 loc ( Ō)) satisfying the estimate (8.45) or (8.46) and being a solution to the transport equation(8.31) in the renormalized sense, namely(8.56)

  in [37, Chap. III], Boyer in [70, Thm. 4.1] and Crippa et al in [120, Thm. 1.1] and [119, Thm. 1.1], where the cases p = 2 or p = ∞ are considered with always the additional assumption a ∈ L ∞ (in the last paper however the present W 1,1 bound on a is relaxed into a BV condition) by adapting the Di Perna-Lions theory developed in [139, Sec. II].

Lemma 8 . 22 (

 822 Existence in L p -optimal assumption). We make the same assumption on a, b and p as in Lemma 8.13. For any λ > λ a,b,p + 1/p and any given functions G ∈ L p (O) and g ∈ L p (Σ -, τ |a • n|dσ), there exists g ∈ L p (O) solution to(8.29).

Proposition 8 . 23 .

 823 We assume that a, b, K and R satisfy the conditions (8.35), (8.59), (8.60), (8.61), (8.62), and (8.63) for some weight function m : Ō → [1, ∞) and some exponent p ∈ [1, ∞).

  02 and g 01 ≤ g 02 . That is an immediate consequence of the way we build the solutions g i thanks to the iterative scheme we present in Step 2 of the proof of Proposition 8.23. (4) Another immediate consequence of the iterative way of building the solution, together with the fact that the characteristics representation (8.47) is the very first step of the construction, is the validity of the Duhamel formula S L = S B + S B A * S L if we denote by S L the semigroup generated by the transport equation (8.58) with G = g = 0, by S B the semigroup when additionally K = 0, and Af = K [f ].

  is the pushforward of the Lebesgue measure λ.From [139, Thm. III.1],[START_REF] Ambrosio | Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields[END_REF] Thm. 31 & Remark 32] and[START_REF] Hauray | Deux remarques sur les flots généralisés d'équations différentielles ordinaires[END_REF] Sec. 3] (see also Theorem 8.33 below), we know the existence and uniqueness of such an a.e. flow for a satisfying (8.73). In the incompressible case (diva = 0), this one furthermore satisfies:

Lemma 8 . 30 .

 830 For any g ∈ L ∞ (O) satisying a • ∇g ∈ L 1 (O), we have (8.83) g = γ α g a.e. on Σ α \{a • n = 0} for a.e. α ∈ [0, α O ],

  and b ∈ C( Ō), and we mostly repeat the proof of [255, Prop. 1]. From the above definition, for a.e. y ∈ O and any t ∈ (0, ∞), we have (8.90) ḡ(t + s, Y s (y))e s 0 b(Yτ (y))dτ = ḡ(t, y), ∀ s ≥ -t.

  with the usual commutator R ε := [a • ∇g + bg, ρ ε ]. We know from a classical time and space variant of Lemma 8.5 (see also[START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF] Lem. II.1], [271, Lem. 1], [70, Lem. 3.1] and the proof in [204, Sec. 3]) that R ε → 0 in L 1 . Because g ε is smooth and Y is an almost everywhere flow, we may set H ♯ (s, y) := H(t + s, Y s (y)), B(s, y) := s 0 b(Y τ (y))dτ and compute d ds [(g ♯ ε e B )(s, y)] = (R ♯ ε e B )(s, y), from what we get

  , y)g 0 (Y -t (y))e B(-t,y) 1 t<t b (y)g(tt b (y), y b (y)e B(-t b(y) ,y) 1 t>t b (y) |dtdy = 0, for any T, ̺ > 0, what is nothing but (8.88). Corollary 8.34 (Representation formula). Under the assumptions of Lemma 8.14 in a smooth domain O = R D , for any λ > λ a,b,p , G ∈ L p (O), the unique solution g ∈ L p (O) to the stationary transport equation (8.29) (with g = 0) satisfies g(y) = ∞ 0 e -λt (S b (t)G)(y)dt, for a.e y ∈ O, (8.91)

Lemma 8 . 35 .

 835 Assume that Y is an almost everywhere flow associated to (8.6) with a satisfying (8.73). The backward exit time t b is the unique renormalized solution in L(O) to the backward exit time problem

( 8 .

 8 103) R ≡ 0, O is bounded and there exists T O such that t b (y) ≤ T O for a.e. y ∈ O.

  H4) and the first identity, we know that |f | > 0 a.e. on O. Using the second identity, we get K [|f |] = ℜe(signf )K [f ]. Writing f = e iα |f |, we deduce O k|f * |(1cos(αα * ))dy * = 0 a.e. on O. Using (8.94), we deduce B(y,rO) |f * |(1cos(αα * ))dy * = 0, and thus α = α * a.e. on O × O. That means f = u|f |, for a constant u = S 1 , that completes the proof of the fact that L satisfies the reverse Kato's inequality condition (H5).

Theorem 8 . 36 .

 836 We assume that a, b, K and R satisfy the conditions (8.35), (8.59), (8.60), (8.61), (8.62) and (8.63) for some weight function m : Ō → [1, ∞) and some exponent p ∈ [1, ∞). Consider the semigroup S L associated to the transport equation (8.1)-(8.3) through Corollary 8.25.

and ( 8

 8 .105) ∀y, y * ∈ O, k 0 ≤ k(y, y * ) ≤ k 1

  we have thus established that L satisfies (H2) with constructive constant κ 0 . Condition (H3) on the dual problem. We define the splitting L * = A * + B * with A * φ := (R * O φ)(y) = φ(0)r O (y). From the first hypothesis in (8.109), for any κ * ≤ 0 there exists y * ∈ [0, ∞) such that K(y) ≥ -κ * for any y ≥ y * . Defining m * := e κ * y 1 [0,y * ) + e κ * y * 1 [y * ,∞) , we compute B * m * = κ * e κ * y 1 [0,y * ) -Km * ≤ κ * m * . Together with Proposition 8.23 and Corollary 8.25, we deduce that the operator Bκ * , with domain D(B) :

Theorem 8 . 38 .

 838 Under the assumptions (8.108) and (8.109), the renewal equation (8.106)-(8.107) enjoys the conclusions (C3) and (E3 1 ) with quantitative rate in L 1 .

  10) is verified by m 1 with κ = κ B . Using the splitting L = A + B with Af = M 1 (x0,R) f , the inequality (9.10) for m 1 reads B * m 1 ≤ κ B m 1 and this ensures (see the proof of Corollary 2.20) that κ -B is invertible in L 1 m1 for any κ > κ B , with positive inverse, and (κ -B) -1

  We may now extend the integral to [x 0 + ε, R 2 ]. The Duhamel formula S * L = S * B0 + S * B0 A 0 * S * L for the splitting L * = A * 0 + B * 0 with A * 0 φ = K * [φ] and B * 0 φ = bφ ′ -Kφ, also reads (9.15)

  and the conclusion follows if R 1 ≤ x 0 + ε. If not, we have with the same argument the existence of T such that for all x ∈ (x 0 , R 1 )T 0 S * t φ(x)dt ≥ c T 2R2 max(R1,2x0+ε)φ(y)dy.

  , v * )dydv * .As x 3 > 2x 2 , we can choose ζ small enough so that x 3 /2 > x 2 + ζ, and we getS * T1+t φ ≥ c T1 c t 1 (x0,R1)×[1Impose additionally that ζ ≤ x 0 . Since the sequence (u n ) defined by u 0 = x 2 and u n+1 = u n /2 + ζ converges to 2ζ ≤ 2x 0 , we deduce by an iteration argument the existence of a time T 2 such that (9.30) S * T2 φ(x) ≥ c T2 1 (x0,R1) (x)

Theorem 10 . 1 .

 101 For the kinetic equation (10.10) in the torus and under conditions (10.5)-(10.6)-(10.7)-(10.8)-(10.9) and (10.11) for some weight functions m, m 0 , m 1 , there exists r * > 0 such that for any r ≥ r * , the conclusions (C3) holds in L 2 m and the conclusion (E3 1 ) holds in L 1 m .

Theorem 10 . 4 .

 104 For the kinetic equation (10.1) in the whole space with confinement force and under conditions (10.5)-(10.6)-(10.7)-(10.8)-(10.19)-(10.20) for some weight function m = ω(H) as discuted above, the conclusion (C3) about existence, uniqueness and positivity of the eigentriplet solution (λ 1 , f 1 , φ 1 ) holds as well as the ergodicity (E2) for the weak convergence in L 1 φ1 .

  a given vector field and c : O → R is a given function. In contrast with the previous part, collisions are typically modelized by a Fokker-Planck operator ∆ v f + div v (vf ) (when b = v and c = d) which takes into account a thermal bath of (Gaussian) whitenoise instead of the integral collisional operator K [f ] -Kf in the linear Boltzmann equation (10.1).

11. 2 .

 2 Well-posedness problem with inflow term at the boundary. We consider the kinetic Fokker-Planck operator L defined in (11.1) and we start revisiting the well posedness problem(11.15) (λ -L)f = F in O, γ -f = g on Σ -,for given data F : O → R and g : Σ -→ R. For a given weight function m : R d → [1, ∞), we define the measure dξ m := m 2 |ν(x) • v| dvdσ x on the boundary Σ. We next define L 2 H 1 m = L 2 H 1 m (O) the space associated to the Hilbert norm defined by f

Theorem 11 . 3 .

 113 Let us fix a vector field b ∈ H 1 loc ( Ō), a function c ∈ L ∞ (O), a weight function m : R d → [1, ∞) and let us assume that b/ v ∈ L ∞ (O), that (11.16) holds and that (11.17) λ

(

  11.29) λ * * := ess sup (cdivb) < ∞, and λ > λ * * , the solution f is unique and f ≥ 0 if F ≥ 0. It is worth emphasizing that the assumptions of Theorem 11.4 hold when b = v and m := M -1/2 .

Theorem 11 . 5 .

 115 Let us make the same assumptions as in Theorem 11.4. For any f 0 ∈ L 2 m , there exists a unique solution f ∈ C([0, T ); L 2 m )∩L 2 (0, T ; H 1 m ) for any T > 0 to the kinetic Fokker-Planck evolution equation(11.36)

2

 2 (v • ν(x)) 2 δ(x) 1/2 dvdx ≤ C F 2 m 2 ,for some constant C = C(b, c, ν, λ). For ε x , ε v , ̺ > 0, let us now define(11.41)U := {(x, v) ∈ O; d(x, ∂Ω) > ε x , |v| < ̺},and compute

(11. 46 )

 46 g C α (Qr 1 ) ≤ C ( g L 2 (Qr 0 ) + s L ∞ (Qr 0 ) )for any solution g on Q r0 from[START_REF] Golse | Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation[END_REF] Thm. 3]. As a consequence of (11.44) and a classical covering argument, for any bounded set U ⊂ Ū ⊂ O, there exist C 0 = C 0 (U) and C 1 = C 1 (U, λ) such that

3 .

 3 From[353, Cor. A.20], we deduce that for any bounded set U ⊂ Ū ⊂ O, there exists a constructive constant δ = δ(δ 0 , r 0 , U) > 0 such thatf (x, v) ≥ δ for any (x, v) ∈ U,where it is worth emphasizing that the hypothesis b, c ∈ C(O) made in[353, Cor. A.20] is not really necessary and can be replaced by b, c ∈ L ∞ (U). Because U may be choosen arbitrary, we have established that f > 0 on O and the strong maximum principle. Condition (H2). For a given function 0 ≤ h 0 ∈ C 2 c (O) normalized by h 0 L 2 m = 1, we define f 0 ∈ D(L) as the solution to (κ 1 -L)f 0 = h 0 in O, γ -f 0 = Rγ + f 0 on Σ -.

  [START_REF] Bernard | Loskot-Rudnicki's Inequality and General Relative Entropy Inequality for Cauchy Problems Preserving Positivity[END_REF])-Bf = F in O, γ -f = Rγ + f on Σ -,which existence follows from Theorem 11.4. From the above discussion (with κ = -1) and the same arguments as in Step 1 of the proof of Theorem 11.3, we have(11.50)

(11. 54 ),

 54 ̟ ♯ (x, v) := sup 1≤p≤∞ w p (x, v) ≤ κ 2 < ∞,and m p := M -1+1/p . Lemma 11.9. For any fixed κ < κ 0 there exists̺ x > 0, ̺ v > 0 and κ 2 ∈ R such that defining Af := ξ ̺v (v)ζ ̺x (x)f with ξ ρv ∈ D(R d ), 1 |v|≤ρv ≤ ξ ρv ≤ 1 |v|≤2ρv , ζ ρx ∈ D(Ω), 1 δ(x)≥ρx/2 ≤ ζ ̺x ≤ 1 δ(x)≥̺x , and next B := L -A, there hold S B (t) B(L 2 m ) e κt , ∀ t ≥ 0, (11.55) S B (t) B(L p mp )e κ2t , ∀ t ≥ 0, ∀ p ∈ (2, ∞].(11.56)

(

  Lf )f p-1 m p p = -Σ (m p γf ) p p ν • v -(p -1) |∇(m p f )| 2 (m p f ) p-2 dx, + f p ̟ p m p p .

1 , j = 1

 11 . . . , k,(11.57) supt∈[τ,T ] AS B (t)f 0 L ∞ ≤ C ∞ p k f 0 L p k , (11.58) sup t∈[τ,T ] S B (t)f 0 C α (V) ≤ C α ∞ f 0 L ∞ . (11.59) Proof of Lemma 11.10. For 0 ≤ f 0 ∈ L 2 m , let us denote f := S B f 0 which thus satisfies the PDE ∂ t f -Bf = s := cf in D ′ ((0, T ) × O).Let us fix two open setsU i such that [τ, T ] × suppξ × suppζ ⊂ U 0 ⊂⊂ U 1 ⊂⊂ (0, T ) × O. From[START_REF] Golse | Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation[END_REF] Thm. 6] and a covering lemma, there exists a constant C0 > 0 and p 1 > 2 such that

Theorem 11 . 11 .Step 1 . 1 ≤ 1 m 1

 11111111 Under the conditions of Theorem 11.6 and the additional assumption(11.54), the conclusion (E3 1 ) holds in L 2 m with non constructive rate. Proof of Theorem 11.11. We introduce the splittingAg := M Υ ε g, Υ ε g := χ ε g, B := L -A, with χ ε ∈ C 2 c (O), 1 U2ε ≤ χ ε ≤ 1 Uε and U ε := {|v| ≤ 1/ε, δ(x) > ε}.We next write the iterated Duhamel formulas (with N := k + 2)S L = V + W * S L ,with the usual notations (3.41) for V and W associated to the integer N := k + 2 and k ≥ 1 has been introduced in Lemma 11.10. Next for T > 0 large, τ ∈ (0, T ) small and two functions (of operators) a and b, we define the modified convolution operator     (a * τ b)(t) := t-τ τ a(ts)b(s) ds if t ∈ [τ, Tτ ] (a * τ b)(t) := 0 if t ∈ [τ, Tτ ] c ,(with these notations * 0 = * ) and by induction a * τ 1 := a, a * τ k := a * τ (k-1) * τ a for k ≥ 2. With these notations, we define the new splittingS L = V + K c 1 + K c 2 + K, with K := Υ ν W τ * τ S L , K c 1 := W * S L -W τ * τ S L , K c 2 := (1 -Υ ν )W τ * τ S L, where W τ := (S B A) * τ N and ν > 0. For later references, we also define recursively Ξ 0 := S L , Ξ ℓ := S B A * τ Ξ ℓ-1 for ℓ ≥ 1, so that K = Υ ε Ξ N . The sequel of the proof is split into two steps. On the one hand, we computeΞ N (T )f 0 L p 1 m S B L ∞ (B(L p 60) Ξ N (T )f 0 L p 1 m 1

K c 2 ( 1 m 1 ≤ 1 3 e

 2113 T )f 0 L 2 m ≤ η(ν) Ξ N (T )f 0 L p κBT f 0 L 2 m .The three last estimates together, we have established(11.61)(V + K c 1 + K c 1 )(T ) B(L 2 m ) ≤ e κBT .Step 2. Performing the same kind of computations as for proving(11.60) and in particular using (11.57), we getT 0 AΞ j (s)f 0 L p j+1 mp j+1 ds ≤ T 0 T -τ τ AS B (t)AΞ j-1 (s)f 0 L p j = 1, . . . ,k, and with p k+1 := ∞. Iterating and using(11.57) with j = 0, we get ts)AΞ k (s)f 0 L ∞ m ds ≤ sup t∈[τ,T ] AS B (s) B(L ∞ m ) T τ AΞ k (s)f 0 L ∞ m ds,thanks to (11.58), andKf 0 C α (O) ≤ T -τ τ S B (Ts)AΞ k+1 (s)f 0 C α (Uν ) ds ≤ C α ∞ T sup [τ,T ]

J = J 1 +

 1 J 2 , J 1 ∈ C 1 c (R d ), κ * := J 2 1 := R ddJ 2 < κ 0 := (a -1)β, (12.5) where we use the notation A β = A ∩ {W ≥ β}. In the sequel, we work in the Banach lattice X := L 1 (R d ). Theorem 12.1. Under the above assumptions, we have (1) The first eigentriplet problem (1.1)-(1.

(12. 8 )

 8 L = A + B, Af := J 1 * f, Bf := J 2 * f -W (x)f.Arguing as in the proof of condition (H1), we see that B is the generator a positive semigroup in L p (R d ), 1 ≤ p ≤ ∞, with growth bound ω(S B ) ≤ κ * and thus (α -B) is invertible for any α ≥ κ 0 > κ * , with (12.9) (α -B) -1 B(L p ) ≤ 1 ακ * .

  ), for W [R] = sup BR W . With that last information and (12.2) again, we have nowJ * f ≥ J * 2 d J * δ W [R] + λ 1 Br (x f 0 ), and, iterating the argument, we deducef ≥ J m * 2 (m-1)d (W [R] + λ) m-1 δ1 B mr/2 (x f 0 ) ≥ γ1 BR , with γ = γ(R) > 0 for m = m(R) large enough.Choosing R an integer, we have proved that(12.16) f ≥ h 0 := γ(R)1 BR + n≥R γ(n + 1)1 Bn+1\Bn > 0.

  ) L|f | = (ℜeλ)|f | and L|f | = ℜe(signf )Lf.The first equality means that ℜeλ is an eigenvalue associated to a positive eigenfunction, and Lemma 4.17 then enforces ℜeλ = λ 1 . Lemma 4.18 subsequently ensures that |f | ∈ (Span f 1 ) + \{0}, and in particular |f | > 0. Throwing away the term W |f | in each side of the second identity in (12.17), we haveℜe f |f | (J * f ) = J * |f |.Integrating this equation, we getR 2d J(xy)ℜe |f (y)| -f (x) |f (x)| f (y) dy = 0.From the positivity condition (12.2) on J, we deduce|f (y)| -f (x) |f (x)| f (y) = ℜe |f (y)| -f (x) |f (x)| f (y) = 0, ∀ x, y ∈ R d , |x -y| < r,and thus f (x)/|f (x)| = ū for any x ∈ R d for a constant u ∈ C. That ends the proof of the reverse Kato's inequality (H5).

S 0 S 0 J

 00 L = S B + • • • + (S B A) * (N -1) * S B + (S B A) ( * N ) * S L .We note that(S B A * S B )f (x) = t B (ts)AS B (s)f ds = t 0 [A(f e W (x)s )]e -W (x)(t-s) ds. For any R > r, x ∈ B R , it is satisfied that A(f e W s )(x) = R d J(xy)f (y)e -W (y)s dy ≥ J * e -W [2R]s Br (x)f (y)dy with W [R] defined as in the proof of(12.16). Then we get(S B A * S B )f (x) ≥ 1 BR (x)J * te -W [2R]t Br (x) f (y)dy.Subsequently, we obtain thatS B A * (S B A * S B )f (x) ≥ 1 BR-r (x) t * se -W [2R]t A 1 BR (x) y)1 BR (y) Br(y) f (z)dzdy ≥ J * Br(x) Br (y) f (z)dzdy.We claim that for all a ≥ r,Br (x) Ba(y) f (z)dzdy ≥ |B r/4 | B a+r/2 (x) f (z)dz.Indeed, we deduceBr(x) Ba(y) f (z)dzdy = Br(x) R d 1 Ba(y) (z)f (z)dzdy = R d f (z) Br(x)1 Ba(z) (y)dy dz and, since for all z ∈ B a+r/2 (x), B r (x) ∩ B a (z), we haveBr (x)1 Ba(z) (y)dy ≥ |B r/4 |1 B a+r/2 (x) (z), and consequently,Br (x) Ba(y) f (z)dzdy ≥ |B r/4 | B a+r/2 (x) f (z)dz.We have obtained thatS B A * (S B A * S B )f (x) ≥ 1 BR-r(x) J 2 * t 2 /2e -W [2R]t B r+r/2 (x)f (y)dy.Iterating the same argument we arrive to(S B A) ( * n) * S B f (x) ≥ 1 BR-nr (x)J n * t n n! e -W [2R]t B r+(n-1)r/2 (x)f (y)dy.In consequence, for R = (n + 1)r, we get(S B A) ( * n) * S B f (x) ≥ 1 Br (x)J n * t n n! e -W [2(n+1)r]t B (n-1)r/2 (0) f(y)dy.Coming back to the Duhamel formula (3.9), we deduce S L f (x) ≥ 1 Br (x) T ) n n! e -W [2(n+1)r]T 1 B (n-1)r/2 and g 0 := 1 Br .

  θκ 1 with θ := (1ζ 2 ) 1/d ∈ (0, 1).Let us fix η > 0 small enough so that1 + (ησ i ) 2 ≤ 2 1 + θ 2 for all i ∈ {1, • • • , d}. We then define f 0 (x) = d j=1 G ε/η (x j ),and we computeLf 0 (x) f 0 (x) = d i=1

2ε 2 0 ≤ η 2 (

 2 ησ i ) 2 (1 + (ησ i ) 2 )C for all i ∈ {1, • • • , d}, we obtain that Lf 0 (x) f 0 (x) ≥ θ d i=1 M i e Cx 2 i ≥ θκ 1 = κ 0for any ε ∈ (0, ε 0 ]. By virtue of Lemma 2.4-(ii), this proves the announced result. Condition (H3) in X ′ = L ∞ . We use the splitting L = A + B with Bφ = -W φ, and we aim at proving that (2.29) holds with N = d in order to apply Lemma 2.19. More precisely, we want to find ϕ ∈ L 1 and γ ∈ (0, 1) such that for any α ≥ κ 0 , there holds(12.21) (R B (α)A) d φ L ∞ ≤ γ φ L ∞ + R d φ(x)ϕ(x) dx for all φ ∈ L ∞ + . We have R B (α)φ = φ α + W ≤ φ κ 0 and, defining A r φ(x) := d ε -d R d φ(xy)J ⊗ (y/ε) dy with J ⊗ (y) := d i=1 J i (y i ),we haveA d = A r + A swith both A r and A s positive operators. Positivity ensures thatA r B(X ′ ) = A r 1 = d d i=1 M i and A s B(X ′ ) = A s 1 = A d -A r ≤ J d 1d d i=1

1 x1+1 x1- 1 φ| + 1 ε x1+1 x1- 1 φ 1 φ 1 φ(x -z 1 e 1 -z 2 e 1 )φ(x -z 1 e 1 -z 2 e 1 )

 111111111 [START_REF] Arlotti | On general transport equations with abstract boundary conditions. The case of divergence free force field[END_REF], namely(12.23) S T φ ≥ φ, g 0 ψ 0 , ∀ φ ∈ X ′ + , where we have used that S * T = S L * (T ) = S L (T ) = S T , since L * = L due to the symmetry of J. The iterated Duhamel formula (3.9) and the positivity of A and S B ensure thatS L ≥ (S B A) ( * d) * S B .We start by estimating (SB A * S B )(t)φ for φ ≥ 0. Since AS B (s)φ(x) ≥ ε -1 R φ(xze 1 )e -sW (x-ze1) J 1 (z/ε) dz ≥ ε -(xze 1 )e -sW (x-ze1) J 1 (z/ε) dz ≥ ε -1 e -sW [|x|+1] J 1 |x 1 (xze 1 ) dz,where we recall the notation W [R] = sup BR W , we get(S B A * S B )(t)φ(x) ≥ t ε e -tW [|x|+1] J 1 |x 1 | + 1 ε x1+1 x1-1 φ(xze 1 ) dz.Using now the part J 2 of J we obtainA(S B A * S B )(s)φ(x) (xz 1 e 1z 2 e 1 ) dz 1 J 2 (z 2 /ε) dz 2 dz 1 dz 2and then ((S B A) ( * 2) * S B )(t)φ(x) dz 1 dz 2 .

S

  L (t)φ(x) ≥ ((S B A) ( * d) * S B )(t)φ(x) ≥ t d d! ε -d e -tW [|x|+d] J ⊗ |x| + 1 ε [-1,1] d φ(y) dy,which yields (12.23), and so(12.22), withψ 0 (x) = T d d! ε -d e -T W [|x|+d] J ⊗ |x| + 1 ε and g 0 = 1 [-1,1] d .

  ′ , h) for any µ ′ ∈ [0, µ]. We finally claim that J is open. Take indeed µ ∈ J and g ∈ K 2 such that (2.36) holds, what is possible due to Step 1. By definition, there would exist A > 0 such that g ≤ Ag 2 . Choosing 0 < ε ≤ 1/(2AC 2 ) and M ≥ 2, we compute

  • = [•] 2 and • 0 = [•] 0 . More specifically, in a Radon measures space framework, the splitting condition (HS2) is obtained by introducing the bounded operator Af := f ψ 0 and the generator B := L -A. Because of (3.6), we have B * ψ 2 ≤ κψ 2 , and arguing as for establishing (3.7), we have [S B (t)f ] 2 ≤ e κt [f ] 2 for any t ≥ 0 and f ∈ X. That last growth condition is equivalent to assuming that Bκ is dissipative for the norm [•] 2 , so that we have established that L enjoys the splitting condition (HS2).

  for a constructive constant ζ ∈ (0, 1), the infimum being taken over all the decreasing function λ : R + → R + , t → λ t , and Θ λ is defined byThe proof follows closely the proof of[START_REF] Cañizo | Harris-type results on geometric and subgeometric convergence to equilibrium for stochastic semigroups[END_REF] Thm. 4.8]. We start with the following key argument of non expansive mapping result on a well chosen norm. Proposition 6.7. Consider a positive semigroup (S t ) which satisfies both above conditions of weak confinement (L) and Doblin-Harris strong irreducibility (H). There exist some equivalent norms ||| • ||| 1 to • 1 and ||| • ||| 3 to • 3 such that S t is a non expansive mapping for the two new norms ||| • ||| 1 and ||| • ||| 3 . More precisely, there exists α > 0 such that

	(6.20)	Θ(t) :≃ inf λ>0	e -λt +	ξ λ λ	.

More precisely, the decay rate functions Θ and Θ are defined by (6.19) Θ(t) := inf λ Θ ζλ (t), Θ(t) := t -1 Θ([t/2]),

  which exists from the above discussion. We next recall some classical results. On the one hand, from[START_REF] Stampacchia | Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus[END_REF] Sec. 3 & Sec. 4] or[START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] Thm. 8.15] (see also the original papers [127,[START_REF] Nash | Continuity of solutions of parabolic and elliptic equations[END_REF][START_REF] Moser | A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations[END_REF]), the following global L ∞ De Gorgi-Nash-Moser type estimate(7.13)

  An alternative and less demanding proof is presented in[START_REF] Chicco | Principio di massimo generalizzato e valutazione del primo autovalore per problemi ellittici del secondo ordine di tipo variazionale[END_REF] Cor. 1] where(7.16) is established without the additionnal assumption(7.15). On the other hand, the following Hölder regularity estimate [339, Théorème 7.1] and[START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] Thm. 8.29] (see also the original papers[127,[START_REF] Nash | Continuity of solutions of parabolic and elliptic equations[END_REF][START_REF] Moser | A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations[END_REF]) of De Gorgi-Nash-Moser type(7.17)

  Prop. 2.4] may be repeated in order to get that solutions to the parabolic equation considered in the present framework fall into De Giorgi classes as defined in[START_REF] Guerand | Quantitative regularity for parabolic De Giorgi classes[END_REF] Definition 2.3], and thus[START_REF] Guerand | Quantitative regularity for parabolic De Giorgi classes[END_REF] Thm. 1.1] applies. On the other hand, in this context and because of the regularity assumptions, we may establish a more accurate regularity estimate. More precisely, by gathering the Sobolev inequality and the Calderon-Zygmond estimate (7.26), we obtain the classical constructive regularity estimate

	(7.31)

  1 ), by choosing ̺ ∈ (0, r Ω /2) small enough. Then, from the Harnack inequality[START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] Cor. 8.21], we deduce inf

	ω̺

  Prop. 3.9], [276, Prop. 2.5] and [231, Lem. 2.4] for details. -Step 4. Lyapunov condition. We may next write

  which is nothing but(6.8) in Harris theorem. -Step 6. Conclusion. Because of the constructive estimates (7.39), (7.48) and (7.49), we may apply the Harris type Theorem 6.3, and we conclude to the exponential stability (E3 1 ) in the norm of L 2 (Ω) with constructive constants.

	7.2. Diffusion in R d with strong potential confinement. We consider in this section the
	elliptic operator
	(7.50)

  There exists a measurable function γg on Σ\Σ 0 such that for a.e. y ∈ O satisfying t

-(y) > -∞, there holds

(8.13)

g(y) = γg(Y (t -(y), y))

+ 0 t-(y) G(Y (t, y)) dt,

and for a.e. y ∈ O satisfying t + (y) < ∞, there holds

(8.14) g(y) = γg(Y (t + (y), y)) -t+(y) 0 G(Y (t, y)) dt.

• Green formula. There exists γg ∈ L r loc (Σ\Σ 0 ), r ∈ [1, ∞], such that (8.15) O G ϕ + g div(aϕ) dy = Σ γg ϕ a(y) • n(y) dσ y , for any ϕ ∈ C 1 c ( Ō\Σ 0 ). • Renormalized Green formula. There exists a measurable function γg on Σ\Σ 0 such that (8.16)

  .29). Lemma 8.12 (Existence in L ∞ m ). We assume that a and b satisfy (8.35) with p = ∞ and some weight function m : Ō → [1, ∞). For any λ > λ * ∞ and any given functions G ∈ L ∞ m (O) and g ∈ L ∞ m (Σ -), there exists g ∈ L ∞ m (O) solution to (8.29) in the distributional sense. This solution satisfies (8.29) in the renormalized sense, the weak maximum principle, namely Proof of Lemma 8.12. The proof follows [270, Lem. 3] using [37, Thm. 2.3]; we only sketch it. Under the stronger regularity assumption a

	(8.52)

g ≥ 0 in O if g ≥ 0 on Σ -and G ≥ 0 in O,

and the L ∞ m estimate (8.43).

  which is nothing but a variant of [139, Lem. II.1]. The function g ε being smooth, for any β ∈ C 2 such that β ′ ∈ C 1

b , we may differentiate β(g ε ) and we get Mβ(g ε

  in the distributional sense D ′ (O), and thus from(11.19), we deduce that f ∈ W 2 . Thanks to the trace Theorem 11.1 and the estimate (11.20), the function f admits a trace γf ∈ L 2 (Σ; dξ 2 m ). Using the Green formula (11.14) with β = id ∈ B 1 ,

	we have
	(11.25)

O

  by f m 2 . As in Step 1 of the proof of Theorem 11.3, we get

	O

  so that the estimate(11.31) holds true for f n (instead of f ) uniformly in n ≥ 1. From the monotonous convergence theorem, there exists f ∈ L 2 H 1 m satisfying (11.31),(11.33),(11.20) and such that f n ր f a.e. Thanks to Proposition 11.2, we have γf n ր γf a.e. on Σ, from what we deduce that Rγ +

  Σ . By the Banach fixed point theorem, we deduce that there exists a solution f ∈ W 2,Σ to the equation(11.28) in that case. Finally, in order to deal with the case ζ * = 1, we consider a sequence (ζ * n ) of [0, 1) such that ζ * n ր 1 and the associated sequence (f n ) of solutions in W 2,Σ associated to the equation (11.28) with the modified reflection kernel R n g := ζ * n Rg. From (11.31) and (11.20), that sequence satisfies

  5 is skipped since it is a mere adaptation of the proof of Theorem 8.23 and Theorem 11.4. We refer to [365, Cor. 2 7, Lem. 2.8 and Cor. 2.8] where similar well-posedness results are established (see also[START_REF] Mischler | Kinetic equations with Maxwell boundary conditions[END_REF] for the existence part).11.4. The first eigenvalue problem in a domain with reflection at the boundary.

	We consider now the first eigenvalue problem for the kinetic Fokker-Planck operator (11.1) in a
	domain with reflection at the boundary, namely
	(11.37)	λf

  |f t0 |φ 1 , ∀ t 1 ≥ t 0 ≥ 0.On the other hand, from the Cauchy-Schwarz inequality, we have(Rγ + f ) 2 ≤ (Rγ + f 1 )R(γ + f 2 /γ + f 1 ), (Rγ + f ) 2 Rγ + f 1 γ -φ 1 |ν • v| ≤ Σ-R(γ + f 2 /γ + f 1 )γ -φ 1 |ν • v|

	from what we deduce the non expansive property
	(11.52)
	so that
	Σ-
	and finally

O |f t1 |φ 1 ≤ O Σ (γf ) 2 (γf 1 ) -1 γφ 1 ν • v ≤ 0.

When H(s) = s 2 , the equation (11.51) and the last inequality imply

(11.53)

Acknowledgments. The authors are very grateful to Philippe Cieutat for the crucial discussions and references about the almost periodic functions, to Boyan Sirakov for having pointed out relevant references about the Krein-Rutman theorem for parabolic equations and to Otared Kavian for the enlightening discussions in many occasions. The authors would also like to warmly thank Jochen Glück for his relevant and useful comments, suggestions, and corrections on the first version of the paper. CFS acknowledge funding by the European Union's Horizon 2020 research and innovation program under the Marie Sk lodowska-Curie grant agreement No 754362. PG has been supported by the ANR project NOLO (ANR-20-CE40-0015), funded by the French Ministry of Research. This work was partially funded by the European Union (ERC, SINGER, 101054787). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

from what we conclude with α := γ/(2β).

Step 2. We now assume β < γ + 2 and we define

so that again w ∼ H . We easily compute

Using Young's inequality similarly as in the step 1, we get that

which in turn implies (10.21) with α := min(γ/2, 1/2 -1/β).

We classically deduce the following resolvent estimate.

Lemma 10.6. For any weight function m 0 := ω(H), there exists a weight function

Proof of Lemma 10.6. We split the proof into two steps.

Step 1. We fix κ B ∈ (-K 0 , κ 0 ) and m 0 := ω 0 (H) with ω 0 a function as defined above. We observe that

Step 2. We take

where w is defined as in Step 1 of Lemma 10.5 when γ ≤ β -2 and as in Step 2 of Lemma 10.5 when γ > β -2. In any cases m m 0 . On the other hand, Lemma 10.5 and T * ω(H) = 0 imply together that

, which is nothing but (10.22).

We argue as during the proof of (H3) in Section 10.1. By a localization argument and the averaging lemma, we have AR B (κ) : L 2 m0 → L 2 (B R × B R ) with compact injection for any R > 0. Togeher with Lemma 10.6, we deduce that R B (κ) ∈ K(L 2 m0 ) for any κ > κ B , and we conclude exactly as in Section 10.1.