Tony Lelièvre
email: tony.lelievre@enpc.fr

Mouad Ramil
email: mouad.ramil@enpc.fr

Julien Reygner
email: julien.reygner@enpc.fr

Quasi-stationary distribution for the Langevin process in cylindrical domains, part I: existence, uniqueness and long-time convergence

Keywords: Mathematics Subject Classification. 35P05, 82C31, 47B07, 60H10 Langevin process, Quasi-stationary distribution, Compactness, Spectral decomposition

Consider the Langevin process, described by a vector (position,momentum)

bounded and connected set of R d . We prove the compactness of the semigroup of the Langevin process absorbed at the boundary of the domain D := O × R d . We then obtain the existence of a unique quasi-stationary distribution (QSD) for the Langevin process on D. We also provide a spectral interpretation of this QSD and obtain an exponential convergence of the Langevin process conditioned on non-absorption towards the QSD.

Introduction

In statistical physics, the evolution of a molecular system at a given temperature is typically modeled by the Langevin dynamics

dq t = M -1 p t dt, dp t = F (q t )dt -γM -1 p t dt + 2γβ -1 dB t , (1)
where d = 3N for a number N of particles, (q t , p t ) ∈ R d × R d denotes the set of positions and momenta of the particles, M ∈ R d×d is the mass matrix, F : R d → R d is the force acting on the particles, γ > 0 is the friction parameter, and β -1 = k B T with k B the Boltzmann constant and T the temperature of the system. Such dynamics are used in particular to compute thermodynamic and dynamic quantities, with numerous applications in biology, chemistry and materials science. In many practical situations of interest, the system remains trapped for very long times in subsets of the phase space, called metastable states, see for example [START_REF] Lelièvre | Partial differential equations and stochastic methods in molecular dynamics[END_REF]Sections 6.3 and 6.4]. This makes the simulation of these systems over the times of interest impossible. Typically, these states are defined in terms of positions only, and are thus cylinders of the form D = O × R d for [START_REF] Champagnat | Criteria for exponential convergence to quasi-stationary distributions and applications to multi-dimensional diffusions[END_REF]. In such a case, it is expected that the process reaches a local equilibrium distribution within the metastable state before leaving it. This distribution is called the quasi-stationary distribution (QSD). Proving the existence of this limiting behavior is in particular important to prove the consistency of accelerated dynamics algorithms, e.g. the parallel replica method, see for example [START_REF] Perez | The parallel replica dynamics method-Coming of age[END_REF]. It is also the building block to justify the use of jump Markov processes among the metastable states (kinetic Monte-Carlo or Markov state Models) to model the evolution over long timescales [START_REF] Voter | Introduction to the kinetic Monte Carlo method[END_REF][START_REF] Schütte | Metastability and Markov State Models in Molecular Dynamics[END_REF].

While several works have already studied the properties of QSD for elliptic diffusion processes on a smooth bounded domain O, to the best of our knowledge there are no available results for the Langevin dynamics [START_REF] Champagnat | Criteria for exponential convergence to quasi-stationary distributions and applications to multi-dimensional diffusions[END_REF], which is not elliptic but only hypoelliptic, and for which the natural domain D = O ×R d is not bounded, even if O is bounded. Building on several analytical results for the Langevin process [START_REF] Champagnat | Criteria for exponential convergence to quasi-stationary distributions and applications to multi-dimensional diffusions[END_REF] obtained in [START_REF] Lelièvre | A probabilistic study of the kinetic Fokker-Planck equation in cylindrical domains[END_REF], including a Gaussian upper-bound satisfied by the transition density of the Langevin process, we obtain the compactness of the semigroup of the Langevin process absorbed at the boundary of D. Applying the Krein-Rutman theorem we then obtain spectral properties on the infinitesimal generator of the Langevin process on D with Dirichlet boundary conditions, and deduce the existence and uniqueness of a QSD µ, as well as the fact that it describes the long-time behavior of the process conditioned on non absorption.

Alternatively, a more probabilistic approach, based on general criteria developed by Champagnat and Villemonais [START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF], is employed to obtain similar results in [START_REF] Ramil | Processus cinétiques dans des domaines à bord et quasi-stationnarité[END_REF]Chapter 4]. As we were finishing this work, we also became aware of the related work [START_REF] Guillin | Quasi-stationary distribution for strongly Feller Markov processes by Lyapunov functions and applications to hypoelliptic Hamiltonian systems[END_REF], using different techniques based on Lyapounov functions.

Outline of the article. In Section 2, we state the main results, which are then proven in Section 3.

Notation. Let us introduce here some notation that will be used in the following. We denote by x = (q, p) generic elements in R 2d , and | • | the Euclidean norm both on R d and on R 2d . For a measurable subset

A of R 2d , R * + × R 2d or R * + × R 2d × R 2d ,
• |A| is the Lebesgue measure of A,

• for 1 ≤ p ≤ ∞, L p (A) is the set of L p scalar-valued functions on A and • L p (A) the associated norm,

• C(A) (resp. C b (A)) is the set of scalar-valued continuous (resp. continuous and bounded) functions on A,

• C ∞ (A) (resp. C ∞ c (A))
is the set of scalar-valued C ∞ (resp. C ∞ with compact support) functions on A.

We denote by • ∞ the sup norm on the Banach space C b (A). For T a linear bounded operator on C b (A), we denote its operator norm by:

|||T ||| C b (A) := sup f ∈C b (A), f ∞≤1 T f ∞ .

Main results

This section presents the main results we obtained.

As a motivation, we first recall in Section 2.1 what is known about the QSD of the overdamped Langevin process.

In order to prepare the presentation of our main results, we state various analytical properties of the Langevin process and the related kinetic Fokker-Planck equation in Section 2.2. The proofs of these auxiliary results are detailed in [START_REF] Lelièvre | A probabilistic study of the kinetic Fokker-Planck equation in cylindrical domains[END_REF].

Our main results, concerning the degenerate case of the Langevin process, are presented in Section 2.3. We first state the compactness of the semigroup of the Langevin process absorbed at the boundary of D. The existence of a unique QSD of the Langevin process in D is then obtained. Besides, this QSD is shown to be the unique solution of an eigenvalue problem related to the infinitesimal generator of the process (q t , p t ) t≥0 with absorbing boundary conditions. Finally, this QSD attracts all probability measures on D, at an exponential rate.

Let us conclude this introduction by recalling the definitions of the quasi-stationary and quasilimiting distributions, which are the central notions of this work, in a general setting. We refer to [START_REF] Collet | Quasi-stationary distributions[END_REF][START_REF] Méléard | Quasi-stationary distributions and population processes[END_REF] for a complete introduction. Let E be a Polish space endowed with its Borel σ-algebra B(E), and let (X t ) t≥0 be a time-homogeneous, strong Markov process in E with continuous sample-paths. For any x ∈ E, we denote by P x the probability measure under which X 0 = x almost surely, and for any probability measure θ on E, we define

P θ (•) := E P x (•)θ(dx).
Let D be an open subset of E and τ ∂ be the stopping time defined by

τ ∂ := inf{t > 0 : X t ∈ D}.
Definition 2.1 (QSD). A probability measure µ on D is said to be a QSD on D of the process (X t ) t≥0 , if for all A ∈ B(D) := {A ∩ D, A ∈ B(E)}, for all t ≥ 0,

P µ (X t ∈ A, τ ∂ > t) = µ(A)P µ (τ ∂ > t).
(

) 2
When P µ (τ ∂ > t) > 0, the identity (2) equivalently writes

P µ (X t ∈ A|τ ∂ > t) = µ(A).
A closely related notion is that of Quasi-Limiting Distribution (QLD), which is a probability measure µ on D such that there exists a probability measure θ on D for which

∀A ∈ B(D), µ(A) = lim t→∞ P θ (X t ∈ A|τ ∂ > t). (3)
A QLD is necessarily a QSD (and the converse is obvious), and we say that µ "attracts" θ when (3) holds. When a QSD attracts all Dirac masses on D, it is called a Yaglom limit. Last, when X 0 is initially distributed according to a QSD on D, the exit event from D of the process (X t ) t≥0 satisfies the following properties, see [4, Theorems 2.2 and 2.6].

Proposition 2.2 (Exit event). Let µ be a QSD on D of the process (X t ) t≥0 , then there exists λ 0 ≥ 0 such that 1. τ ∂ follows the exponential law of parameter λ 0 , that is to say P µ (τ ∂ > t) = e -λ0t for all t ≥ 0,

2. if λ 0 > 0, X τ ∂ is independent of τ ∂ .
In the former statement, the case λ 0 = 0 means that τ ∂ = ∞, P µ -almost surely.

Elliptic case and the overdamped Langevin process

Quasi-stationary distributions on smooth bounded domains for elliptic diffusion processes, have been widely studied in the literature. We refer for example to [START_REF] Gong | Killed diffusions and their conditioning[END_REF][START_REF] Bris | A mathematical formalization of the parallel replica dynamics[END_REF][START_REF] Champagnat | Lyapunov criteria for uniform convergence of conditional distributions of absorbed Markov processes[END_REF][START_REF] Champagnat | Criteria for exponential convergence to quasi-stationary distributions and applications to multi-dimensional diffusions[END_REF]. Let us recall here some of their important results.

Let β > 0 and F : R d → R d satisfying the following assumption.

Assumption (F1). F ∈ C ∞ (R d , R d ).
Let (Ω, F , (F t ) t≥0 , P) be a filtered probability space and (B t ) t≥0 a d-dimensional (F t ) t≥0 -Brownian motion. Consider the overdamped Langevin process defined by

dq t = F (q t )dt + 2β -1 dB t . (4)
Under Assumption (F1), the vector field F is locally Lipschitz continuous and therefore the stochastic differential equation (4) possesses a unique strong solution (q t ) 0≤t<τ ∞ defined up to some explosion time τ ∞ ∈ (0, +∞]. The overdamped Langevin process admits the following infinitesimal generator:

L = F • ∇ + β -1 ∆, (5)
with formal adjoint L * in L 2 (dx) given by:

L * = -div(F •) + β -1 ∆.
Let O be an open set of R d satisfying the following assumption.

Assumption (O1).

O is an open C 2 bounded connected set of R d .

Let τ ∂ := inf{t > 0 : q t / ∈ O} be the first exit time from O of the process (q t ) 0≤t<τ ∞ . Under Assumption (O1), the vector field F is Lipschitz continuous on O and therefore τ ∂ ≤ τ ∞ .

It has been shown in [START_REF] Champagnat | Criteria for exponential convergence to quasi-stationary distributions and applications to multi-dimensional diffusions[END_REF][START_REF] Gong | Killed diffusions and their conditioning[END_REF][START_REF] Bris | A mathematical formalization of the parallel replica dynamics[END_REF][START_REF] Knobloch | Uniform conditional ergodicity and intrinsic ultracontractivity[END_REF] that the overdamped Langevin process admits a unique QSD on O, which moreover satisfies the following properties.

Theorem 2.3 (QSD of the overdamped Langevin process). Under Assumptions (F1) and (O1), there exists a unique QSD µ on O of the process (q t ) t≥0 . Furthermore, (i) there exists ψ ∈ C 2 (O) ∩ C b (O) such that µ(dq) = ψ(q)dq, where dq is the Lebesgue measure on R d , (ii) Span(ψ) is the eigenspace associated with the smallest eigenvalue λ of the operator -L * with homogeneous Dirichlet boundary conditions on ∂O, (iii) there exist C > 0 and α > 0 such that for all probability measures θ on O, for all t ≥ 0,

P θ (q t ∈ •|τ ∂ > t) -µ(•) T V ≤ Ce -αt ,
where • T V is the total-variation norm on the space of bounded signed measures on R d .

Multiple approaches are used in the literature to obtain the properties above. In the conservative case F = -∇V , under suitable assumptions on V we have τ ∞ = ∞ and the process (q t ) t≥0 is reversible with respect to the measure e -βV (q) dq. As a consequence, L is symmetric with respect to the canonical scalar product on L 2 (e -βV (q) dq) and since the inverse of the operator L with homogeneous Dirichlet boundary condition on ∂O is compact from L 2 (e -βV (q) dq) to L 2 (e -βV (q) dq), one can obtain a discrete spectral decomposition of L with this boundary condition. This then yields the theorem above, see [START_REF] Bris | A mathematical formalization of the parallel replica dynamics[END_REF].

In the general case when F is non conservative, the process (q t ) t≥0 is not necessarily reversible but a spectral approach can still be used. In [START_REF] Gong | Killed diffusions and their conditioning[END_REF] the authors prove the compactness of the semigroup

(P O t ) t≥0 defined on the Banach space {f ∈ C b (O) : ∀q ∈ O, f (q) = d ∂ (q)g(q) s.t. g is uniformly continuous on O},
where d ∂ is the Euclidean distance to the boundary ∂O, by

P O t f : x ∈ D → E q [f (q t )1 τ ∂ >t ] ,
using sharp estimates of the Green function of L shown in [START_REF] Grüter | The Green function for uniformly elliptic equations[END_REF]. Then, applying Krein-Rutman theorem to the operator P O t , the authors manage to deduce Theorem 2.3. Last, a more probabilistic approach is developed in [START_REF] Champagnat | Criteria for exponential convergence to quasi-stationary distributions and applications to multi-dimensional diffusions[END_REF] where the authors prove that the semigroup (P O t ) t≥0 satisfies a gradient estimate, irreducibility conditions and a controlled probability of absorption near the boundary ∂O which also yields Theorem 2.3.

Analytical properties of the Langevin process

In this section we recall some results from [START_REF] Lelièvre | A probabilistic study of the kinetic Fokker-Planck equation in cylindrical domains[END_REF] that will be used henceforth. Let γ ∈ R, σ > 0. Under Assumption (F1), the stochastic differential equation

dq t = p t dt, dp t = F (q t )dt -γp t dt + σdB t , (6)
possesses a unique strong solution (X t = (q t , p t )) 0≤t<τ∞ , defined up to some explosion time τ ∞ ∈ (0, +∞]. Notice that, compared to (1), we consider here and henceforth the mass to be identity without loss of generality (see the change of variables in [13, Equation (3.117)]), so that momentum is identified with velocity. Besides, we consider the general case γ ∈ R and σ > 0 not necessarily related to γ.

The infinitesimal generator of the Langevin process is the operator L, defined for (q, p) ∈ R d × R d by:

L = p • ∇ q + F (q) • ∇ p -γp • ∇ p + σ 2 2 ∆ p , (7)
with formal adjoint L * in L 2 (dx) given by:

L * = -p • ∇ q -F (q) • ∇ p + γdiv p (p•) + σ 2 2 ∆ p . (8)
Consider now the following strenghtening of Assumption (F1). Theorem 2.4 (Gaussian upper-bound). Under Assumption (F2), the transition density p t (x, y) of the Langevin process (X t ) t≥0 satisfying (6) is such that for all α ∈ (0, 1), there exists c α > 0, depending only on α, such that for all T > 0 and t ∈ (0, T ], for all x, y ∈ R 2d ,

Assumption (F2). F ∈ C ∞ (R d , R d )
p t (x, y) ≤ C t,T p (α) t (x, y), (9)
where C t,T := 1

α d ∞ j=0 ( F ∞cα(1+ √ γ-T ) √ πt) j σ j Γ( j+1 2 )
, γ -= max(-γ, 0) is the negative part of γ ∈ R, Γ is the Gamma function and p (α) t (x, y) is the transition density of the Gaussian process ( q

(α) t , p (α) t ) t≥0 defined by    d q (α) t = p (α) t dt, d p (α) t = -γ p (α) t dt + σ √ α dB t . (10)
We now let O ⊂ R d satisfy Assumption (O1) and consider the following domain of R 2d ,

D := O × R d ,
where the first coordinate (position) is constrained to remain on the bounded open set O and the second one (velocity) remains free. This is the natural phase space domain of the Langevin process absorbed when leaving O.

For q ∈ ∂O, let n(q) ∈ R d be the unitary outward normal vector to O at q ∈ ∂O. We introduce the following partition of ∂D:

Γ + = {(q, p) ∈ ∂O × R d : p • n(q) > 0}, Γ -= {(q, p) ∈ ∂O × R d : p • n(q) < 0}, Γ 0 = {(q, p) ∈ ∂O × R d : p • n(q) = 0}.
Let τ ∂ be the first exit time from D of the Langevin process (X t ) t≥0 in (6), i.e.

τ ∂ = inf{t > 0 : X t / ∈ D}.
Under Assumptions (F1) and (O1), F is Lipschitz continuous on O and therefore τ ∂ ≤ τ ∞ .

Remark 2.5. Friedman's uniqueness result [5, Theorem 5.2.1.] ensures that the trajectories (X t ) 0≤t≤τ ∂ do not depend on the values of F outside of O. Therefore, whenever we are interested in quantities which only depend on the absorbed Langevin process, there is no loss of generality in modifying F outside of O so that it satisfies Assumption (F2).

The Langevin process absorbed outside of the domain D has been thoroughly studied in [START_REF] Lelièvre | A probabilistic study of the kinetic Fokker-Planck equation in cylindrical domains[END_REF]. Some of the results associated to its transition density are reminded below and can be found in [15, Theorem 2.20, Corollary 2.21].

Theorem 2.6 (Transition density of the absorbed Langevin process). Under Assumptions (F1) and (O1), there exists a function

(t, x, y) → p D t (x, y) ∈ C ∞ (R * + × D × D) ∩ C(R * + × D × D)
which satisfies for all t > 0,

• p D t (x, y) > 0 for all x / ∈ Γ + ∪ Γ 0 and y / ∈ Γ -∪ Γ 0 ,

• p D t (x, y) = 0 if x ∈ Γ + ∪ Γ 0 or if y ∈ Γ -∪ Γ 0 ,
and is such that for all t > 0, x ∈ D and A ∈ B(D),

P x (X t ∈ A, τ ∂ > t) = A p D t (x, y)dy.
Moreover, for f ∈ C b (D), the functions u, v defined by:

∀t > 0, ∀x ∈ D, u(t, x) := D p D t (x, y)f (y)dy, v(t, x) := D p D t (y, x)f (y)dy, are in C ∞ (R * + × D) and satisfy: ∀t > 0, ∀x ∈ D, ∂ t u(t, x) = Lu(t, x), ∂ t v(t, x) = L * v(t, x).
Finally, p D t (x, y) also satisfies the Gaussian upper-bound [START_REF] Guillin | Quasi-stationary distribution for strongly Feller Markov processes by Lyapunov functions and applications to hypoelliptic Hamiltonian systems[END_REF] where

F ∞ is replaced with F L ∞ (D) in C t,T .
We conclude this subsection with a time-reversibility result from [15, Section 6.1] linking the transition densities of the Langevin process ( 6) and of the process called "adjoint" Langevin process ( Xt = (q t , pt )) t≥0 with infinitesimal generator L := L *dγ, and satisfying the following SDE:

dq t = -p t dt, dp t = -F (q t )dt + γ pt dt + σdB t . (11)
Let τ∂ be the first exit time from D of Xt , i.e. τ∂ := inf{t > 0 : Xt / ∈ D}. The transition kernel P x ( Xt ∈ •, τ∂ > t) admits a transition density pD t (x, y) which satisfies the following equality, see [15, Theorem 6.2].

Theorem 2.7 (Time-reversibility). Under Assumptions (F1) and (O1),

∀t > 0, ∀x, y ∈ D, p D t (x, y) = e dγt pD t (y, x). ( 12
)

Compactness and QSD of the Langevin process

In this section we state the main results proven in this work. Let us emphasize the fact that the results stated in the present section hold for any γ ∈ R, σ > 0 and F satisfying Assumption (F1). The first result states the compactness of the semigroup (P D t ) t≥0 , defined below, based on the Gaussian estimate from Theorem 2.6, which implies that

p D t ∈ L ∞ (D × D) ∩ L 1 (D × D) for all t > 0 (see Lemma 3.1).
Theorem 2.8 (Semigroup of the absorbed Langevin process). Let Assumptions (O1) and (F1) hold. For any t ≥ 0, p ∈ [1, +∞] and f ∈ L p (D), the quantity

P D t f : x ∈ D → E x [1 τ ∂ >t f (X t )] (13)
is well-defined. Besides, let p, q ∈ [1, +∞].

(i) The family of operators (P D t ) t≥0 is a semigroup on L p (D) and on C b (D). (ii) For any t > 0, the operator P D t maps L p (D) into L q (D) and into C b (D) continuously. (iii) For any t > 0, the operator P D t is compact from L p (D) to L p (D), and from C b (D) to C b (D). Remark 2.9. As an alternative to our probabilistic approach, we expect that a similar statement might also be deduced from the subelliptic estimates on the kinetic Fokker-Planck operator recently obtained by Nier [START_REF] Nier | Boundary conditions and subelliptic estimates for geometric Kramers-Fokker-Planck operators on manifolds with boundaries[END_REF] in a very general framework (both on the geometry of the underlying phase space and on the boundary conditions).

Similarly, one can define the family of operators ( P D t ) t≥0 associated to the transition density pD t , given for f ∈ L ∞ (D) by

P D t f : x ∈ D → E x 1 τ∂ >t f ( Xt ) . ( 14
)
Remark 2.10. It follows from Theorem 2.7 and Lemma 3.

1 that pD t ∈ L ∞ (D × D) ∩ L 1 (D × D
) for all t > 0. Therefore, following the proof of Theorem 2.8 in Section 3.1, one can also obtain that the family ( P D t ) t≥0 defined in [START_REF] Lelièvre | Partial differential equations and stochastic methods in molecular dynamics[END_REF] satisfies the properties detailed in Theorem 2.8. In particular,

P D t : C b (D) → C b (D) and P D t : L p (D) → L p (D) for p ∈ [1, +∞] are compact.
Our second result focuses on the spectral radii of (P D t ) t≥0 and ( P D t ) t≥0 . Let us recall here the definition of the spectral radius of a bounded operator T on the Banach space C b (D), which can be found in [20, p. 192]. Definition 2.11 (Spectrum and spectral radius). Let T be a bounded real operator on C b (D) and I the identity operator. Let us call σ(T ) the spectrum of T which is defined by: σ(T ) := {λ ∈ C : T -λI does not have an inverse that is a bounded linear operator}.

The spectral radius r(T ) of T is then defined as:

r(T ) := sup λ∈σ(T ) |λ|.
We obtain the following result on the operators P D t and P D t (defined in ( 13) and ( 14)) and their spectral radius.

Theorem 2.12 (Spectral properties of P D t and P D t ). Under Assumptions (F1) and (O1), there exists λ 0 > 0 such that for all t ≥ 0,

r(P D t ) = e -λ0t , r( P D t ) = e -(λ0+dγ)t .
Besides, there exist unique functions φ, ψ ∈ C b (D), up to a multiplicative constant, such that for all t ≥ 0, P D t φ = e -λ0t φ and

P D t ψ = e -(λ0+dγ)t ψ. Last, φ, ψ ∈ L 1 (D) ∩ C ∞ (D) and • φ > 0 on D ∪ Γ -, φ = 0 on Γ + ∪ Γ 0 and Lφ = -λ 0 φ on D, • ψ > 0 on D ∪ Γ + , ψ = 0 on Γ -∪ Γ 0 and L * ψ = -λ 0 ψ on D.
In the following, we choose φ and ψ such that D φ(x)dx = D ψ(x)dx = 1. The proof of this theorem relies on the application of the Krein-Rutman theorem [22, p. 313] to the compact operators P D t and P D t on the Banach space C b (D). We are able to deduce from this result the existence of a unique QSD on D for the processes (X t ) t≥0 and ( Xt ) t≥0 .

Theorem 2.13 (Existence and uniqueness of a QSD). Let Assumptions (F1) and (O1) hold. Let µ and μ be the following probability measures on D:

∀A ∈ B(D), µ(A) = A ψ(x)dx, μ(A) = A φ(x)dx. ( 15
)
Then µ (resp. μ) is the unique QSD on D of (X t ) t≥0 (resp. ( Xt ) t≥0 ) and for all t ≥ 0,

P µ (τ ∂ > t) = e -λ0t , P μ(τ ∂ > t) = e -(λ0+dγ)t . (16)
Moreover, one can characterize the law of X τ ∂ (resp. Xτ ∂ ) when X 0 ∼ µ (resp. X0 ∼ μ) using the following theorem.

Theorem 2.14. For all t ≥ 0, f ∈ L ∞ (∂D), D ψ(q, p)E (q,p) [f (q τ ∂ , p τ ∂ )1 τ ∂ ≤t ] dqdp = 1 -e -λ0t λ 0 ∂D ψ(q, p)f (q, p) |p • n(q)| σ ∂O (dq)dp, D φ(q, p)E (q,p) [f (q τ∂ , pτ ∂ )1 τ∂ ≤t ] dqdp = 1 -e -(λ0+dγ)t λ 0 + dγ ∂D φ(q, p)f (q, p) |p • n(q)| σ ∂O (dq)dp,
where σ ∂O is the surface measure on ∂O.

In fact, taking t → ∞, one obtain the following description of the first exit event starting from the QSD on D.

Corollary 2.15 (First exit point starting from the QSD).

If X 0 ∼ µ, X τ ∂ ∼ 1 λ 0 |p • n(q)| ψ(q, p)σ ∂O (dq)dp. Similarly, if X0 ∼ μ, Xτ ∂ ∼ 1 λ 0 + dγ |p • n(q)| φ(q, p)σ ∂O (dq)dp.
Furthermore, the densities of the QSD are the unique classical solutions to an eigenvalue problem.

Theorem 2.16 (Spectral interpretation of the QSD). Under Assumptions (F1) and (O1), there exists a unique couple (λ, η) (resp. (λ * , η * )), up to a multiplicative constant on η (resp.

η * ) such that η ∈ C 2 (D) ∩ C b (D ∪ Γ + ) (resp. η * ∈ C 2 (D) ∩ C b (D ∪ Γ -)
) is a non-zero, non-negative classical solution to the following problem

Lη(x) = -λη(x) x ∈ D, η(x) = 0 x ∈ Γ + , resp. L * η * (x) = -λ * η * (x) x ∈ D, η * (x) = 0 x ∈ Γ -. (17)
Moreover, η ∈ L 1 (D), λ = λ 0 and η D η = φ (resp. η * ∈ L 1 (D), λ * = λ 0 and η * D η * = ψ). Remark 2.17. In particular, it follows from the expression of the spectral radii in Theorem 2.12 that λ 0 is the smallest eigenvalue associated with the operators -L and -L * .

Last, we are able to obtain the following long-time asymptotics of the operator P D t on the Banach space L ∞ (D).

Theorem 2.18 (Long-time asymptotics). Let Assumptions (F1) and (O1) hold. Let α * be defined by e -(λ0+α * ) := sup

z∈σ(P D 1 )\{e -λ 0 } |z|. ( 18
)
Then α * ∈ (0, +∞], and for all α ∈ [0, α * ), there exists C α > 0 such that for all t ≥ 0, for all f ∈ L ∞ (D),

P D t f -e -λ0t φ ⊗ ψ(f ) D φψ ∞ ≤ C α f L ∞ (D) e -(λ0+α)t , (19)
where the tensor product φ ⊗ ψ is defined by: for any

f ∈ L ∞ (D), φ ⊗ ψ(f ) = ( D ψf )φ.
Remark 2.19. In particular, for f constant equal to 1, Theorem 2.18 ensures the existence of a constant C > 0 such that for all x ∈ D, t > 0,

P x (τ ∂ > t) ≤ Ce -λ0t .
This ensures in particular that for any λ ∈ [0, λ 0 ),

sup x∈D E x e λτ ∂ < ∞.
Remark 2.20. A similar result can be deduced in the Banach space L p (D) for any p ≥ 1 using the compactness property of P D t in L p (D) obtained in Theorem 2.8.

We deduce from the previous result that the QSD µ attracts all probability measures θ on D at an exponential rate.

Theorem 2.21 (Convergence to the QSD in total variation). Under the assumptions of Theorem 2.18, for all α ∈ [0, α * ), there exists C ′ α > 0 such that, for all t ≥ 0, for any probability measure θ on D, P θ (τ ∂ > t) > 0, and

P θ (X t ∈ •|τ ∂ > t) -µ T V ≤ C ′ α D φdθ e -αt , (20)
where • T V denotes the total-variation norm on the space of bounded signed measures on R 2d .

Remark [START_REF] Knobloch | Uniform conditional ergodicity and intrinsic ultracontractivity[END_REF] following the exact same proofs and using the equality in Theorem 2.7 satisfied by its transition density pD t (x, y).

The proofs of the theorems above are provided in the next Section.

Proofs

This section is organized as follows. In Section 3.1, we prove Theorem 2.8 and we prove Theorem 2.12 in Section 3.2. In Section 3.3 we obtain Theorems 2.13, 2.14 and 2.16 and finally, Section 3.4 is devoted to the proofs of Theorems 2.18 and 2.21.

Proof of Theorem 2.8

We prove in this section the results of Theorem 2.8 and in particular the compactness of the semigroup (P D t ) t≥0 . The crucial ingredient is the Gaussian upper-bound satisfied by the transition density p D t (see Theorem 2.6): for any α ∈ (0, 1) and for t > 0, there exists C t,t > 0 such that for all x, y ∈ D,

p D t (x, y) ≤ C t,t p (α) t (x, y), (21)
where p (α) t (x, y) is the transition density of the Gaussian process ( q

(α) t , p (α)
t ) t≥0 defined in [START_REF] Herzog | A practical criterion for positivity of transition densities[END_REF]. Let Φ 1 , Φ 2 be the following positive continuous functions on R:

Φ 1 : ρ ∈ R → 1-e -ρ ρ if ρ = 0, 1 if ρ = 0, (22)
Φ 2 : ρ ∈ R → 3 2ρ 3 2ρ -3 + 4e -ρ -e -2ρ if ρ = 0, 1 if ρ = 0. ( 23
)
One can show, see [15, Section 5.1], that for all t ≥ 0 and α ∈ (0, 1], the vector ( q

(α) t , p (α)
t ) admits the following law under P (q,p)

q (α) t p (α) t ∼ N 2d m q (t) m p (t) , C(t) α , (24)
where the mean vector is m q (t) := q + tpΦ 1 (γt), m p (t) := pe -γt , and the matrix C(t) is

C(t) := c qq (t)I d c qp (t)I d c qp (t)I d c pp (t)I d ,
where I d is the identity matrix in R d×d and

c qq (t) := σ 2 t 3 3 Φ 2 (γt), c qp (t) := σ 2 t 2 2 Φ 1 (γt) 2 , c pp (t) := σ 2 tΦ 1 (2γt). (25)
The determinant of the covariance matrix

C(t) α is det C(t) α = σ 4 t 4 12α 2 φ(γt) d
where φ is the positive continuous function defined by

φ : ρ ∈ R → 4Φ 2 (ρ)Φ 1 (2ρ) -3Φ 1 (ρ) 4 = 6(1-e -ρ ) ρ 4 [-2 + ρ + (2 + ρ)e -ρ ] if ρ = 0, 1 if ρ = 0. ( 26
)
Using the Gaussian upper-bound [START_REF] Rey-Bellet | Ergodic properties of Markov processes[END_REF] let us now prove that

p D t ∈ L ∞ (D × D) ∩ L 1 (D × D). Lemma 3.1 (p D t ∈ L ∞ (D × D) ∩ L 1 (D × D)).
Let Assumptions (O1) and (F1) hold. For all t > 0, p D t is bounded on D × D and

D×D p D t (x, y)dxdy < ∞. (27)
Proof. Let t > 0 and α ∈ (0, 1). By [START_REF] Rey-Bellet | Ergodic properties of Markov processes[END_REF], for all x, y ∈ D,

p D t (x, y) ≤ C t,t (2π) 2d det C(t) α = C t,t (2π) 2d σ 4 t 4 12α 2 φ(γt) d , which ensures that p D t ∈ L ∞ (D × D). To obtain that p D t ∈ L 1 (D × D), we prove that D×D p (α) t (x, y)dxdy < ∞.
Let us first integrate p (α) t ((q, p), (q ′ , p ′ )) with respect to p, p ′ ∈ R d using Fubini-Tonelli's theorem. Since p (α) t ((q, p), (q ′ , p ′ )) is the transition density of the Gaussian process ( q

(α) t , p (α)
t ) t≥0 , one can obtain an explicit expression of R d p (α) t ((q, p), (q ′ , p ′ ))dp ′ which corresponds to the marginal density of q (α) t under P (q,p) . Since, under P (q,p) ,

q (α) t ∼ N d q + tpΦ 1 (γt), c qq (t) α I d , c qq (t) α = σ 2 t 3 3α Φ 2 (γt), so that R d p (α) t ((q, p), (q ′ , p ′ ))dp ′ = (3α) d/2 (2πσ 2 t 3 Φ 2 (γt)) d/2 e - 3α 2σ 2 t 3 Φ 2 (γt) |q ′ -q-tpΦ1(γt)| 2
where Φ 1 and Φ 2 are defined in ( 22) and ( 23). Then,

R d R d p (α) t ((q, p), (q ′ , p ′ ))dpdp ′ = (3α) d/2 (2πσ 2 t 3 Φ 2 (γt)) d/2 R d e - 3α 2σ 2 t 3 Φ 2 (γt) |q ′ -q-tpΦ1(γt)| 2 dp = 1 t d Φ 1 (γt) d . Consequently, D×D p (α) t (x, y)dxdy = |O| 2 t d Φ 1 (γt) d ,
which concludes the proof. Remark 3.2. This ensures in particular that p D t ∈ L q (D × D) for any q ∈ [1, +∞] and t > 0. Let us now prove Theorem 2.8.

Proof of Theorem 2.8. It follows from Remark 3.2 that for all t > 0,

D×D p D t (x, y) 2 dxdy < ∞.
Therefore, by Theorems VI.22 and VI.23 in [START_REF] Reed | Methods of modern mathematical physics. I. Functional analysis[END_REF], the operator

P D t is a compact operator from L 2 (D) to L 2 (D).
Let s > 0. In Step 1, we show that P D s maps L p (D), p ∈ [1, +∞), continuously into L ∞ (D). In Step 2, we show that P D s maps L ∞ (D) continuously into L q (D), q ∈ [1, +∞). In Step 3, we show that P D s maps L ∞ (D) continuously into C b (D), and that P D s satisfies a semigroup property. We conclude the proof in Step 4.

Step 1. Let s > 0, p ≥ 1. Let us prove that P D s maps L p (D) continuously into L ∞ (D). Recall that by Lemma 3.1, p D s L ∞ (D×D) < +∞. Therefore if p = 1, then for any f ∈ L 1 (D) we have

P D s f L ∞ (D) ≤ p D s L ∞ (D×D) f L 1 (D) ,
while if p ∈ (1, +∞), then for any f ∈ L p (D) and x ∈ D, letting q ∈ (1, +∞) be such that 1/p+1/q = 1, we get by Hölder's inequality

P D s f (x) ≤ p D s (x, •) L q (D) f L p (D) ≤ p D s q-1 L ∞ (D×D) P(τ x ∂ > s) 1/q f L p (D) ≤ p D s q-1 q L ∞ (D×D) f L p (D) , which yields P D s f L ∞ (D) ≤ p D s q-1 q L ∞ (D×D) f L p (D) .
Step 2. Let s > 0, q ≥ 1. Let us prove that P D s maps L ∞ (D) continuously into L q (D). Let f ∈ L ∞ (D). For x ∈ D, one has that

P D s f (x) q = D p D s (x, y)f (y)dy q ≤ P(τ x ∂ > s) q f q L ∞ (D) ≤ P(τ x ∂ > s) f q L ∞ (D) .
Therefore, using Lemma 3.1 we get

P D s f L q (D) ≤ p D s 1/q L 1 (D×D) f L ∞ (D) .
Step 3. We first deduce from the Markov property that

P D t+s f (x) = E x [f (X t+s )1 τ ∂ >t+s ] = E x 1 τ ∂ >t P D s f (X t ) = P D t P D s f,
which together with the obvious observation that

P D 0 f = f , shows that (P D t ) t≥0 is a semigroup on L p (D)
P D s f C b (D) := P D s f ∞ ≤ f L ∞ (D) , so that P D s maps L ∞ (D) into C b (D) continuously. Notice that (P D t ) t≥0 is thus also a semigroup on C b (D).
Step 4. In order to study compactness, we shall use repeatedly the fact that the composition of a continuous operator with a compact operator is a compact operator.

Let p ∈ [1, +∞] and t > 0. Writing P D t = P D t/3 P D t/3 P D t/3 , using the continuity of the mappings P

Proof of Theorem 2.12

Using the compactness properties obtained in Theorem 2.8 and Remark 2.10, we apply the Krein-Rutman theorem to the operators P D t and P D t defined in ( 13) and [START_REF] Lelièvre | Partial differential equations and stochastic methods in molecular dynamics[END_REF]. In order to do that let us first recall an important property satisfied by the spectral radius of a bounded operator, see [20, p. 192 The Krein-Rutman theorem, recalled below, basically states that under some conditions on the bounded operator T , the spectral radius r(T ) is also an eigenvalue of the operator T . The following version of the Krein-Rutman theorem can be found in [22, p. 313].

Theorem 3.4 (Krein-Rutman). Let K ⊂ C b (D) be a convex cone such that the set {fg : f, g ∈ K} is dense in C b (D). Let T : C b (D) → C b (D) be a compact operator such that T (K) ⊂ K, and assume that its spectral radius r(T ) is strictly positive. Then, r(T ) is an eigenvalue of T with an eigenvector u in K \ {0} such that T (u) = r(T )u.

Let K be the following convex cone of C b (D),

K := {f ∈ C b (D) : f ≥ 0}. ( 28
)
The density of {fg : f, g ∈ K} in C b (D) is immediate. Our goal now is to apply the Krein-Rutman theorem above to the compact operators P D t and P D t , for t > 0, on the cone K. In order to do that we need to prove the positivity of the spectral radii r(P D t ) and r( P D t ) for t > 0.

Proposition 3.5 (Spectral radius positivity). Under Assumptions (F1) and (O1), for all t > 0, r(P D t ) > 0 and r( P D t ) > 0.

Proof. Let t > 0. We prove here that r(P D t ) > 0 which relies merely on the positivity of its transition density p D t (•, •) on D × D, see Theorem 2.6. Besides, the positivity of the transition density pD t (•, •) easily follows from the equality in Theorem 2.7, therefore the exact same proof applies to P D t and ensures that r( P D t ) > 0. As a result, we omit here the case of P D t to avoid repetition. Following Proposition 3.3, it is sufficient to prove that there exists a constant β > 0 such that for all n ≥ 1, (

P D t ) n 1/n C b (D) ≥ β. Let C ⊂ D
≥ P D t n 1 D ∞ .
Moreover, for all x ∈ C,

P D t n 1 D ∞ ≥ P D t n 1 D (x) = D n p D t (x, y 1 ) . . . p D t (y n-1 , y n )dy 1 . . . dy n ≥ C n p D t (x, y 1 ) . . . p D t (y n-1 , y n )dy 1 . . . dy n ≥ (α|C|) n .
Consequently, for all n ≥ 1,

P D t n 1/n C b (D) ≥ α|C| > 0,
which concludes the proof.

We now apply Theorem 3.4 to the operators P D t and P D t on the cone K to obtain that r(P D t ) and r( P D t ) are eigenvalues of their respective operators. In addition, these eigenvalues are shown to be simple. The following proof is inspired from unpublished lecture notes by P. Collet. Proposition 3.6 (Consequence of the Krein-Rutman theorem). Let Assumptions (F1) and (O1) hold. For all t > 0, r(P D t ) (resp. r( P D t )) is a simple eigenvalue of the operator P D t (resp. P D t ) with eigenspace generated by an element φ t (resp.

ψ t ) of K ∩ L 1 (D) such that φ t > 0 (resp. ψ t > 0) on D. Furthermore, r( P D t ) = r(P D t )e -dγt .
Proof. Let t > 0. The compactness of P D t and P D t on C b (D) follows from Theorem 2.8 and Remark 2.10. Besides, the cone K defined in (28) evidently satisfies the assumptions of Theorem 3.4. Therefore, by Theorem 3.4 and Proposition 3.5 we obtain the existence of φ t , ψ t ∈ K \ {0} such that

P D t φ t =
Let us now prove that D φ t (x)ψ t (x)dx ∈ (0, ∞). First, for x ∈ D, r(P D t )φ t (x) = P D t φ t (x) = D p D t (x, y)φ t (y)dy > 0 since φ t ∈ K \ {0} and p D t > 0 on D × D by Theorem 2.6. Therefore, φ t > 0 on D. Likewise ψ t > 0 on D so that D φ t (x)ψ t (x)dx > 0. Second, using the boundedness of φ t along with the double integral estimate from Lemma 3.1 in the left equality in (29), one obtains that φ t ∈ L 1 (D). Using Theorem 2.7 one also has that ψ t ∈ L 1 (D). In particular, since φ t and ψ t are in L ∞ (D), this yields that D φ t (x)ψ t (x)dx < ∞. As a result, the equalities (31) and ( 32) yield (30).

Step ) is also a simple eigenvalue. This concludes the proof of Proposition 3.6.

To prove Theorem 2.12, we finally need the following technical lemma. Lemma 3.7 (High velocity exit event). Under Assumptions (F1) and (O1),

∀t > 0, sup q∈O P (q,p) (τ ∂ > t) -→ |p|→∞ 0.
Proof. Let t > 0 and α ∈ (0, 1).

(α) t , p (α)
t ) t≥0 defined in [START_REF] Herzog | A practical criterion for positivity of transition densities[END_REF].

Furthermore, for x = (q, p) ∈ D, the law of q (α) t is Gaussian with mean q + tpΦ 1 (γt) and covariance matrix σ 2 t 3 3α Φ 2 (γt)I d , where Φ 1 are Φ 2 are defined in ( 22) and [START_REF] Schütte | Metastability and Markov State Models in Molecular Dynamics[END_REF]. Therefore,

R d p (α) t ((q, p), (q ′ , p ′ ))dp ′ = (3α) d/2 (2πσ 2 t 3 Φ 2 (γt)) d/2 e - 3αt 2 Φ 1 (γt) 2 2σ 2 t 3 Φ 2 (γt) p-q ′ -q tΦ 1 (γt) 2 .
Let δ := sup q,q ′ ∈O |qq ′ | (which is finite since O is bounded), then for t > 0 and q, q

′ ∈ O, if |p| ≥ 2δ tΦ1(γt) (Φ 1 is positive), R d p (α) t ((q, p), (q ′ , p ′ ))dp ′ ≤ (3α) d/2 (2πσ 2 t 3 Φ 2 (γt)) d/2 e - 3αt 2 Φ 1 (γt) 2 8σ 2 t 3 Φ 2 (γt) |p| 2 .
As a consequence,

sup q∈O P (q,p) (τ ∂ > t) = sup q∈O O R d p D t ((q, p), (q ′ , p ′ ))dp ′ dq ′ ≤ sup q∈O C t,t O R d p (α) t ((q, p), (q ′ , p ′ ))dp ′ dq ′ ≤ C t,t (3α) d/2 |O| (2πσ 2 t 3 Φ 2 (γt)) d/2 e - 3αt 2 Φ 1 (γt) 2 8σ 2 t 3 Φ 2 (γt) |p| 2 -→ |p|→∞ 0,
which concludes the proof.

We are now in position to prove Theorem 2.12.

Proof of Theorem 2.12. For t > 0, let φ t (resp. ψ t ) be an eigenvector of P D t (resp. P D t ) in K \ {0} associated with the eigenvalue r(P D t ) (resp. r( P D t )) and such that φ t > 0 (resp. ψ t > 0) on D, whose existence is ensured by Proposition 3.6. We will prove Theorem 2.12 for φ t and P D t , but the exact same reasoning with the operator P D t instead yields the proof for ψ t and P D t .

Step 1. Let us start by proving that Span(φ t , t > 0) is a one-dimensional space generated by a function φ ∈ K. This is the case if one can prove that for all s, t > 0, φ s ∈ Span(φ t ).

For s, t > 0, P D s φ s = r(P D s )φ s . Furthermore, by the semigroup property satisfied by (P D r ) r≥0 , P D s P D t φ s = P D t P D s φ s = r(P D s )P D t φ s .

Since r(P D s ) is a simple eigenvalue of P D s by Proposition 3.6 then P D t φ s ∈ Span(φ s ), i.e. there exists α t,s > 0 such that P D t φ s = α t,s φ s . Let us prove that α t,s = r(P D t ). Consider the integral D P D t φ s (x)ψ t (x)dx. One has that

D P D t φ s (x)ψ t (x)dx = α t,s D φ s (x)ψ t (x)dx.
Furthermore, Theorem 2.7 and Proposition 3.6 also ensure that

D P D t φ s (x)ψ t (x)dx = e dγt D φ s (x) P D t ψ t (x)dx = r(P D t ) D φ s (x)ψ t (x)dx.
Since φ s , ψ t are positive on D and belong to C b (D)∩L 1 (D) then D φ s (x)ψ t (x)dx ∈ (0, ∞). Therefore, the equalities above ensure that α t,s = r(P D t ). In particular, this yields that φ s ∈ Span(φ t ) since r(P D t ) is a simple eigenvalue for P D t . Let us now denote by φ ∈ K a function generating Span(φ t , t > 0). Step 2. Let us now show that there exists λ 0 ≥ 0 such that for all t > 0, r(P D t ) = e -λ0t . For s, t > 0, P D t+s φ = r(P D t+s )φ. Besides,

P D t+s φ = P D t P D s φ = r(P D t )r(P D s )φ.
Therefore, r(P D t+s ) = r(P D t )r(P D s ) since φ > 0. Since P D s C b (D) ≤ 1 for all s > 0, then r(P D s ) ≤ 1. As a result, for all s, t > 0, r(P D t+s ) ≤ r(P D t ). Consequently, the function v : t > 0 → log(r(P D t )) is a decreasing function which satisfies the Cauchy equation:

∀s, t > 0, v t+s = v t + v s .
Classical results for Cauchy equations ensure that v t is linear. This implies that there exists λ 0 ≥ 0 such that for all t > 0, r(P D t ) = e -λ0t . Step 3. Let us prove that λ 0 > 0. Assume that λ 0 = 0, then for all (q, p) ∈ D, t > 0,

P D t φ(q, p) = E (q,p) [1 τ ∂ >t φ(X t )] = φ(q, p).
Hence, sup q∈O φ(q, p) ≤ φ ∞ sup q∈O P (q,p) (τ ∂ > t) -→ |p|→∞ 0 by Lemma 3.7. As a consequence,

φ ∈ C b (D) attains its maximum φ ∞ at some x 0 ∈ D. Then, φ ∞ = φ(x 0 ) ≤ φ ∞ P x0 (τ ∂ > t).
Hence, P x0 (τ ∂ > t) = 1, which leads to a contradiction since P x0 (X t ∈ R 2d \ D) > 0.

Step 4. Let us finally prove the properties on φ stated in Theorem 2.12. First, φ ∈ C b (D) ∩ L 1 (D) by Proposition 3.6. In addition, φ > 0 on D ∪ Γ -and φ = 0 on Γ + ∪ Γ 0 using Theorem 2.6 and the fact that P D t φ = e -λ0t φ. Last, letting u(t, x) = P D t φ(x) in Theorem 2.6, we get that u ∈ C ∞ (R * + × D) and ∂ t u = Lu, but since u(t, x) also writes e -λ0t φ(x), we conclude that φ ∈ C ∞ (D) and Lφ = -λ 0 φ on D.

Proof of Theorem 2.13 and Theorem 2.16

Let us now prove the existence of a unique QSD on the domain D for the processes (X t ) t≥0 in ( 6) and ( Xt ) t≥0 in [START_REF] Knobloch | Uniform conditional ergodicity and intrinsic ultracontractivity[END_REF].

Proof of Theorem 2.13. We prove this theorem for the process (X t ) t≥0 and notice that the exact same proof with the process ( Xt ) t≥0 instead of (X t ) t≥0 yields the result for ( Xt ) t≥0 with the function φ instead of ψ.

Step 1. Let us prove that the measure µ defined in ( 15) is a QSD on D of the Langevin process (X t ) t≥0 .

Let t > 0, A ∈ B(D). Integrating the equality P D t ψ = e -(λ0+dγ)t ψ over A, one obtains that

A P D t ψ(x)dx = e -(λ0+dγ)t A ψ(x)dx. (34)
Furthermore, using Fubini-Tonelli's theorem along with Theorem 2.7, one has that

A P D t ψ(x)dx = e -dγt D ψ(x)P x (X t ∈ A, τ ∂ > t)dx.
Therefore, reinjecting into (34) we obtain since µ(dx) = ψ(x)dx that

P µ (X t ∈ A, τ ∂ > t) = e -λ0t µ(A),
which gives in particular for A = D that P µ (τ ∂ > t) = e -λ0t and thus µ is a QSD on D for the process (X t ) t≥0 by Definition 2.1.

Step 2. Let μ be a QSD on D for the process (X t ) t≥0 . Let us prove that μ = µ, where µ is defined in [START_REF] Lelièvre | A probabilistic study of the kinetic Fokker-Planck equation in cylindrical domains[END_REF]. We start by proving that μ admits a density with respect to the Lebesgue measure on D and that its density is an eigenvector of the semigroup ( P D t ) t>0 . By Definition 2.1, for all A ∈ B(D) and t > 0,

P μ(X t ∈ A, τ ∂ > t) = P μ(τ ∂ > t)μ(A). (35)
Moreover, by Proposition 2.2 and the positivity of the transition density p D t on D × D stated in Theorem 2.6, there exists λ0 ∈ [0, ∞) such that P μ(τ ∂ > t) = e -λ0t .

Let A ∈ B(D) with zero Lebesgue measure, then for all x ∈ D, P x (X t ∈ A, τ ∂ > t) = 0 by Theorem 2.6. As a result, P μ(X t ∈ A, τ ∂ > t) = 0 and μ(A) = 0 by (35). Therefore, by Radon-Nikodym's theorem, μ admits a measurable non-negative density ψ with respect to the Lebesgue measure on D. Therefore, by (35), for all t > 0, A ∈ B(D),

D ψ(x)P x (X t ∈ A, τ ∂ > t)dx = e -λ0t A ψ(y)dy. (36)
By Fubini-Tonelli's theorem,

D ψ(x)P x (X t ∈ A, τ ∂ > t)dx = D D ψ(x)p D t (x, y)1 A (y)dydx = D 1 A (y) D ψ(x)p D t (x, y)dx dy.
As a result, it follows from (36) that for almost every y ∈ D,

D ψ(x)p D t (x, y)dx = ψ(y)e -λ0t .
Then, Theorem 2.7 ensures that

D ψ(x)p D t (y, x)dx = ψ(y)e -( λ0+dγ)t ,
which can be rewritten as: for almost every y ∈ D,

P D t ψ(y) = ψ(y)e -( λ0+dγ)t . (37)
Since ψ ∈ L 1 (D) and P D t : L 1 (D) → C b (D) by Remark 2.10, then ψ can be chosen in C b (D) by (37) and in particular (37) holds for all y ∈ D.

Step 3. Let us prove that λ0 = λ 0 . In order to do that let us compute the integral D P D t ψ(x)φ(x)dx in two different ways. On the one hand, it follows from (37) that

D P D t ψ(x)φ(x)dx = e - ( λ0+dγ)t D ψ(x)φ(x)dx.
On the other hand, using Fubini-Tonelli's theorem, Theorem 2.7 and Theorem 2.12,

D P D t ψ(x)φ(x)dx = e -dγt D ψ(x)P D t φ(x)dx = e -(λ0+dγ)t D ψ(x)φ(x)dx.
Since φ > 0, ψ ≥ 0 on D and ψ satisfies D ψ(x)dx = 1, then D ψ(x)φ(x)dx > 0 and λ0 = λ 0 .

Step 4. For t > 0, r( P D t ) = e -(λ0+dγ)t is a simple eigenvalue of P D t (seen as an operator on C b (D)) with eigenvector ψ by Theorem 2.12. As a result, since D ψ(x)dx = D ψ(x)dx = 1, then ψ = ψ and thus μ = µ.

Let us now prove Theorem 2.14.

Proof of Theorem 2.14. We shall only prove the first equality, the reasoning is exactly similar for the second equality regarding the QSD of the adjoint process ( Xt ) t≥0 . It is also sufficient to prove the result for f ∈ C b (∂D), the extension to f ∈ L ∞ (∂D) being straightforward using a dominated convergence argument. Besides, Tietze extension's theorem guarantees that f ∈ C b (∂D) can be extended to a continuous bounded function on R 2d . Such a function can then be approached by a smooth and compactly supported function on R 2d . Therefore, it is enough to look at the case f ∈ C ∞ c (D) and conclude with a dominated convergence argument.

Let Φ be a non-negative C 2 function in R + such that Φ(0) = 1, Φ(ρ) = 0 for ρ ≥ 1 and for all ρ ≥ 0, Φ(ρ) ∈ [0, 1].

Since O is C 2 there exists δ > 0 small enough such that the Euclidean distance to the boundary ∂O, dist(•, ∂O), is C 2 on the set O δ := {q ∈ O : dist(•, ∂O) ≤ δ}. Let d ∂ be defined as the distance function dist(•, ∂O) on O δ and extended to a C 2 function on O.

For k ≥ 1, we define the function

g k : (q, p) ∈ D → f (q, p)Φ(kd ∂ (q)) ∈ C 2 c (D). (38)
For t > 0, let us compute the limit when k → ∞ of the following integral in two different ways, D ψ(q, p)E (q,p) [g k (q τ ∂ ∧t , p τ ∂ ∧t )] dqdp.

Step 1: For (q, p) ∈ D,

E (q,p) [g k (q τ ∂ ∧t , p τ ∂ ∧t )] = E (q,p) [f (q τ ∂ , p τ ∂ )1 τ ∂ ≤t ] + E (q,p) [g k (q t , p t )1 τ ∂ >t ] .
Notice that the sequence (g k ) k≥1 is uniformly bounded by f ∞ and that g k (q, p) converges to 0 everywhere in D. Therefore, by the dominated convergence theorem, since ψ is in L 1 (D), D ψ(q, p)E (q,p) [g k (q t , p t )

1 τ ∂ >t ] -→ k→∞ 0.
As a result,

D ψ(q, p)E (q,p) [g k (q τ ∂ ∧t , p τ ∂ ∧t )] dqdp -→ k→∞ D ψ(q, p)E (q,p) [f (q τ ∂ , p τ ∂ )1 τ ∂ ≤t ] dqdp. ( 39
)
Step 2: Second, by the Itô formula, we have that

E (q,p) [g k (q τ ∂ ∧t , p τ ∂ ∧t )] = g k (q, p) + E (q,p) t 0 Lg k (q s , p s )1 τ ∂ >s ds . ( 40
)
By definition of g k (38) and L (7), for all (q, p)

∈ O × R d , Lg k (q, p) = kp • ∇ q d ∂ (q)Φ ′ (kd ∂ (q))f (q, p) + Φ(kd ∂ (q))Lf (q, p). (41)
Besides, by definition of Φ, Φ(kd

∂ (q)) ≤ 1 [0,1/k] (d ∂ (q)).
As a result, integrating in the first term in the right-hand side of (40) against ψ ,

D ψ(q, p)g k (q, p)dqdp ≤ f ∞ D ψ(q, p)1 [0,1/k] (d ∂ (q))dqdp. (42)
Furthermore, considering the second term in the right-hand side of the equality (41) one has since

f ∈ C ∞ c (D), D ψ(q, p)E (q,p) t 0 Φ(kd ∂ (q s ))Lf (q s , p s )1 τ ∂ >s ds dqdp ≤ Lf ∞ D ψ(q, p)E (q,p) t 0 1 [0,1/k] (d ∂ (q s ))1 τ ∂ >s ds dqdp = Lf ∞ t 0 D ψ(q, p)E (q,p) 1 [0,1/k] (d ∂ (q s ))1 τ ∂ >s dqdpds = Lf ∞ 1 -e -λ0t λ 0 D ψ(q, p)1 [0,1/k] (d ∂ (q))dqdp (43)
since ψ is the QSD on D of (q t , p t ) t≥0 . It remains to estimate the integral against ψ of the first term in the right-hand side of the equality (41) D ψ(q, p)E (q,p)

t 0 kp • ∇ q d ∂ (q s )Φ ′ (kd ∂ (q s ))f (q s , p s )1 τ ∂ >s ds dqdp = 1 -e -λ0t λ 0 D ψ(q, p)kp • ∇ q d ∂ (q)Φ ′ (kd ∂ (q))f (q, p)dqdp. ( 44
)
Let us study the limit when k goes to infinity of D ψ(q, p)kp • ∇ q d ∂ (q)Φ ′ (kd ∂ (q))f (q, p)dqdp.

Recall that Φ ′ has support in [0, 1]. Besides, d ∂ is C 2 on O and coincides with the Euclidean distance to the boundary on O δ . Therefore, for k ≥ 1/δ, one can apply the following change of variable involving the Weingarten map W q and detailed for example in [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Lemma 14.16], D ψ(q, p)kp • ∇ q d ∂ (q)Φ ′ (kd ∂ (q))f (q, p)dqdp = ∂D 1/k 0 ψ(qλn(q), p)kp • ∇ q d ∂ (qλn(q))Φ ′ (kd ∂ (qλn(q)))f (qλn(q), p) det(I + λW q )dλσ ∂O (dq)dp.

(45) Moreover, by the change of variable s → λk, 1/k 0 ψ(qλn(q), p)kp • ∇ q d ∂ (qλn(q))Φ ′ (kd ∂ (qλn(q)))f (qλn(q), p) det(I + λW q )dλ = 1 0 ψ(qsn(q)/k, p)p • ∇ q d ∂ (qsn(q)/k)Φ ′ (kd ∂ (qsn(q)/k))f (qsn(q)/k, p) det(I + sW q /k)ds (46) All the terms inside the integral are bounded. Therefore, using the dominated convergence theorem one easily shows that 1 0 ψ(qsn(q)/k, p)p • ∇ q d ∂ (qsn(q)/k)Φ ′ (kd ∂ (qsn(q)/k))f (qsn(q)/k, p) det(I + sW q /k)ds

= ψ(q, p)f (q, p)p • ∇ q d ∂ (q) 1 0 Φ ′ (kd ∂ (q -sn(q)/k))ds + o(k) k→∞ . (47)
Besides, for q ∈ ∂O, ∇ q d ∂ (q) = -n(q). In addition, for s ∈ (0, 1),

kd ∂ (q -sn(q)/k) -→ k→∞ ∇ q d ∂ (q) • (-sn(q)) = s.
As a result,

1 0 Φ ′ (kd ∂ (q -sn(q)/k))ds -→ k→∞ 1 0 Φ ′ (s)ds = Φ(1) -Φ(0) = -1.
Then, one concludes from (45),( 46),(47) that

D ψ(q, p)kp • ∇ q d ∂ (q)Φ ′ (kd ∂ (q))f (q, p)dqdp -→ k→∞ ∂D ψ(q, p)f (q, p)p • n(q)σ ∂O (dq)dp. (48)
Consequently, it follows from (40),( 41),( 42),( 43),( 44) and (48) that

D ψ(q, p)E (q,p) [g k (q τ ∂ ∧t , p τ ∂ ∧t )] dqdp -→ k→∞ 1 -e -λ0t λ 0 ∂D ψ(q, p)f (q, p)p • n(q)σ ∂O (dq)dp. (49)
Finally, it follows from (39) and (49) that

D ψ(q, p)E (q,p) [f (q τ ∂ , p τ ∂ )1 τ ∂ ≤t ] dqdp = 1 -e -λ0t λ 0 ∂D ψ(q, p)f (q, p)p • n(q)σ ∂O (dq)dp
which concludes the proof since ψ = 0 on Γ -and positive on Γ + by Theorem 2.12.

Let us conclude this subsection by proving Theorem 2.16. This will provide a spectral interpretation of the QSD on D of the Langevin process, similarly to the spectral interpretation obtained in the overdamped Langevin case, cf. Theorem 2.3.

Proof of Theorem 2.16. The couple (λ 0 , φ), defined in Theorem 2.12, is clearly a solution to the left eigenvalue problem in [START_REF] Nier | Boundary conditions and subelliptic estimates for geometric Kramers-Fokker-Planck operators on manifolds with boundaries[END_REF]. Let us prove that such a couple (λ 0 , φ) is unique, up to a multiplicative constant for φ. Since the reasoning for the right eigenvalue problem with solution (λ 0 , ψ) is the same, with the process ( Xt ) t≥0 instead of (X t ) t≥0 , it will not be detailed.

Let λ ∈ R and η ∈ C 2 (D) ∩ C b (D ∪ Γ + ) be a non-zero and non-negative classical solution of the left eigenvalue problem in [START_REF] Nier | Boundary conditions and subelliptic estimates for geometric Kramers-Fokker-Planck operators on manifolds with boundaries[END_REF]. Let

τ V c k := inf{t > 0 : X t / ∈ V k }, where V k := {(q, p) ∈ D : |p| < k, d ∂ (q) > 1
k } and we recall that d ∂ refers to the distance to ∂O. Applying Itô's formula to the process (e λs η(X s )) s≥0 at the stopping time t ∧ τ V c k , one gets, for x ∈ D, P x -almost surely, for all t ≥ 0, e

λ(t∧τ V c k ) η X t∧τ V c k = η(x) + σ t 0 1 s≤τ V c k e λs ∇ p η(X s ) • dB s , (50)
since Lη + λη = 0 on D. Moreover, ∇ p η is bounded on the compact V k since η ∈ C 2 (D). Therefore, the stochastic integral in the right-hand side of the equality (50) is a martingale and its expectation vanishes. Hence,

E x e λ(t∧τ V c k ) η X t∧τ V c k = η(x),
which can be rewritten as

η(x) = e λt E x η X t 1 τ V c k >t + E x e λτ V c k η X τ V c k 1 τ V c k ≤t . (51)
Now we would like to let k → ∞. Let us prove the following limit, P x -almost surely,

lim k→∞ τ V c k = τ ∂ ,
using the same reasoning as in [15, Section 3.1].

The sequence (τ V c k ) k≥1 is an increasing sequence of random variables, therefore it converges almost surely to sup k≥1 τ V c k . Besides, using the continuity of the trajectories of (X t ) t≥0 , one gets for all r > 0,

sup k≥1 τ V c k > r = ∃k ≥ 1 : τ V c k > r = ∃k ≥ 1 : sup u∈[0,r] |p u | < k, inf u∈[0,r] d ∂ (q u ) > 1 k = sup u∈[0,r] |p u | < ∞, inf u∈[0,r] d ∂ (q u ) > 0 = sup u∈[0,r] |p u | < ∞, τ ∂ > r .
For all r > 0, we have that sup u∈[0,r] |p u | < ∞, almost surely. Therefore, sup k≥1 τ V c k > r if and only if τ ∂ > r, that is to say sup k≥1 τ V c k = τ ∂ almost surely. As a result, one gets lim k→∞ τ V c k = τ ∂ almost surely. Besides, since (τ V c k ) k≥1 is an increasing sequence, then for all s > 0, almost surely,

1 τ V c k >s -→ k→∞ 1 τ ∂ >s .
As a result, by continuity of the trajectories of (X t ) t≥0 , P x -almost surely,

1 τ V c k ≤t η(X τ V c k ) -→ k→∞ 1 τ ∂ ≤t η(X τ ∂ ).
Moreover η(X τ ∂ ) = 0 almost surely on the event {τ ∂ ≤ t} since X τ ∂ ∈ Γ + P x -almost surely by [15, Assertion (ii) of Proposition 2.8]. Therefore, taking the limit k → ∞ in (51), one gets by the dominated convergence theorem that

∀t > 0, ∀x ∈ D, E x η X t 1 τ ∂ >t = e -λt η(x), (52)
which ensures in particular that necessarily λ ≥ 0. This also writes ∀t > 0, ∀x ∈ D, D p D t (x, y)η(y)dy = e -λt η(x).

Using the boundedness of η along with (ii) in Theorem 2.8, we deduce that η ∈ L 1 (D). Now let η = η/ D η, then using Theorem 2.7 one obtains that ∀t > 0, ∀x ∈ D, D η(y)p D t (y, x)dy = e -(λ+dγ)t η(x).

Integrating over D we obtain that P ν (τ ∂ > t) = e -(λ+dγ)t with ν(dx) = η(x)dx. Then, integrating over any A ∈ B(D), we obtain that ν is a QSD on D of the process ( Xt ) t≥0 . Consequently, the uniqueness of such a QSD, by Theorem 2.13, ensures that ν = μ where μ is defined in Theorem 2. [START_REF] Lelièvre | Free energy computations[END_REF]. In addition, it implies that λ = λ 0 , which concludes the proof.

Proof of Theorems 2.18 and 2.21

This section is devoted to the study of the long-time convergence of the semigroup (P D t ) t>0 . Note that a similar study could be performed for the semigroup ( P D t ) t>0 , using the duality between the two semigroups as stated in Theorem 2.7.

We start this subsection by ensuring the existence of a spectral gap for the operator P D t . In the next statement, we denote by C b (D, C) the space of complex-valued continuous bounded functions on D.

Lemma 3.8 (Spectral gap). Under Assumptions (F1) and (O1), for all t > 0, the operator P D t admits a unique complex eigenvalue with modulus equal to r(P D t ) = e -λ0t and eigenvector in C b (D, C). Proof. Assume that there exists an eigenvector h t ∈ C b (D, C) of P D t with eigenvalue z ∈ C \ {e -λ0t } such that |z| = e -λ0t . Let ψ ∈ C b (D) be the eigenvector of P D t from Theorem 2.12. First, let us prove that D h t (x)ψ(x)dx = 0 by computing the integral D h t (x) P D t ψ(x)dx in two different ways. On the one hand, by Theorem 2.12, which leads to a contradiction, therefore such an eigenvalue does not exist.

D
We are now able to prove Theorem 2.18.

Proof of Theorem 2.18.

Step 1: We shall first prove [START_REF] Ramil | Processus cinétiques dans des domaines à bord et quasi-stationnarité[END_REF] On the one hand, it is clear that this is a closed subset of C b (D), and thus a Banach space. On the other hand, it follows from Theorems 2.7 and 2.12 that Span(ψ) ⊥ is stable by P D 1 . As a consequence, we may consider in the sequel the operator P D 1 | Span(ψ) ⊥ .

P D N f -e -λ0N φ ⊗ ψ D φψ (f ) ∞ = P D N f - φ ⊗ ψ D φψ (f ) ∞ ≤ e -(λ0+α)N f - φ ⊗ ψ D φψ (f ) ∞ .
Let t ≥ N 0 , then ⌊t⌋ ≥ N 0 , and we have that

P D t f -e -λ0t φ ⊗ ψ D φψ (f ) ∞ = P D t-⌊t⌋ P D ⌊t⌋ f - φ ⊗ ψ D φψ (f ) ∞ ≤ P D ⌊t⌋ f - φ ⊗ ψ D φψ (f ) ∞ ≤ e -(λ0+α)⌊t⌋ f - φ ⊗ ψ D φψ (f ) ∞ ≤ e -(λ0+α)t e λ0+α f - φ ⊗ ψ D φψ (f )
∞ ≤ e -(λ0+α)t e λ0+α 1 + φ ∞ D φψ f ∞ , which concludes the proof of (19) when f ∈ C b (D), since the behavior of the left-hand side for t ≤ N 0 can easily be bounded appropriately.

Step 2: Let us extend [START_REF] Ramil | Processus cinétiques dans des domaines à bord et quasi-stationnarité[END_REF] for f ∈ L ∞ (D). It follows from Theorem 2.8 that for any s > 0 and any f ∈ L ∞ (D), P D s f ∈ C b (D). Therefore, by Step 1, for all α ∈ [0, α * ), there exists C α > 0 such that for all t > 0, ǫ ∈ (0, t) and f ∈ L ∞ (D),

P D t-ǫ (P D ǫ f ) -e -λ0(t-ǫ) φ ⊗ ψ(P D ǫ f ) D φψ ∞ ≤ C α P D ǫ f ∞ e -(λ0+α)(t-ǫ) .
Furthermore, As a result, for all α ∈ [0, α * ), there exists C α > 0 such that for all t > 0, ǫ ∈ (0, t) and f ∈ L ∞ (D),

P D t-ǫ (P D ǫ f ) = P D t f , P D ǫ f ∞ ≤ f L ∞
P D t f -e -λ0t φ ⊗ ψ(f ) D φψ ∞ ≤ C α f L ∞ (D) e -(λ0+α)(t-ǫ) ,
which yields the proof of Step 2.

We conclude this section with a proof of the long-time convergence, in total variation, of the distribution of the Langevin process conditioned to remain in D, towards its QSD on D.

Proof of Theorem 2.21. We first recall that for any probability measure θ on D and t ≥ 0,

P θ (X t ∈ •|τ ∂ > t) -µ T V = sup f ∈L ∞ (D), f L ∞ (D) ≤1 E θ [f (X t )|τ ∂ > t] - D f dµ .
Let us fix α ∈ [0, α * ) and show that there exists C ′ α such that for any initial distribution θ, t ≥ 0 and f ∈ L ∞ (D), E θ (f (X t )1 τ ∂ >t )

P θ (τ ∂ > t) - D f ψ ≤ C ′ α D φdθ e -αt f L ∞ (D) .
First, let us prove that

P θ (τ ∂ > t) ≥ D φdθ φ ∞ e -λ0t .
For x ∈ D, t > 0 one has

P x (τ ∂ > t) = D p D t (

  for any p ∈ [1, +∞] by Steps 1 and 2. Let s > 0 and f ∈ L ∞ (D). Since for all x ∈ D, P D s f (x) = D p D s/2 (x, y)P D s/2 f (y)dy, and P D s/2 maps L ∞ (D) into L 1 (D) by Step 2, then it is an immediate application of the dominated convergence theorem along with the continuity and boundedness of x ∈ D → p D s/2 (x, y) in Theorem 2.6 that P D s f ∈ C b (D). Besides, one has obviously

D t/ 3 :

 3 L p (D) → L 2 (D) and P D t/3 : L 2 (D) → L p (D), and the compactness of P D t/3 : L 2 (D) → L 2 (D), we obtain that P D t is a compact operator from L p (D) to L p (D). Similarly, writing P D t = ιP D t/2 P D t/2 , where ι is the injection from C b (D) to L ∞ (D), using the continuity of the operators ι and P t/2 : L ∞ (D) → C b (D), as well as the compactness of the operator P D t/2 : L ∞ (D) → L ∞ (D) that we have just proven, we conclude that P D t is a compact operator from C b (D) to C b (D).

  , Theorem VI.6]. Proposition 3.3 (Gelfand's formula). Let T be a bounded real operator on C b (D). One has that r(T ) = lim n→∞ |||T n ||| 1/n C b (D) .

  for f ∈ C b (D) then use a regularization argument of the semigroup (P D t ) t≥0 in the next step to conclude. Let us define the following vector space of C b (D), Span(ψ) ⊥ := f ∈ C b (D) : D f (x)ψ(x)dx = 0 .

  (D) and by the Fubini permutation along with Theorem 2.7 sincef, ψ ∈ L ∞ (D), e λ0ǫ φ ⊗ ψ(P D ǫ f ) = e λ0ǫ φ D ψP D ǫ f = e (λ0+dγ)ǫ φ D f P D ǫ ψ = φ ⊗ ψ(f ).

  and F is bounded and globally Lipschitz continuous on R d .Under Assumption (F2), τ ∞ = ∞ almost surely and the Langevin process (6) admits a smooth transition density p t (x, y) [21, Corollary 7.2], which is positive[START_REF] Herzog | A practical criterion for positivity of transition densities[END_REF] Corollary 3.3]. In addition, this density admits an explicit Gaussian upper-bound, see[START_REF] Lelièvre | A probabilistic study of the kinetic Fokker-Planck equation in cylindrical domains[END_REF] Theorem 2.19].

  2.22. The convergence speed and the prefactor are similar to what is obtained in[START_REF] Ramil | Processus cinétiques dans des domaines à bord et quasi-stationnarité[END_REF] Chapter 4] using the results by Champagnat and Villemonais in[START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF].

	Remark 2.23. Similar statements for Theorems 2.18 and 2.21 can be obtained for the adjoint Langevin process ( Xt ) t≥0

  be a compact set with positive Lebesgue measure, i.e. |C| > 0. It follows from Theorem 2.6 that P D t admits a smooth transition density p D t , which is positive on C × C. Therefore, there exists α > 0 such that for all x, y ∈ C, p D

t (x, y) ≥ α. Besides, for n ≥ 1, P D t n C b (D)

  2. Let us prove that every real-valued eigenvector of P D t associated with the eigenvalue r(P D t ) has a constant sign. Assume that there exists h t ∈ C b (D) such that P D t h t = r(P D t )h t and h t changes sign on D. Then, by the positivity of p D t one has for x ∈ D, Up to changing h t to -h t , we can assume that h t ∈ K by Step 2. Notice that, φtht is an eigenvector of P D t with eigenvalue r(P D t ), therefore it has a constant sign. By (33), one concludes that necessarily φtht = 0 on D since ψ t > 0 on D. Hence h t ∈ Span(φ t ) and r(P D t ) is a simple eigenvalue. Step 4. Applying this time Step 2 and Step 3 to the operator P D t , one also obtains that r( P D t

		r(P D t )|h t (x)| = |P D t h t (x)|
			=	D	p D t (x, y)h t (y)dy
			<
				= e dγt	D	P D t ψ t (x)|h t (x)|dx
				= r(P D t )
				Let us define
	for x ∈ D,	ht (x) :=	h t (x)
				(33)

D p D t (x, y)|h t (y)|dy = P D t |h t |(x). As a result, since ψ t > 0 on D, by Theorem 2.7 one has that r(P D t ) D ψ t (x)|h t (x)|dx < D ψ t (x)P D t |h t |(x)dx D ψ t (x)|h t (x)|dx, by (30), which leads to a contradiction. Therefore, h t has a constant sign on D. Step 3. Let h t be a real-valued eigenvector of P D t associated with the eigenvalue r(P D t ), let us prove that h t ∈ Span(φ t ). D ψ t (y)h t (y)dy , φt (x) := φ t (x) D ψ t (y)φ t (y)dy , so that D ht (x)ψ t (x)dx = D φt (x)ψ t (x)dx = 1.

  (x, y)|h t (y)|dy = P D t |h t |(x), by the triangle inequality since the equality case requires that Re(h t ) and Im(h t ) have constant signs on D, which would imply h t = 0 from (53) since ψ > 0 on D. As a result,

	h t (x) P D t ψ(x)dx = e -(λ0+dγ)t	D	h t (x)ψ(x)dx.
	On the other hand, by Theorem 2.7,
				(53)
	Besides, one has for x ∈ D,
	r(P D t )|h t (x)| = |P D t h t (x)|
	=	D	p D t (x, y)h t (y)dy
	< t r(P D D p D t )
			= e dγt	D	P D t ψ(x)|h t (x)|dx
			= r(P D t )

D h t (x) P D t ψ(x)dx = e -dγt D P D t h t (x)ψ(x)dx = ze -dγt D h t (x)ψ(x)dx.

Therefore, since z = e -λ0t , D h t (x)ψ(x)dx = 0, and in particular

D Re(h t (x))ψ(x)dx = D Im(h t (x))ψ(x)dx = 0. D ψ(x)|h t (x)|dx < D ψ(x)P D t |h t |(x)dx D ψ(x)|h t (x)|dx

  x, y)dy P θ (τ ∂ > t) e -(λ0+α)t f L ∞ (D) + f L ∞ (D)C α e -(λ0+α)t P θ (τ ∂ > t)

		=	D	p D t (x, y)	φ(y) φ(y)	dy
		≥ D p D t (x, y)φ(y)dy φ ∞	=	φ(x)e -λ0 t φ ∞	,
	by Theorem 2.12. Therefore,
	P P θ (τ ∂ > t)	θ(dx)
	+		ψf	e -λ0t D φdθ
		D
	≤	2C α D φdθ	e -αt f L

θ (τ ∂ > t) = D θ(dx)P x (τ ∂ > t) ≥ D φdθ φ ∞ e -λ0t .

(54)

As a result, it follows from Theorem 2.18 and the inequality (54), the existence of C α > 0 such that for all t > 0,

E θ (f (X t )1 τ ∂ >t ) P θ (τ ∂ > t) -D f ψ = D E x (f (X t )1 τ ∂ >t ) -D f ψ P θ (τ ∂ > t) θ(dx) P θ (τ ∂ > t) ≤ D E x (f (X t )1 τ ∂ >t )e -λ0t D ψf D φψ φ(x) D φψ -P θ (τ ∂ > t) P θ (τ ∂ > t) ≤ C α ∞ (D) φ ∞ .

Acknowledgments : Mouad Ramil is supported by the Région Ile-de-France through a PhD fellowship of the Domaine d'Intérêt Majeur (DIM) Math Innov. This work also benefited from the support of the projects ANR EFI (ANR-17-CE40-0030) and ANR QuAMProcs (ANR-19-CE40-0010) from the French National Research Agency. Finally, Tony Lelièvre has received funding from the European Research-Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 810367), project EMC2. The authors would also like to thank Etienne Bernard (CERMICS) for useful discussions on analytical properties of semigroups.

The compactness of P D 1 ensures the compactness of P D 1 | Span(ψ) ⊥ as well. Therefore, any nonzero element of the spectrum σ(P ) is a simple eigenvalue associated to a positive function φ which thus does not belong to Span(ψ) ⊥ .

In any case, we thus have r(P D 1 | Span(ψ) ⊥ ) < r(P D 1 ), so that there exists α * ∈ (0, +∞] such that r(P D 1 | Span(ψ) ⊥ ) = e -λ0-α * . In addition, for α ∈ [0, α * ), by Proposition 3.3 there exists N 0 ≥ 1 such that all N ≥ N 0 ,

where we have used the semigroup property to write


Hirsh Majid
email: hirshmajid@yahoo.com

Hassane Abouaïssa

Daniel Jolly
email: daniel.jolly@univ-artois.fr

Gildas Morvan
email: gildas.morvan@univ-artois.fr

M Gildas

Advanced Control for Real-Time Dynamic Traffic Routing Problem

Keywords: Traffic modelling, Dynamic traffic assignment, Flatness-based control

The aim of this paper is to present a new algorithm to deal with the real-time Dynamic Traffic Routing (DTR) problem using a second order traffic flow model (METANET). The proposed approach rests on the use of the concept of differential flatness for which the structure of the trajectories of the dynamics can be completely characterized, without integration of any differential equations. The objective is to achieve user-equilibrium on alternate routes in a network setting. The relevance of the algorithm is demonstrated via a set of numerical simulations.

I. INTRODUCTION

Traffic control represents the main way to improve the freeway throughput and to ensure an efficient, safe and less polluting transportation of goods and persons [START_REF] Kostialos | Motorway network traffic control systems[END_REF]. It also contributes to a large reduction of direct and indirect costs.

Freeway traffic control can be achieved via a set of actions and measurements such as: dynamic speed limits, route guidance, dynamic assignment and dynamic traffic routing , ramp metering, ... etc.

Dynamic Traffic Assignment (DTA) and Dynamic Traffic Routing (DTR) represents one of the most efficient solutions to steer the congestion problems especially the no-recurrent one. DTR/DTA or control of traffic diversion rests on the determination of a time-dependent split variables at the diversion point in order to achieve a user-equilibrium traffic pattern [6]. Several algorithms have been developed for this problem. Most of them rest on optimization approaches such those proposed by [START_REF] Petta | Multiple users classes real time traffic assignement for on-line operations: a rolling horizon solution framework[END_REF]. As stated in [7], the proposed optimization algorithm attempts to solve the DTR problem by optimizing the objective functions for the nominal model over the planning horizon and it is not adapted for on-line DTR control. Papageorgiou [START_REF] Papageorgiou | Dynamic modeling, assignment and route guidance traffic networks[END_REF] and Messmer & Papageorgiou [START_REF] Messmer | Route diversion control in motorway networks via nonlinear optimization[END_REF] have proposed algorithms for Dynamic Traffic Assignment (DTA) based on the linear quadratic regulator, and nonlinear optimization techniques. Other techniques use expert systems to deal with the diversion problem [START_REF] Gupta | Development of prototype kbes in the management of non-recurrent congestion of the massachusetts turnpike[END_REF]. Liu & al. [9] have proposed a strategy based on a model reference adaptive control in order to guide the real-world traffic flow to evolve towards the desired states especially under emergency evacuation. Kachroo & Özbay [5] have designed feedback and fuzzy control laws for an on-line diversion problem. Another algorithm for DTR and DTA problems based on the nonlinear H ∞ was proposed in [4]. These algorithms are based on a simple conservation law and a linear fundamental diagram proposed by [2] that do not reflect all the traffic flow phenomena.

In this paper, we use a specific class of complex systems called "Differentially flat systems" to solve the userequilibrium DTA/DTR. The main characteristics of flatness is particularly well tuned for allowing one to solve inverse dynamics problems and one builds off of that fundamental solution in using the structure of flatness to solve more general control problems [START_REF] Ph | Flat systems[END_REF]. A system is flat if it does exist a set of variables called "flat outputs", such that the system is (nondifferentially) algebraic over the differential field generated by the set of flat outputs. In other words, a system is flat if we can find a set of outputs (equal in number to the number of inputs) such that all states and inputs can be determined from these outputs without integration. Moreover, in this paper the non-destination oriented METANET model [START_REF] Messemer | [END_REF] [1] is used which is a second-order, deterministic, macroscopic traffic flow model that consists of the Payne model [START_REF] Payne | Models of freeway traffic and control[END_REF] discretized in both space and time with some model enhancements proposed by Papageorgiou added. The paper is organized as follows:

• Section II, presents the main principles of the DTR problem and its mathematical formulation. • Section III, recalls the main definitions of flat systems.

• Section IV shows the control design methodology using the concept of flatness.

• Section V provides some numerical simulations for a sample network. • Section VI concludes the paper and outlines some tracks for future developments.

II. DYNAMIC TRAFFIC ROUTING FORMULATION

In this section, we first present a mathematical formulation which is usable for the design of DTA/DTR flat controller. We expose a second order traffic flow model ( METANET). Notice that there are two ways to describe the system dynamics for DTA/DTR problems: the link-based model and the route-based model. In this paper we focus on the route-based model.

A. DTR model

For the sake of simplicity, we consider the case of two alternate routes divided into n 1 and n 2 sections respectively, (1-α)qe qe qs 2,n2 2,1 ( )q The METANET model is used to design the studied system; The equation of dynamic density is:

ρi,j (t) = 1 L i,j [q i,j-1 (t) -q i,j (t)] (1)
where;

(i, j) = ((1, 1), (1, 2), . . . , (1, n 1 ), (2, 1), (2, 2), . . . , (2, n 2 )). L i,j is the section length and ρ i,j represents the traffic density at section j of the route i.

The relationships between the traffic flow q i,j and the mean speed v i,j is:

q i,j (t) = ρ i,j (t)v i,j (t)λ i,j (2)
λ i,j is the number of lanes. v i,j is a dynamic mean speed of section j on the route i.

The equation of dynamic mean speed is:

vi,j (t) = 1 τ (V ρi,j (t) -v i,j (t)) + 1 Li,j v i,j (t)(v i,j-1 (t) -v i,j (t)) -ν τ Li,j
ρi,j+1(t)-ρi,j (t) ρi,j (t)+κ

(3) V ρi,j is defined as a nonlinear expression (See [START_REF] May | Traffic Flow Fundamentals[END_REF]):

V ρi,j (t) = vf i,j exp - 1 a ρ i,j (t) ρ ci,j a (4)
ρ ci,j represents the critical density and a, a model parameter. vf i,j , the free-flow speed.

The control input α(t) ∈ [0, 1] is defined as an exogenous variable of the system [START_REF] Pavlis | Simple decentralized feedback strategies for route guidance in traffic networks[END_REF], it is a split rate which allows to reach a user equilibrium traffic pattern: For the case of two routes, the equations of entry flow in each route are:

α(t)q e (t) = q 1,in (t) (1 -α)q e (t) = q 2,in (t) (5)
q e (t) is the incoming measured flow. The control objective is to find the split rates α in order to minimize the total travel time T T . Kachroo and Özbay [7] have formulates the DTR problem as follow: find α 0 , the optimal α(t), which mininizes

J(α) = t f 0 m i=1 T T (ρ i ) - m+p m+1 T T (ρ i ) 2 dt (6)
where, T T is the travel time function and t f the final time.

In the general case, consider a traffic flow system with n alternate routes. The system is described by the same ordinary differential equation (1) where, (i, j) = ((1, 1), . . . , (1, n 1 ), (2, 1), . . . , (2, n 2 ), (n, 1), . . . , (n, n n )). The problem consists then on finding a set of split variables, α n-1 , where, n-1 i=1 α i = 1 which minimize the total travel time T T , (See [7], for optimal formulation of the DTR problem for a general case)

         α 1 (t)q e (t) = q 1,in (t) α 2 (t)q e (t) = q 2,in (t) . . . (1 -α 1 -α 2 -, . . . , -α n-1 )q e (t) = q n,in (t) (7)
Note that for such complex problem, an open loop control structure is not sufficient [4], it calls for a more robust feedback control one. In this context, we highlight the interest of an attractive approach based on the concept of differential flatness [START_REF] Fliess | Sur les systèmes non linéaires différentiellements plats[END_REF] [START_REF] Fliess | Flatness and defect of non-linear systems: Introductory theory and examples[END_REF]. The following section, recalls the main definitions and principles of differentially flat systems.

III. DIFFERENTIALLY FLAT SYSTEMS

The DTR problem can be solved by a typical use of control theory which consists the system dynamics inversion to compute the required inputs to achieve a user-equilibrium traffic pattern. As stated in [START_REF] Ph | Flat systems[END_REF], This inversion may involve finding appropriate inputs to steer a control system from one state to another or may imply to find inputs to follow a desired trajectory for some or all of the state variables of the system. Generally, if it does exist, the solution to such problems will not be unique. In this case a trade off between the system performance for the stability and actuation effort must be done. Frequently, this trade off is described as in [6], as a cost function balancing the desired performance objectives with stability and actuation effort, resulting in an optimal control problem as solved in [START_REF] Messmer | Route diversion control in motorway networks via nonlinear optimization[END_REF].

The basic paradigm used in most, if not all, control techniques is to exploit the mathematical structure of the system to obtain solutions to the inverse dynamics and feedback regulation problems. The most common structure to exploit is linear structure, where one approximates the given system by its linearization and then uses properties of linear control systems combined with appropriate cost function to give closed form (or at least numerically computable) solutions. By using different linearizations around different operating points, it is even possible to obtain good results when the system is nonlinear by "scheduling" the gains depending on the operating point. As the systems that we seek to control become more complex, the use of linear structure alone is often not sufficient to solve the control problems that are arising in applications. This is especially true of inverse dynamics problems, where the desired task may span multiple operating regions and hence the use of a single linear system is inappropriate.

In this paper, we concentrate on a specific class of systems, called "differentially flat systems", for which the structure of the trajectories of the (nonlinear) dynamics can be completely characterized.

The concept of Flat systems was first introduced by Fliess and coworkers ( [11] [10]). more than 10 years ago using the formalism of differential algebra (See [START_REF] Ph | Contribution à l'étude des systèmes Différentiellement plat[END_REF]) for a slightly different approach of differentially flat systems). This special class of non-linear control systems described by ordinary differential equations: differentially flat systems form a special class of nonlinear control systems for which systematic control methods are available once a flat-output is explicitly known. The flatness-based concept was developed in a differentially algebraic context and was later expressed using Lie-Bäcklund transformation [START_REF] Fliess | A lie-bäcklund approach to equivalence and flatness of nonlinear systems[END_REF].

The flatness-based control methods may be expected to play a very significant role in high technology applications in the next few years, similar to what happened for nonlinear control in the last decade [START_REF] Sira-Ramirez | Differentially flat systems[END_REF]. The main property of flat systems is that all the state and input variables can be expressed directly, without integration of any differential equation, in term of the set of so-called "flat output" and a number of its time derivatives. More precisely, the entire system behavior is determined by the trajectory of a finite collection of quantities: flat outputs. This leads to a simple and elegant trajectory design. For a given system, the number of flat outputs is equal to the number of the system inputs. The flatness concept is closely related to the state feedback linearization.

A. Flat Systems Definition

In this section, We just sketch a tutorial definition of flatness for state-space control system. Consider the smooth system defined using the following equation:

ẋ = f (x, u); x ∈ R n , u ∈ R m (8)
where x = (x 1 , x 2 , ..., x n ) vector of state variables and u = (u 1 , u 2 , ..., u m ), m scalar control. The system (8) is flat if and only if, there exist m real smooth functions h = (h 1 , h 2 , ..., h m ) depending on x and a finite number of u derivatives, says β, such that, generically, the solution (x, u) of the square differential-algebraic system (t → y(t) is given)

ẋ = f (x, u), y(t) = h(x, u, u, ..., u (β) ) (9)
does not involve any differential equation and thus is of the form:

x = Φ(y, ẏ, ..., y (β) ), u = Ψ(y, ẏ, ..., y (β+1) )

where, Φ and Ψ are smooth functions, and β is some finite number [START_REF] Rouchon | Flatness based control of oscillators[END_REF]. The quantity y is of fundamental importance: it is called "flat output" or linearizing output. In the control language, the flat output y is such that, the inverse of ẋ = f (x, u), y = h(x, u, u, ..., u (β) ) has no dynamics ( [17] [23]).

Differentially flat systems are very useful in situations where the explicit generation of trajectories is required. Since the behavior of flat systems is determined by the flat outputs, one can plan the trajectories in the outputs space and then connect these to appropriate inputs. More precisely, from the trajectories of the flat outputs y , we can deduce immediately the trajectories of the state x and the input u variables. Applications of the flatness concept to engineering problems field have grown steadily in recent years and a variety of case studies have been shown to be flat and flatness based controllers based on trajectories generation by polynomial interpolation and then closing the loop on the obtained trajectories have been developed.

B. Trajectories planning for Flat systems

The main advantage of flat systems is there ability for solving control problems. The synthesis of control laws is divided into: "trajectory planning" corresponding to the open-loop control synthesis and "trajectory tracking" which corresponds to the closed-loop control design.

For flat systems, there is a systematic method for trajectory planning (See [START_REF] Rudolph | Flatness based control of distributed parameter systems[END_REF]). Often these trajectories are required to parameterize a finite time transition between two stationary regimes. These equilibria can be computed (locally) from relations such that: u = Ψ(y, ẏ, ..., y (β+1) ) by substituting y (i) = 0, i > 0. Thus, they are, specified by constant values of the flat output.

The transition between two states is determined by choosing a reference trajectory for the flat output, and y being differentially independent. The trajectories of its components can be chosen independently, and freely. In principle, any sufficiently smooth curve t → y * i (t) may be used. A particularly simple method uses polynomials to define reference trajectories. Then the coefficients of the polynomials, follow from the initial (t = 0) and final values (t = T ) of y by solving a linear system of equations (See [START_REF] Rudolph | Flatness based control of distributed parameter systems[END_REF] for the exact method of coefficients calculation).

IV. DIFFERENTIAL FLATNESS FOR THE DTR PROBLEM

In [3], authors introduce flatness-based control for dynamic traffic routing for two alternate routes with a single section using first order traffic flow model. Here we propose the same type of control using the METANET model. Moreover, in order to illustrate the relevance of the proposed algorithm, we extend it to a case of two alternate routes with two sections. The simulated freeway portion is shown in figure (2).

A. DTR control for two alternate routes with two sections

The space discretized equations of the system is

ρ11 (t) = 1 L 11 [αq e -ρ 11 (t)v 11 (t)] (11)
v11 (t) = 1 τ (V ρ11 (t) -v 11 (t)) + 1 L11 v 11 (t)(v 10 (t) -v 11 (t)) -ν τ L11 ρ12(t)-ρ11(t) ρ11(t)+κ ( 12
)
ρ12 (t) = 1 L 12 [ρ 11 (t)v 11 (t) -ρ 12 (t)v 12 (t)] ( 13
)
v12 (t) = 1 τ (V ρ12 (t) -v 12 (t)) + 1 L12 v 12 (t)(v 11 (t) -v 12 (t)) -ν τ L12 ρ13(t)-ρ12(t) ρ12(t)+κ ( 14
)
ρ21 (t) = 1 L 21 [(1 -α)q e -ρ 21 (t)v 21 (t)] ( 15
)
v21 (t) = 1 τ (V ρ21 (t) -v 21 (t)) + 1 L21 v 21 (t)(v 20 (t) -v 21 (t)) -ν τ L21 ρ22(t)-ρ21(t) ρ21(t)+κ (16)
ρ22 (t) = 1 L 22 [ρ 21 (t)v 21 (t) -ρ 22 (t)v 22 (t)] (17)
v22 (t) = 1 τ (V ρ22 (t) -v 22 (t)) + 1 L22 v 22 (t)(v 21 (t) -v 22 (t)) -ν τ L22 ρ23(t)-ρ22(t) ρ22(t)+κ (18)
V ρ11 (t) = vf 11 exp -1 a ρ11(t) ρc 11 a V ρ12 (t) = vf 12 exp -1 a ρ12(t) ρc 12 a V ρ21 (t) = vf 21 exp -1 a ρ21(t) ρc 21 a V ρ22 (t) = vf 22 exp -1 a ρ22(t) ρc 22 a
The travel time function is obtained as follow:

y 1 (t) = L 11 v 11 (t) + L 12 v 12 (t) (19)
y 2 (t) = L 21 v 21 (t) + L 22 v 22 (t) (20)
The variable y is equal to the difference in the travel time on the two sections.

y(t) = L 11 v 11 (t) + L 12 v 12 (t) - L 21 v 21 (t) + L 22 v 22 (t) (21)
The first time derivative leads to:

ẏ(t) = A + B + C + D + αE (22)
The terms A, B, C, and D ... represents derivatives of sections 21, 11, 12 and 22 respectively. And the term E consists of all the terms in which α does exist. It comes from sections 11 and 21. The term A is composed of three terms (A 1 , A 2 and A 3 );

A 1 = L 21 (v 21 ) 2    v21- vf 21 σ 4 τ - νσ6 L21τ (κ+ρ21) τ - ν(ρ21v21-ρ22v22) L22 L 21 τ (κ + ρ 21 )    A 2 = L 21 v 21 ρ 21   vf21σ8 L21ρc 21 σ4 τ - ν L21 L 21 τ (κ + ρ 21 ) + νσ 6 (L 21 ) 2 τ σ 9   A 3 = L 21 q e (v 21 ) 2   -vf21σ8 L21ρc 21 σ4 τ + ν L21 L 21 τ (κ + ρ 21 ) - νσ 6 (L 21 ) 2 τ σ 9  
The term B is composed of two terms (B 1 and B 2 );

B 1 = -L 11 (v 11 ) 2    v11- vf 11 σ 3 τ - νσ5 L11τ (κ+ρ11) τ - ν(ρ11v11-ρ12v12) L12 L 11 τ (κ + ρ 11 )    B 2 = -L 11 v 11 ρ 11   vf11σ7 L11ρc 11 σ3 τ - ν L11 L 11 τ (κ + ρ 11 ) + νσ 5 (L 11 ) 2 τ σ 10  
The term C is composed of three terms (C 1 , C 2 and C 3 );

C 1 = -L 12 (v 12 ) 2    -σ 13 + v12- vf 12 σ 11 τ -vf12σ15(ρ11v11-ρ12v12) L12ρc 12 σ11 τ    C 2 = 1 v 12 v 11 -vf11 σ19 τ - v 12 -vf12 σ11 τ + σ 13 - ν(ρ 11 -ρ 12 ) L 11 τ (κ + ρ 11 ) C 3 = σ 17 (v 12 ) 2 v 12 -vf12 σ11 τ -σ 13
The term D is composed of three terms (D 1 , D 2 and D 3 );

D 1 = L 22 (v 22 ) 2    -σ 14 + v22- vf 22 σ 12 τ -vf22σ16(ρ21v21-ρ22v22) L22ρc 22 σ12 τ    D 2 = -1 v 22 v 21 -vf21 σ20 τ - v 22 -vf22 σ12 τ + σ 14 - ν(ρ 21 -ρ 22 ) L 21 τ (κ + ρ 21 ) D 3 = - σ 18 (v 22 ) 2 v 22 -vf22 σ12 τ -σ 14
The term E is composed of two terms one concerns section 11 and the other concerns section 21;

E 11 = L 11 q e (v 11 ) 2   vf11σ7 L11ρc 11 σ3 τ - ν L11 L 11 τ (κ + ρ 11 ) + νσ 5 (L 11 ) 2 τ σ 10   E 21 = L 21 q e (v 21 ) 2   vf21σ8 L21ρc 21 σ4 τ - ν L21 L 21 τ (κ + ρ 21 ) + νσ 6 (L 21 ) 2 τ σ 9   σ 3 = exp 1 a ( ρ11 ρc 11 ) a , σ 4 = exp 1 a ( ρ21 ρc 21 ) a , σ 5 = ρ 11 -ρ 12 , σ 6 = ρ 21 -ρ 22 , σ 7 = ( ρ11 ρc 11
) (a-1) , σ 8 = ( ρ21 ρc 21

) (a-1) ,

σ 9 = (κ + ρ 21 ) 2 , σ 10 = (κ + ρ 11 ) 2 , σ 11 = exp 1 a ( ρ12 ρc 12
) a ,

σ 12 = exp 1 a ( ρ22 ρc 22
) a , σ 13 = v12σ17 L12 , σ 14 = v22σ18 L22 , σ 15 = ( ρ12 ρc 12

) (a-1) , σ 16 = ( ρ22 ρc 22

) (a-1) , σ 17 = v 11 -v 12 , σ 18 = v 21 -v 22 , σ 19 = exp( 1 a ( ρ11 ρc 11
) a ) and

σ 20 = exp( 1 a ( ρ21 ρc 21
) a ).

The studied system is characterized by one input (control) variable, we have then one flat output F = y, which represents the difference in travel time function between the two routes. Thus the equation ( 22) can be rewritten as:

Ḟ (t) = A + B + C + D + αE (23)
from which

α(t) = Ḟ (t) -(A + B + C + D) E (24)
The expression of the state variable allows to choose a suitable trajectory of the travel time (the flat output). The equation of the control (input) variable allows to add additional constraints to this travel time trajectory. This means that all important properties of the system [START_REF] Petta | Multiple users classes real time traffic assignement for on-line operations: a rolling horizon solution framework[END_REF] are contained in such a differential parametrization.

B. Trajectory planning:

The equation ( 24), corresponds to an open loop control algorithm. In order to define the trajectory planning, a suitable desired trajectory F * has to be defined. According to the expression of the control variable in [START_REF] Rudolph | Flatness based control of distributed parameter systems[END_REF], this trajectory must have smooth derivatives up to order two. In order to reduce computational effort in real time situation, one can build this reference trajectory for the travel time (flat output) using a polynomial interpolation [START_REF] Sira-Ramirez | Differentially flat systems[END_REF] from the initial and final conditions of the travel time (F (t i ) = F i , Ḟ (t i ) = 0) and (F (t f ) = F f , Ḟ (t f )) = 0). This is accomplished by prescribing the following desired trajectory for the flat output F : where ϕ(t, t i , t f ) is a polynomial function of time, exhibiting a sufficient number of zero derivatives at times t i and t f . (For the polynomial calculation see e.g. [START_REF] Rudolph | Flatness based control of distributed parameter systems[END_REF] [25]):

F * (t) =    F ti for t < t i F ti + (F t f -F ti )ϕ(t, t i , t f ) for t i ≤ t ≤ t f F t f for t > t f (25)
ϕ(t) =      0 if t < t i 3 t-ti t f -ti 2 -2 t-ti t f -ti 3 if t i ≤ t ≤ t f 1 if t > t f (26
) Thus, by replacing the term Ḟ (t) in equation ( 24) by the term of desired trajectory Ḟ * (t), we obtain the nominal open loop control:

α * (t) = Ḟ * (t) -(A + B + C + D) E (27)
C. Trajectory tracking:

Due to the parameter variations and disturbances in traffic flow, and changes in the traffic conditions, the open loop control is not sufficient to control the traffic flow. To ensure a steady state and reduce the influence of parameter variations, the traffic flow has to be operated in closed loop.

Using the flatness-based open loop presented in algorithm equation [START_REF] Rudolph | Flatness based control of distributed parameter systems[END_REF], an additional feedback can be determined in order to achieve desired dynamic behaviour and to compensate the external disturbances. Here, a PI regulator is used to feedback the system (See Fig. 3). Thus, the closed-loop control schema including the flatness-based loop can be obtained as follow:

ν(t) = Ḟ * (t) -k 1 (F (t) -F * (t)) -k 2 (F (t) -F * (t))dt (28)
where k 1 and k 2 are parameters that must be selected so as to satisfy the desired performances of the closed loop system and to ensure asymptotically stabilization of the input variable. The control law α now reads:

α(t) = Ḟ * (t) -k1(F (t) -F * (t)) -k2 (F (t) -F * (t))dt E - A + B + C + D E (29)
V. NUMERICAL SIMULATIONS The macroscopic simulation has been done during 165 minutes. Traffic incoming flow for the simulation is shown in figure (4). In figures (5, 6, 7 and 8) densities and mean speeds of the two sections and the two routes are shown respectively under control algorithm (see section IV-C). From figures, it is clear that both density and mean speed of the system in each reciprocal cell are equal, it is mean that the travel time difference between these two routes are zero. In figure [START_REF] Liu | Model reference adaptive control framework for real-time traffic management under emergency evacuation[END_REF] the control value for each route is given. In figure 10 the travel time of each routes and their difference is shown after applying the control algorithm.

VI. CONCLUSION

In this paper, a closed loop control algorithm was simulated in order to control the incoming flow of two routes that have the same length and geometry. The control algorithm is designed to minimize the travel time and optimize the

Figure 1 .

 1 Figure 1. Sample example of two alternate routes

Figure 2 .

 2 Figure 2. Simulated freeway section

Figure 3 .

 3 Figure 3. flow chart (Reduced-order estimator)

Figure ( 2

 2 Figure (2) is a schematic diagram of the freeway portion used in the simulation. It shows two routes with the same

Figure 4 .

 4 Figure 4. Traffic income flow

Figure 6 .

 6 Figure 5. Traffic density of section 1