Dorian Le Peutrec

AND Laurent Michel
email: laurent.michel@math.u-bordeaux.fr

Boris Nectoux
email: boris.nectoux@uca.fr

EYRING-KRAMERS TYPE FORMULAS FOR SOME PIECEWISE DETERMINISTIC MARKOV PROCESSES

Keywords: Piecewise Deterministic Markov Processes, small temperature regime, Eyring-Kramers formulas, metastability, spectral theory, semiclassical analysis AMS classification. 35P15, 47F05, 35P20, 35Q82, 35Q92

In this work, we give sharp asymptotic equivalents in the small temperature regime of the smallest eigenvalues of the generator of some piecewise deterministic Markov processes (including the Zig-Zag process and the Bouncy Particle Sampler process) with refreshment rate α on the one-dimensional torus T. These asymptotic equivalents are usually called Eyring-Kramers type formulas in the literature. The case when the refreshment rate α vanishes on T is also considered.

1. Introduction 1.1. Purpose and motivation. Piecewise Deterministic Markov Processes [START_REF] Davis | Piecewise-deterministic Markov processes: A general class of non-diffusion stochastic models[END_REF] (PDMP hereafter) have recently attracted a lot of attention for their use within the Markov Chain Monte Carlo methodology. Given a potential U ∶ M → R on a d-dimensional manifold M without boundary, such processes are indeed ergodic with respect to the Gibbs measure

π(dx) = e -2 h U (x)
∫ M e -2 h U dx, dx being the Lebesgue measure on the position space M. Here, the parameter h > 0 is proportional to the Boltzmann constant k B through the relation h = k B T , T being the temperature of the underlying system. When h > 0 is fixed, the ergodic properties and the rate of convergence of such processes with or without a refreshment rate α, have for instance been studied in [START_REF] Bierkens | Ergodicity of the zigzag process[END_REF][START_REF] Deligiannidis | Exponential Ergodicity of the Bouncy Particle Sampler[END_REF][START_REF]Geometric ergodicity of the Bouncy Particle Sampler[END_REF][START_REF] Dolbeault | Hypocoercivity for linear kinetic equations conserving mass[END_REF][START_REF] Andrieu | Hypercoercivity of Piecewise Deterministic Markov Process-Monte Carlo[END_REF][START_REF] Benaim | Qualitative properties of certain piecewise deterministic Markov processes[END_REF][START_REF] Miclo | Étude spectrale minutieuse de processus moins indécis que les autres[END_REF][START_REF] Monmarché | Piecewise deterministic simulated annealing[END_REF] (see also references therein). We also refer to [START_REF] Bierkens | Spectral analysis of the zigzag process[END_REF] for a spectral analysis of the generator of the one-dimensional Zig-Zag process and its corresponding semigroup when h is fixed.

In many applications in statistical physics where one needs to sample from π, the positive constant h is very small compared to the energetic barriers of U . Recently, the long time behavior of the semigroups generated by the generators L h of these processes on L 2 have been investigated in [START_REF] Guillin | Low-Lying Eigenvalues and Convergence to the Equilibrium of Some Piecewise Deterministic Markov Processes Generators in the Small Temperature Regime[END_REF] when h ≪ 1, where it has also been proved that in the subset {Re(z) ≥ -0 h} of the complex plane, L h has exactly n 0 eigenvalues (n 0 being the number of local minima of U ), which are non positive, real, and exponentially small as h → 0. Such results exhibit a metastable behavior of the PDMP when h ≪ 1, as it is the case for diffusion processes (see e.g. [START_REF] Di Gesù | Jump markov models and transition state theory: the quasi-stationary distribution approach[END_REF] and references therein). In this work, we want to push the analysis of the metastability of the PDMP further by proving that each of these n 0 1 eigenvalues satisfies a so-called Eyring-Kramers type formula when h → 0. Such sharp formulas describe completely the successive timescales involved in the convergence of the semigroup generated by -L h to π, as it has been done in [START_REF] Bovier | Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues[END_REF][START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach[END_REF] for elliptic reversible diffusions.

Let us be more precise on our results. We work in the following setting: d = 1 (i.e. the dimension of the position space is equal to 1) and M = T, the one-dimensional torus. In addition, we work with the operator P h defined below in (1.1) which is (up to a multiplication by -h) unitary equivalent to L h , see (1.2). Thus, all our results are easily translated in terms of L h . The purpose of this work is to compute sharp asymptotic equivalents of the n 0 smallest eigenvalues of P h (or equivalently, those of L h ) in the limit h → 0, see Theorems 2 and 3. In particular these asymptotic equivalents allow us to provide a sharp exponential decay rate of the semigroup associated with -P h in L 2 when h ≪ 1, see Corollary 1.5. These results hold when (1.4) is satisfied, which implies that the refreshment rate α does not vanish at critical points of U (i.e. where ∂ x U = 0), see (1.5). The case α = 0 when ∂ x U = 0 is investigated in Section 1. [START_REF] Bierkens | Ergodicity of the zigzag process[END_REF], where we show that the smallest eigenvalues of P h satisfy different asymptotic equivalents as h → 0, see Theorem 4 To compute sharp asymptotic equivalents of the n 0 smallest eigenvalues of P h , we proceed as follows. We first introduce a suitable change of variables to turn the eigenvalue problem (P h -λ)u = 0 into a nonlinear eigenvalue problem:

(∆ V,h -λW + λ 2 )g = 0,
where V = -U , ∆ V,h is the Witten Laplacian associated with V , and W = 2 ∂ x U + α (see (2.7) and Lemma 2.2). Using a Grushin problem and known results on the low-lying spectrum of ∆ V,h , we then prove that when λ is small enough, the kernel of ∆ V,h + λW + λ 2 is composed of the singularities of a holomorphic function λ ↦ E -+ (λ) ∈ C n 0 ×n 0 (see (2.23)). We finally investigate the localization of the singularities of λ ↦ E -1 -+ (λ) to deduce sharp asymptotic equivalents of the n 0 smallest eigenvalues of P h . 1.2. Setting. Let E = {(x, v) ∈ T × {±1}}, where T = R Z is the one-dimensional torus. Consider the following unbounded operator P h on L 2 (E) associated with a smooth function U ∶ T → R defined by (1.1)

P h = -v d U,h + 2(v ∂ x U ) + (I -B) + α(I -π v ),
where d U,h = h∂ x +∂ x U , α ∶ T → R is a C ∞ non-negative function (the refreshment rate), h > 0 is a parameter proportional to the temperature of the underlying statistical system, I is the identity operator, and B and π v are defined by

∀(x, v) ∈ E, Bf (x, v) = f (x, -v)
and

π v f (x, v) = 1 2 (f (x, +1) + f (x, -1)) for all f ∈ L 2 (E).
Here and in the following, for u ∈ L 2 (E), we denote by u ± = max(0, ±u).

The operator P h is linked to the Zig-Zag (or the Bouncy Particle Sampler generator) process generator L h , where

L h = v∂ x -( 2 h v∂ x U ) + (I -B) -1 h α(I -π v ), through the relation (1.2) P h = -h e -1 h U L h e 1 h U
. We refer to [START_REF] Bierkens | The zig-zag process and super-efficient sampling for Bayesian analysis of big data[END_REF][START_REF] Peters | Rejection-free Monte Carlo sampling for general potentials[END_REF][START_REF] Bouchard-Côté | The bouncy particle sampler: A nonreversible rejection-free markov chain Monte Carlo method[END_REF] and references therein for more details on these two processes (see also [START_REF] Faggionato | Non-equilibrium thermodynamics of piecewise deterministic Markov processes[END_REF] and [START_REF] Durmus | Piecewise Deterministic Markov Processes and their invariant measure[END_REF]Section 3.1]). The space E = T × {±1} is endowed with the natural scalar product

(1.3) ⟨f, g⟩ L 2 (E) = 1 2 v=±1 T f (x, v)g(x, v)dx.
Throughout this work, we will assume that

U ∶ T → R is a smooth Morse function.
By definition, this means that ∂ 2 x U (x) ≠ 0 when ∂ x U (x) = 0, x ∈ T. In particular, U has a finite number of critical points on T. It is proved in [START_REF] Guillin | Low-Lying Eigenvalues and Convergence to the Equilibrium of Some Piecewise Deterministic Markov Processes Generators in the Small Temperature Regime[END_REF] that P h with domain

D(P h ) = {f ∈ L 2 (E), v∂ x f ∈ L 2 (E)
} is maximal accretive. The following spectral result on P h in the limit h → 0 is also proved in [START_REF] Guillin | Low-Lying Eigenvalues and Convergence to the Equilibrium of Some Piecewise Deterministic Markov Processes Generators in the Small Temperature Regime[END_REF].

Theorem 1. ( [16, Theorem 1 and Proposition 13]) Assume that U is a Morse function with n 0 local minimum points. Assume also that

(1.4) min T 2 ∂ x U + α > 0.
Then, there exist 0 > 0 and h 0 > 0 such that for all h ∈]0, h 0 ], σ(P h ) ∩ {Re(z) ≤ 0 h 2 } is made of n 0 real nonnegative eigenvalues λ 1,h ≤ . . . ≤ λ n 0 ,h (counted with algebraic multiplicity). Moreover, their algebraic multiplicities equal their geometric multiplicities and there exist C > 0 and h 0 > 0 such that for all h ∈ (0, h 0 ], λ n 0 ,h ≤ e -C h . Finally, λ 1,h = 0 and has algebraic multiplicity 1.

Let us mention that dim Ker(P h ) = 1 holds for all h > 0 (see [START_REF] Guillin | Low-Lying Eigenvalues and Convergence to the Equilibrium of Some Piecewise Deterministic Markov Processes Generators in the Small Temperature Regime[END_REF]). In [START_REF] Guillin | Low-Lying Eigenvalues and Convergence to the Equilibrium of Some Piecewise Deterministic Markov Processes Generators in the Small Temperature Regime[END_REF], it is assumed that min T α > 0, but all the results of [START_REF] Guillin | Low-Lying Eigenvalues and Convergence to the Equilibrium of Some Piecewise Deterministic Markov Processes Generators in the Small Temperature Regime[END_REF] still hold when the less stringent assumption (1.4) is satisfied. Indeed, if (1.4) holds and u ∈ D(P h ), then Re ⟨P h u, u⟩

L 2 (E) ≥ r m (I -π v )u 2 ,
where r m = min E 2 ∂ x U + α. This follows from the following computations. By [START_REF] Guillin | Low-Lying Eigenvalues and Convergence to the Equilibrium of Some Piecewise Deterministic Markov Processes Generators in the Small Temperature Regime[END_REF]Lemma 5], since (I -B) = 2(I -π v ), one has for all u ∈ D(P h ), denoting by w = (I -π v )u (notice that w(⋅, 1) = w(⋅, -1) ):

Re ⟨P h u, u⟩ L 2 (E) = 1 2 E (2v ∂ x U ) + (I -B)u 2 + E α (I -π v )u 2 = E 4(v ∂ x U ) + w 2 + E α w 2 = 1 2 T 4(∂ x U ) + w 2 + 1 2 T 4(∂ x U ) -w 2 + E α w 2 = E (2 ∂ x U + α) w 2 .
Notice that (1.4) allows α to vanish on T, but not everywhere since (1.4) is equivalent to:

(1.5) for any x ∈ T, ∂ x U (x) = 0 ⇒ α(x) > 0.
The case when there exists x ∈ T such that ∂ x U (x) = α(x) = 0 is considered in Section 1.4.

For our analysis, it is convenient to introduce the function

V = -U,
which is also a Morse function on T. We shall denote by U (0) the set of local minima of V (i.e. of local maxima of U ) and by U (1) the set of its local maxima (i.e. of local minima of U ). Since we are on the torus, the sets U (0) and U (1) have the same cardinality n 0 .

Remark 1.1. As mentioned in the introduction, we use the spectral properties of the Witten Laplacian

∆ V,h = -h 2 ∆ + ∂ x V (x) 2 -h∂ 2 x V (x)
to study the spectrum of P h . Let us emphasize that, considering the Witten Laplacian acting on the full space of differential forms, ∆ V,h is simply

∆ (1)
U,h acting on 1-forms (through the identification between functions and 1-forms) on the one-dimensional torus. In addition, the Witten complex structure on the torus shows that σ(∆

(1) U,h ) = σ(∆ U,h ) and thus σ(∆ V,h ) = σ(∆ U,h ), counting the multiplicities. Furthermore, ∆ U,h = d * U,h d U,h
, where d * U,h = -h∂ x + ∂ x U , and d U,h induces a one-to-one correspondance between the eigenfunctions of ∆ U,h associated with its nonzero eigenvalues (i.e. which are not in Span(e -U h )) and those of ∆ V,h (i.e. which are not in Span(e U h )).

In the present setting, the application Ψ defined in (2.8) realizes a correspondance between the Witten Laplacian and the generator L h of the PDMP. We refer e.g. to the introduction of [START_REF] Helffer | Puits multiples en mécanique semi-classique. IV. Étude du complexe de Witten[END_REF] and references therein on the Witten complex structure.

Throughout the paper, we will say that a family of complex numbers (a h ) h>0 admits a classical expansion in power of h β (where β > 0) if there exists a sequence (a k ) k≥0 such that, for all K ≥ 0, one has a h = ∑ K k=0 a k h βk + O(h β(K+1) ) when h → 0. In that case, we will denote a h ∼ ∑ k≥0 a k h βk . 1.3. Main results. In this section we give sharp asymptotics of the exponentially small eigenvalues λ j,h , j = 1, . . . , n 0 . Observe that if n 0 = 1, there is only one small eigenvalue by Theorem 1 which is 0, and there is thus nothing to compute. We then consider the case when n 0 ≥ 2. We start with the following theorem, which gives the result in the simplified setting of a non-symmetric double well potential V , or equivalently of a non-symmetric double well potential U (see Theorem 3 below for the general case n 0 ≥ 2). Theorem 2. Let U be a Morse function. Assume that (1.4) holds and that U (0) is made of two elements m 1 and m 2 such that V (m 1 ) < V (m 2 ). Assume also that the two elements s 1 , s 2 of U (1) satisfy V (s 1 ) > V (s 2 ). Let 0 > 0 be as in Theorem 1. Then, the second smallest eigenvalue λ 2,h of P h satisfies as h → 0:

λ 2,h = ζ h h e -2 h (V (s 2 )-V (m 2 )) , where ζ h ∼ k≥0 h k 2 ζ k and ζ 0 = 1 π V ′′ (m 2 ) V ′′ (s 2 ) α(m 2 ) .
The situation where V (m 1 ) = V (m 2 ) and/or V (s 1 ) = V (s 2 ) could be handled easily by constructing adapted quasimodes in the spirit of [START_REF] Michel | About small eigenvalues of the Witten Laplacian[END_REF]. Here, we decided to state our result in the above simplified setting in order to lighten the formulas. Note that, in terms of U , s 1 and s 2 are the local minima and U (s 1 ) < U (s 2 ), while m 1 and m 2 are the local maxima and U (m 1 ) > U (m 2 ). In addition, the energetic barrier V (s 2 ) -V (m 2 ) for V is the energetic barrier U (m 2 ) -U (s 2 ) for U . Moreover,

ζ 0 = 1 π U ′′ (m 2 ) U ′′ (s 2 ) α(m 2 ) .
Let us now state our results in the general setting n 0 ≥ 2. To this end, we need to label the local minima and maxima of V in a suitable way. The following construction is inspired by [START_REF] Hérau | Tunnel effect and symmetries for Kramers-Fokker-Planck type operators[END_REF] (see also [START_REF] Michel | About small eigenvalues of the Witten Laplacian[END_REF] and [START_REF] Peutrec | Sharp spectral asymptotics for non-reversible diffusion processes[END_REF]). In order to simplify, we assume from now that V uniquely attains its maximum at the point s 1 ∈ U (1) , i.e. that (1.6) argmax T V = {s 1 }.

Then, set U (1) = U (1) ∖ {s 1 }. Since n 0 ≥ 2, U (1) ≠ ∅. We denote the elements of V (U (1) ) by σ 2 > σ 3 > . . . > σ N , where N ≥ 2. For convenience, we also introduce a fictive infinite saddle value σ 1 = +∞ and we denote Σ = {σ 1 , . . . , σ N }. Starting from σ 1 , we will recursively associate to each σ i a finite family of local minima (m i,j ) j and a finite family (C i,j ) j of connected components of {V < σ i } in the following way

⋆ Let X σ 1 = {x ∈ T; V (x) < σ 1 = +∞} = T.
We let m 1,1 be an arbitrary global minimum of V and C 1,1 = T. ⋆ Next we consider X σ 2 = {x ∈ T; V (x) < σ 2 }. This is the union of its finitely many connected components. Exactly one of these components contains m 1,1 and the other components are denoted by C 2,1 , . . . , C 2,N 2 . In each component C 2,j , we pick up a point m 2,j which is a global minimum of V C 2,j . ⋆ Suppose now that the families (m k,j ) j and (C k,j ) j have been constructed until rank k = i -1. The set X σ i = {x ∈ T; V (x) < σ i } has again finitely many connected components and we label C i,j , j = 1, . . . , N i , those of these components which do not contain any m k, with k < i. In each C i,j , we pick up a point m i,j which is a global minimum of V C i,j .

We run the procedure until all the minima have been labeled.

Remark 1.2. Since we work on T, using the terminology of [START_REF] Hérau | Tunnel effect and symmetries for Kramers-Fokker-Planck type operators[END_REF], every maximum point s ∈ U (1) is a separating saddle points whereas s 1 is not. In the case where V attains its global maximum at several distinct points s 1,1 , . . . , s 1,k , every maximum point is separating and the situation can be handled easily by a modification of the above construction. However, this would lead to a slightly more complicated presentation of the results that we prefer to avoid in this work.

We now recall some constructions of [START_REF] Michel | About small eigenvalues of the Witten Laplacian[END_REF] and [START_REF] Peutrec | Sharp spectral asymptotics for non-reversible diffusion processes[END_REF] that will be useful in the sequel. Throughout, we denote by s ∞ a fictive saddle point such that V (s ∞ ) = σ 1 = +∞. For any set A, P(A) denotes the power set of A. From the above labelling, we define two mappings

C ∶ U (0) → P(R d ) and j ∶ U (0) → P(U (1) ∪ {s ∞ })
as follows: for every i ∈ {1, . . . , N } and j ∈ {1, . . . , N i },

(1.7) C(m i,j ) ∶= C i,j ,

and

(1.8) j(m 1,1 ) ∶= {s ∞ } and j(m i,j ) ∶= ∂C i,j ∩ U (1) for i ≥ 2.

In particular, we have

C(m 1,1 ) = T and ∅ ≠ j(m i,j ) ⊂ {V = σ i } for all i, j ∈ {1, . . . , N }.
Note also that, in our 1-dimensional setting, the cardinality of every j(m), m ∈ U (0) , is 1 or 2.

We then define the mappings

σ ∶ U (0) → Σ and S ∶ U (0) → (0, +∞],
by

(1.9) ∀m ∈ U (0) , σ(m) ∶= V (j(m)) and S(m) ∶= σ(m) -V (m),
where, with a slight abuse of notation, we have identified the set V (j(m)) with its unique element. Note that S(m) = +∞ if and only if m = m 1,1 . With the above notations, our last assumption is the following:

(1.10)

⋆ Equation (1.6) is satisfied. ⋆ For any m ∈ U (0) , m is the unique global minimum of V C(m) . ⋆ For all m ′ ≠ m ∈ U (0) , j(m) ∩ j(m ′ ) = ∅.
⋆ The map S:

U (0) → (0, +∞] is injective. m 1,1 j(m 3,2 ) j(m 3,1 ) j(m 2,1 ) m 1,1 m 2,1 m 3,2 m 3,1 C 1,1 = T C 2,1 C 3,2 C 3,1 s 1 s 1 m 3,1 S(m 2,1 ) S(m 3,1 ) S(m 3,2 ) Figure 1.1.
Schematic representation of the maps C and j. In this example, the function V satisfies (1.10) and V (j(m 3,1 )) = V (j(m 3,2 )).

In particular, (1.10) implies that V uniquely attains its global minimum on T at m 1,1 . Moreover, the dimension being 1, (1.10) actually implies that every j(m), m ∈ U (0) , has cardinality 1. This follows for example from [START_REF] Peutrec | Sharp spectral asymptotics for non-reversible diffusion processes[END_REF]Remark 1.7], by taking into account the dimension. Hence, identifying every j(m) with its unique element, the map j ∶ U (0) → P(U (1) ∪ {s 1 }) defined in (1.8) becomes an injective function

(1.11) j ∶ U (0) → U (1) ∪ {s ∞ } such that j(m 1,1 ) = s ∞ .
In the following, when (1.10) holds, we label the local minima m 1 , . . . , m n 0 of V such that (S(m j )) j∈{1,...,n 0 } is decreasing (see (1.9)), that is such that (1.12) for all j ∈ {1, . . . , n 0 -1}, S(m j+1 ) < S(m j ) and S(m 1 ) = +∞ (i.e. m 1 = m 1,1 ).

Remark 1.3. Note that, in the geometrical setting of Theorem 2, one has by construction of S (see (1.9)) and of j (see

(1.11)), S(m 2 ) = V (s 2 ) -V (m 2 ) and j(m 2 ) = s 2 .
The main result of this work is the following. Theorem 3. Let V = -U be a Morse function. Assume that (1.4) and (1.10) are satisfied. Let 0 > 0 be given by Theorem 1. Then, there exists h 0 > 0 such that, for all h ∈]0, h 0 ], the n 0 eigenvalues λ 1,h ≤ . . . ≤ λ n 0 ,h of P h in {Re(z) ≤ 0 h 2 } (counted with algebraic multiplicity) satisfy: λ 1,h = 0 and, for all j ∈ {2, . . . , n 0 },

(1.13) λ j,h = ζ h (m j ) h e -2 h S(m j ) , where S ∶ U (0) → (0, +∞] is defined in (1.9) (see also (1.12)) and ζ h admits a classical expansion ζ h (m j ) ∼ ∑ k≥0 h k 2 ζ k (m j ) with (1.14) ζ 0 (m j ) = 1 π V ′′ (m j ) V ′′ (j(m j )) α(m j ) ,
where j ∶ U (0) → U (1) ∪ {s ∞ } is defined in (1.11).

Notice that, according to Theorem 3, λ j,h is a simple eigenvalue of P h for all j ∈ {1, . . . , n 0 } and h small enough (that is dim Ker(P h -λ j,h ) m = 1 for every m ∈ N * ).

Remark 1.4. The assumptions (1.6) and S: U (0) → (0, +∞] is injective in (1.10) are generic. They can be relaxed following the procedure of [START_REF] Peutrec | Sharp spectral asymptotics for non-reversible diffusion processes[END_REF]. The whole assumption (1.10) could also be relaxed by following the strategy of [START_REF] Michel | About small eigenvalues of the Witten Laplacian[END_REF].

Let us recall that by the Hille-Yosida Theorem, -P h generates a strongly continuous contraction semigroup (e -tP h ) t≥0 on L 2 (E). Let us recall that under the assumptions of Theorem 3, according to [16, Theorem 2], we have, for some c > 0 and every h > 0 small enough:

e -tP h = n 0 j=1 e -tλ j,h Π j,h + O(e -c t h 2 ) in L(L 2 (E)),
where, for j = 1, . . . , n 0 , Π j,h is the spectral projector associated with the eigenvalue λ j,h of P h , and Π j,h = O(1) in L(L 2 (E)). Using in addition Theorem 3, we get sharp asymptotic equivalents of the different timescales 1 λ j,h involved in the return to equilibrium. This leads in particular to the following accurate exponential decay rate in L 2 (E) of the semigroup (e -tP h ) t≥0 as h ≪ 1.

Corollary 1.5. Let U be a Morse function. Assume that (1.4) and (1.10) are satisfied. Then, there exist C > 0 and h 0 > 0 such that, for all h ∈ (0, h 0 ), it holds for all t ≥ 0:

e -tP h -Π 1,h L(L 2 (E)) ≤ Ce -tλ 2,h , where, as h → 0, λ 2,h = ζ h (m 2 ) h e -2S(m 2 ) h with ζ h (m 2 ) ∼ ∑ k≥0 h k 2 ζ k (m 2
) and ζ 0 (m 2 ) given by (1.14), and where Π 1,h is the L 2 (E) orthogonal projection on Span(e -U h 1 {±1} ), where 1 {±1} is the constant function on {±1} which equals 1.

1.4. Extension of the results to the case when (1.4) is not satisfied. In this section, we assume that (1.15) for any x ∈ T,

∂ x U (x) = 0 ⇒ α(x) = 0,
and we give asymptotic equivalents of the n 0 first eigenvalues of P h when (1.15) holds (and then, in particular, when there is no refreshment at all, i.e. when α = 0). Then, for any c > 0, there exists h 0 > 0 such that, for all h ∈ (0, h 0 ], σ(P h ) ∩ D(0, ch 2 ) is made of n 0 real eigenvalues λ 1,h ≤ . . . ≤ λ n 0 ,h (counted with geometric multiplicity), which all have geometric multiplicity one. In addition, λ 1,h = 0 and, for all j ∈ {2, . . . , n 0 },

(1.16) λ j,h = ζ h (m j ) √ h e -2 h S(m j ) ,
where

ζ h admits a classical expansion ζ h (m j ) ∼ ∑ k≥0 h k 2 ζ k (m j ) with (1.17) ζ 0 (m j ) = 1 2 √ π V ′′ (j(m j ))
,

where j ∶ U (0) → U (1) ∪ {s ∞ } is defined in (1.11).
Compared with Theorem 3, we cannot exclude the existence of other eigenvalues in the strip 0 < Re(z) < ch 2 with large imaginary part. The reason for this is that Theorem 1 does not apply when α and ∂ x U vanish simultaneously.

The prefactor (1.17) is consistent with the one obtained in [23, Theorem 1.1] for the expected hitting time of a 1-dimensional PDMP with no refreshment. Indeed, when α = 0 and V , and thus U = -V , is a non-degenerate double well potential (i.e. when

U (0) = {m 1 , m 2 } with V (m 1 ) < V (m 2 ) and U (1) = {s 1 , s 2 } with V (s 1 ) > V (s 2 )), Theorem 4 and [23, Theorem 1.1] imply that h -1 λ 2,h E[τ ] = 1 + o(1) when h → 0, where τ is the first time X t reaches {m 2 } when (X 0 , Y 0 ) = (s 2 , -1), where (X t , Y t ) is the process with generator L h on T × {±1}. Let us recall here that h -1 λ 2,h is the first nonzero eigenvalue of -L h (see (1.2)).
Of course, the situation where the refreshing function α vanishes at some critical points x ∈ T of U but not at all could easily be handled and would lead, for each eigenvalue λ j,h , either to the formula given in Theorem 3 or to the formula given in Theorem 4, depending on whether α(m j ) > 0 or α(m j ) = 0.

Reduction to a finite dimensional problem

In this section, we prove that, if λ is small enough, we can reduce the infinite dimensional problem (P h -λ h )u = 0 into a finite dimensional nonlinear eigenvalue problem.

2.1.

A suitable change of variables. Let us introduce some notation. For λ ∈ C and r > 0 we denote D(λ, r) = {z ∈ C, z -λ < r}. For two families of numbers a = (a h ) h>0 and b = (b h ) h>0 , we say that a ∈ E cl (b) if there exists a family c = (c h ) h>0 such that a h = b h c h and c admits a classical expansion c h ∼ ∑ j≥0 c j h j with c 0 = 1 as h → 0. In all this work, C > 0 and c > 0 are constant which are independent of h and which can change from one occurence to another.

Recall that E = T × {±1} is endowed with the natural scalar product (1.3). Let F = L 2 (T) × L 2 (T) and let ⟨, ⟩ F be the Hilbertian structure induced by the isomorphism

Ω 1 ∶ L 2 (T×{±1}) → L 2 (T) × L 2 (T) f → f (., +1) f (., -1) . (2.1) Then, Ω 1 is unitary from (E, ⟨, ⟩ L 2 (E) ) onto (F, ⟨, ⟩ F ),
and

Ω 1 D(P h ) = H 1 (T) × H 1 (T). Direct computations show that Ω 1 v d U,h Ω -1 1 = d U,h 0 0 -d U,h , Ω 1 (v ∂ x U ) + Ω -1 1 = (∂ x U ) + 0 0 (∂ x U ) - Ω 1 BΩ -1 1 = 0 1 1 0 , and Ω 1 π v Ω -1 1 = 1 2 1 1 1 1 .
Combining these identities with (1.1), we get

(2.2) Ω 1 P h Ω -1 1 = -d U,h 0 0 d U,h + 2 (∂ x U ) + -(∂ x U ) + -(∂ x U ) -(∂ x U ) - + α 2 1 -1 -1 1 .
We now change the variables in F and consider the unitary transformation Ω 2 ∶ F → F defined by

Ω 2 (f, g) = 1 √ 2 (f + g, g -f ).
Consider the two vectors of R 2 given by e 1 = 1 1 and e 2 = -1 1 . Then, one has:

(2.3) Ω 1 P h Ω -1 1 (f e 1 ) = d U,h f e 2 . On the other hand, 2 (∂ x U ) + -(∂ x U ) + -(∂ x U ) -(∂ x U ) - -f f = 4f -(∂ x U ) + (∂ x U ) - = 2((∂ x U ) --(∂ x U ) + ) f f + 2((∂ x U ) -+ (∂ x U ) + ) -f f = -2∂ x U f e 1 + 2 ∂ x U f e 2 .
It follows that:

Ω 1 P h Ω -1 1 (f e 2 ) = d U,h f e 1 -2∂ x U f e 1 + (α + 2 ∂ x U )f e 2 = d -U,h f e 1 + (α + 2 ∂ x U )f e 2 (2.4)
Combining (2.3) and (2.4), we get

Ω 2 Ω 1 P h Ω -1 1 Ω -1 2 = 0 d -U,h d U,h α + 2 ∂ x U . Set Q h ∶= Ω 2 Ω 1 P h Ω -1 1 Ω -1 2 with domain D(Q h ) = H 1 (T) × H 1 (T) on L 2 (T) 2 , i.e.
(2.5)

Q h = 0 d -U,h d U,h W = 0 d V,h -d * V,h W
where we recall that V = -U and where we denote

(2.6) W (x) ∶= α(x) + 2 ∂ x U (x) = α(x) + 2 ∂ x V (x) , x ∈ T.
Since the transformation Ω 2 Ω 1 is unitary from E onto F,

σ(P h ) = σ(Q h ).
Remark 2.1. Note that the identity (I -B) = 2(I -π v ), which is specific to the onedimensional setting, is crucial to derive the operator Q h .

We will in the following study the spectrum of Q h to prove Theorems 2 and 3. Consider the semiclassical Witten Laplacian ∆ V,h associated with V on T, that is (see Remark 1.1)

∆ V,h = d * V,h d V,h = -h 2 ∆ + ∂ x V (x) 2 -h∂ 2 x V (x), with domain D(∆ V,h ) = H 2 (T). For λ ∈ C, we finally define (2.7) T h (λ) = ∆ V,h -λW + λ 2 with domain D(∆ V,h ).
The following result is the key point of our analysis: it establishes an equivalence between the spectrum of Q h and the kernels of the operators

T h (λ), λ ∈ C. Lemma 2.2. The operator Q h with domain D(Q h ) = H 1 (T) × H 1 (T)
is closed and has compact resolvent. In particular, it has only discrete spectrum. Moreover, λ = 0 is a simple eigenvalue of Q h and dim(Ker

T h (0)) = dim(Ker Q h ) = 1.
Besides, for every λ ∈ C ∖ {0}, the application

(2.8) Ψ ∶ Ker(T h (λ)) → Ker(Q h -λ) g → 1 λ d V,h g g is a linear isomorphism. Eventually, any µ ∈ C is a singularity of λ ↦ T h (λ) -1 if and only if there exists g ∈ H 2 (T), g ≠ 0, such that T h (µ)g = 0.
Proof. The operator Q h is closed on L 2 (T) 2 since it is a bounded perturbation of the closed operator (0, h∂ x ; h∂ x , 0). The resolvent of Q h is compact since the injection

H 1 (T) ⊂ L 2 (T) is compact. Since Q h is unitarily equivalent to P h (see (2.5)), it follows from Theorem 1 that λ = 0 is a simple eigenvalue of Q h . Moreover, since T h (0) = ∆ V,h , one gets Ker T h (0) = Ce -V h
and then dim(Ker T h (0)) = 1.

Let us now consider λ ∈ C ∖ {0}. For any g ∈ Ker(T h (λ)), one has g ∈ H 2 (T) and thus

( 1 λ d V,h g, g) ∈ D(Q h ). It follows moreover from T h (λ)g = 0 that Q h Ψ(g) = 0 d V,h -d * V,h W 1 λ d V,h g g = d V,h g (-1 λ ∆ V,h + W )g = λΨ(g).
This proves that Ψ is well defined. The linearity and the injectivity of Ψ are obvious. To prove its surjectivity, consider u

= f g ∈ D(Q h ) such that Q h u = λu, i.e. such that d V,h g = λf, -d * V,h f + W g = λg.
It follows from the first equation that g ∈ H 2 (T) and, since λ ≠ 0, that u =

1 λ d V,h g g . Moreover, applying d * V,h to the second equation leads to ∆ V,h g = λd * V,h f = λW g -λ 2 g. Consequently, g ∈ D(∆ V,h
) and T h (λ)g = 0, which proves the surjectivity.

It remains to prove the last statement of Lemma 2.2. To this end, let us consider

µ ∈ C. It is a singularity of λ ↦ T h (λ) -1 if and only if T h (µ) ∶ H 2 (T) → L 2 (T) is not invertible. Furthermore, since ∆ V,h + 1 ∶ H 2 (T) → L 2 (T) is invertible, T h (λ) = 1 -(1 + λW -λ 2 )(∆ V,h + 1) -1 (∆ V,h + 1) is not invertible if and only if 1 -B(λ) ∶ L 2 (T) → L 2 (T) is not invertible, where B(λ) = (1 + λW -λ 2 )(∆ V,h + 1) -1 . Since (∆ V,h + 1) -1 : L 2 (T) → L 2 (T) is compact, so is B(λ).
The Fredholm alternative then implies that T h (µ) is not invertible if and only if there exists u ∈ L 2 (T), u ≠ 0 such that (1 -B(λ))u = 0, that is if and only if there exists g ∈ H 2 (T), g ≠ 0 such that T h (µ)g = 0. This concludes the proof of Lemma 2.2. ◻

According to Lemma 2.2, for every µ ∈ C:

(2.9) µ ∈ σ d (P h ) = σ(P h ) if and only if µ is a singularity of λ ↦ T h (λ) -1
and, for such a µ, one has

(2.10) dim(Ker(P h -µ)) = dim(Ker(T h (µ)).
To prove Theorems 2 and 3, we will thus investigate the singularities of λ ↦ T h (λ) -1 near 0.

2.2. Finite dimensional reduction, a Grushin problem. In this section, we show that the singularities of T h (λ) near 0 are those of a matrix valued holomorphic function E -+ (λ). To this end, we will construct a so-called Grushin problem.

This construction relies on well known results on the low-lying spectrum of ∆ V,h . From the early works of Witten [START_REF] Witten | Supersymmetry and Morse theory[END_REF] and Helffer-Sjöstrand [START_REF] Helffer | Puits multiples en mécanique semi-classique. IV. Étude du complexe de Witten[END_REF], we know that there is a one-to-one correspondence between the local minima of V and the smallest eigenvalues of ∆ V,h . This correspondence was further investigated by several authors and sharp asymptotic equivalents of these small eigenvalues were finally obtained in [START_REF] Bovier | Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues[END_REF] and [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach[END_REF] under assumptions on the relative positions of the minima and saddle points, and in [START_REF] Michel | About small eigenvalues of the Witten Laplacian[END_REF] in the general case. The following version gives the general form of these asymptotic equivalents in a non-degenerate setting.

Theorem 5. [START_REF] Bovier | Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues[END_REF], [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach[END_REF]. Let U = -V be a Morse function. There exist * > 0, C > 0, and h 0 > 0 such that, for all h ∈]0, h 0 ], the nonnegative self-adjoint operator (∆ V,h , H 2 (T)) admits exactly n 0 eigenvalues (counted with algebraic multiplicity) in [0, * h]:

(2.11) σ(∆ V,h ) ∩ [0, ε * h] = {0, µ ∆ 2,h , . . . , µ ∆ n 0 ,h }, where 0 is a simple eigenvalue of ∆ V,h . Let us order {µ ∆ 2,h , . . . , µ ∆ n 0 ,h } such that µ ∆ j,h ≤ µ ∆ j+1,h
for j = 1, . . . , n 0 -1. Then, when (1.10) holds, we have for all j = 2, . . . , n 0 :

(2.12) µ ∆ j,h = a h (m j ) h e -2 h S(m j ) , where S ∶ U (0) → (0, +∞] is defined in (1.9) (see also (1.12)) and a h (m j ) admits a classical expansion a h (m j ) ∼ ∑ k≥0 h k a k (m j ) with

(2.13) a 0 (m j ) = 1 π V ′′ (m j ) V ′′ (j(m j )) ,
where j ∶ U (0) → U (1) ∪ {s ∞ } is defined in (1.11).

Remark 2.3. We have decided to work with ∆ V,h , which appeared naturally in the above derivation of T h (λ). However, according to Remark 1.1, we could equivalently have worked with ∆ U,h .

We are now in position to construct a Grushin problem. Set According to Theorem 5, the space E 0 has dimension n 0 for all h > 0 small enough. Let (Ψ j ) j=1,...n 0 be an orthonormal basis of the space E 0 . We assume without loss of generality that Ψ 1 is proportional to e -V h . Introduce the operators

R -∶ C n 0 → L 2 (T) u ↦ n 0 j=1 u j Ψ j (2.15) and R + ∶ L 2 (T) → C n 0 u ↦ (⟨u, Ψ j ⟩ L 2 ) j . (2.16)
We equip C n 0 with the 2 norm. Notice that R + R -= I C n 0 and R -R + = Π. In addition, R + ≤ 1 and R -≤ 1, for all h > 0. These inequalities will be used many times in what follows. From now on, we denote Π = I -Π. Moreover, there exist C, 0 , h 0 > 0 such that for all h ∈]0, h 0 ], and for all λ ∈ D(0, 0 h), Th (λ) is invertible, holomorphic with respect to λ, and Th (λ

) -1 ≤ Ch -1 .
Proof. The proof is very close to the proof of Lemma 2.1 in [START_REF] Peutrec | Sharp spectral asymptotics for non-reversible diffusion processes[END_REF]. We sketch it for reader's convenience. We first observe that since W is bounded, the operator T h (λ) with domain

D(T h (λ)) = D(∆ V,h ) = H 2 (T) is closed and densely defined. Its adjoint T h (λ) * satisfies D(T h (λ) * ) = H 2 (T). Suppose that (u n , Th (λ)u n ) ∈ D( Th (λ)) × L 2 (T) converges to (u, v) ∈ L 2 (T) × L 2 (T)
. For any j = 1, . . . , n 0 , one has Ψ j ∈ D(∆ V,h ) = D(T h (λ) * ) and hence for all n ∈ N, one has

(2.17) ΠT h (λ)u n = n 0 j=1 ⟨T h (λ)u n , Ψ j ⟩Ψ j = n 0 j=1 ⟨u n , T h (λ) * Ψ j ⟩Ψ j .
Consequently, the sequence (ΠT h (λ)u n ) n converges and using the identity

T h (λ)u n = ΠT h (λ)u n + Th (λ)u n ,
it follows that (T h (λ)u n ) converges. Since T h (λ) is closed as a bounded perturbation of a closed operator, and

u n ∈ H 2 (T) = D(T h (λ)), one deduces that u ∈ D(T h (λ)) and T h (λ)u = lim T h (λ)u n . Since Π = 1 -Π is bounded, this implies that u ∈ D( Th (λ))
and Th (λ)u = v which proves that Th (λ) is closed. In addition, the operator Th (λ) is clearly densely defined on ΠL 2 . Let us now study the invertibility of Th (λ). For any u ∈ D( Th (λ)), one has by definition

Re⟨ Th (λ)u, u⟩ = Re⟨(∆ V,h -λW + λ 2 ) Πu, Πu⟩.
Using Theorem 5, this implies that for λ < 0 h, one has Re⟨ Th (λ)u, u⟩

≥ * h Πu 2 -Re(λ)⟨W Πu, Πu⟩ + Re(λ 2 ) Πu 2 ≥ ( * -0 W L ∞ )h Πu 2 -2 0 h 2 Πu 2 ≥ * 2 h Πu 2
for 0 small enough. This proves that Th (λ) is injective when λ < 0 h. We observe that the same proof shows that Th (λ) * = ΠT h (λ) * Π is injective. Moreover, the Cauchy-Schwarz inequality implies that for all u ∈ D( Th (λ)),

(2.18) Th (λ)u ≥ * 2 h Πu .
Let us now prove that Th (λ) is surjective if λ < 0 h. We first observe that Ran( Th (λ)) is closed since Th (λ) is closed and the convergence of any sequence ( Th (λ)u n ) implies the convergence of (u n ) thanks to (2.18). Hence it is sufficient to prove that Ran( Th (λ)) = {0}, which is consequence of the injectivity of the operator Th (λ) * . ◻

We now introduce the Grushin operator

(2.19) K h (λ) = T h (λ) R - R + 0 .
Proposition 2.5. Let U be a Morse function. There exist 0 > 0 and h 0 > 0 such that, for all h ∈]0, h 0 ] and all λ ∈ D(0, 0 h), the operator K h (λ) is invertible. Moreover, its inverse

E h (λ) writes E h (λ) = E(λ) E + (λ) E -(λ) E -+ (λ) ,
where E, E -, E + , E -+ are holomorphic in D(0, 0 h) and satisfy the following formulas:

E + (λ) = R --Th (λ) -1 ΠT h (λ)R -, E -(λ) = R + -R + T h (λ) Th (λ) -1 Π , (2.20) E -+ (λ) = -R + T h (λ)R -+ R + T h (λ) ΠT h (λ) -1 ΠT h (λ)R -∶ C n 0 → C n 0 , and
(2.21) E(λ) = Th (λ) -1 Π .
Moreover, for λ ∈ D(0, 0 h), T h (λ) is invertible if and only if the matrix E -+ (λ) is invertible, in which case it holds

T h (λ) -1 = E(λ) -E + (λ)E -1 -+ (λ)E -(λ).
Proof. Thanks to Lemma 2.4, the proof is reduced to an algebraic computation which is completely analogous to the one used in the proof of [START_REF] Peutrec | Sharp spectral asymptotics for non-reversible diffusion processes[END_REF]Lemma 2.2]. ◻

We now give some direct consequences of Proposition 2.5 and Lemma 2.4, which will be used in the following. By definition, one has R + T h (λ) Π = -λR + W Π. Hence (2.20) becomes

E -+ (λ) = -R + T h (λ)R -+ λ 2 R + W ΠT h (λ) -1 ΠW R -.
Introducing the matrices

(2.22) M V,h = R + ∆ V,h R -, W h = R + W R -, and G h (λ) = λ 2 (I C n 0 -R + W ΠT h (λ) -1 ΠW R -), this rewrites (2.23) -E -+ (λ) = M V,h -λW h + G h (λ).
Moreover, it follows from Lemma 2.4 that λ ∈ D(0, 0 h) ↦ G h (λ) is holomorphic and that there exists C > 0 such that for all h small enough and all λ ∈ D(0, 0 h),

(2.24) G h (λ) ≤ C λ 2 h -1 .
According to (2.9) and to Proposition 2.5,

(2.25) λ ∈ D(0, 0 h) ∩ σ(P h ) if and only if λ ∈ D(0, 0 h) is a singularity of E -1 -+ (λ).
To prove Theorems 2 and 3, the strategy consists in studying the singularities of E -1 -+ (λ) in D(0, 0 h), which first requires to compute asymptotic equivalents of M V,h and W h as h → 0 (see (2.23)).

The double well case

In this section we prove Theorem 2. To this end, we assume throughout this section that U (0) has exactly two elements m 1 and m 2 such that V (m 1 ) < V (m 2 ). Assume also that the two elements s 1 , s 2 of U (1) satisfy V (s 1 ) > V (s 2 ). Recall that in this case, one has by construction of S (see (1.9)) and j (see (1.11)), S(m 2 ) = V (s 2 ) -V (m 2 ) and j(m 2 ) = s 2 (see Remark 1.3). Thus, according to Theorem 5, it holds Hence, in order to compute an asymptotic equivalent of W h in the limit h → 0, we first consider a suitable basis (Ψ 1 , Ψ 2 ), which is chosen as follows.

(3.1) µ ∆ 2,h = a h (m 2 ) h e -2(V (s 2 )-V (m 2 )) h , with a h (m 2 ) ∼ ∑ k≥0 h k a k (m 2 ) and a 0 (m 2 ) = 1 π V ′′ (m 2 ) V ′′ (s 2 ) . 3.1. Asymptotic equivalent of W h , as h → 0. Recall that W h = R + W R -(
We choose Ψ 1 as the L 2 (T)-normalized solution of ∆ V,h Ψ 1 = 0, namely:

Ψ 1 = 1 Z 1,h e -(V -V (m 1 )) h with Z 1,h = e -(V -V (m 1 )) h L 2 (T) ,
where by Laplace's method: Z 1,h ∈ E cl ((πh V ′′ (m 1 )) 1 4 ), since m 1 is the unique global minimum of V on T. We recall that Z 1,h ∈ E cl ((πh V ′′ (m 1 )) 1 4 ) means that Z 1,h = (πh V ′′ (m 1 )) 1 4 z h where z h ∼ 1 + ∑ k≥1 w k h k , see the beginning of Section 2.

To construct Ψ 2 , we first choose a L 2 (T)-normalized function ϕ 2 such that both ∆ V,h ϕ 2 and ⟨Ψ 1 , ϕ 2 ⟩ L 2 (T) are exponentially small as h → 0. Since Ψ 1 concentrates around m 1 and V (m 2 ) > V (m 1 ), a good candidate is given by:

(3.2) ϕ 2 = 1 Z 2,h χ 2 e -(V -V (m 2 )) h with Z 2,h = e -(V -V (m 2 )) h L 2 (T) ,
where r > 0 and

χ 2 ∈ C ∞ c (T, [0, 1]) satisfies 1l B(m 2 ,r) ≤ χ 2 ≤ 1l B(m 2 ,2r
) . When r > 0 is small enough, m 2 is the unique global minimum of V on the closure of B(m 2 , 2r) (because V is a Morse function). Thus, by Laplace's method: Z 2,h ∈ E cl ((πh V ′′ (m 2 )) 1 4 ). In addition, for such a fixed r > 0, V > V (m 2 ) on supp(∇χ 2 ), and therefore, since supp([∆, χ 2 ]) ⊂ supp(∇χ 2 ) and ∆ V,h (e -V h ) = 0, there exists C > 0 such that for h small enough:

(3.3) ∆ V,h ϕ 2 = -h 2 [∆, χ 2 ]ϕ 2 =O(e -C h ) in L 2 (T).
On the other hand, since ∆ V,h is self-adjoint, it follows from the localisation of the spectrum stated in Theorem 5 (with n 0 = 2 there) that

(3.4) ∀z ∈ ∂D(0, * h 2), (∆ V,h -z) -1 L(L 2 (T)) = 1 d(z, σ(∆ V,h )) = O(h -1 ).
By definition of Π and according to Theorem 5,

Π = 1 2iπ ∂D(0, * 2 h) (z -∆ V,h ) -1 dz
and, using (3.3) and (3.4) together with the second resolvent identity, it follows that Πϕ 2 satisfies

Πϕ 2 -ϕ 2 = 1 2iπ ∂D(0, * 2 h) (z -∆ V,h ) -1 -z -1 ϕ 2 dz = 1 2iπ ∂D(0, * 2 h) z -1 (z -∆ V,h ) -1 ∆ V,h ϕ 2 dz = O(e -C h ). (3.5)
Moreover, since V (m 2 ) > V (m 1 ) and ΠΨ 1 = Ψ 1 , one has for h small enough:

(3.6) ⟨Πϕ 2 , Ψ 1 ⟩ L 2 (T) = ⟨ϕ 2 , Ψ 1 ⟩ L 2 (T) = O(e -C h ).
We finally set

Ψ 2 = Πϕ 2 -⟨Πϕ 2 , Ψ 1 ⟩ L 2 (T) Ψ 1 Πϕ 2 -⟨Πϕ 2 , Ψ 1 ⟩Ψ 1 L 2 (T) .
The function Ψ 2 belongs to E 0 , is orthogonal to Ψ 1 , and Ψ 2 L 2 (T) = 1. From now on, we consider the above orthonormal basis (Ψ 1 , Ψ 2 ) of E 0 . Notice that, using (3.5) and (3.6), one has ⟨Πϕ 2 ,

Ψ 1 ⟩Ψ 1 = O(e -C h ) and Πϕ 2 -⟨Πϕ 2 , Ψ 1 ⟩Ψ 1 L 2 (T) = 1 + O(e -C h ). Consequently, (3.7)
Ψ 2 = ϕ 2 + O(e -C h ) in L 2 (T). Lemma 3.1.
With the above choice of (Ψ 1 , Ψ 2 ), there exists C > 0 such that for all h small enough,

W h = γ 1,h 0 0 γ 2,h + O(e -C h ),
where we recall that W h is defined in (2.22) and where, for i ∈ {1, 2}, γ i,h satisfies as h → 0:

γ i,h ∼ ∑ k≥0 h k γ α,k (m i ) + √ h ∑ k≥0 h k γ V,k (m i ), with γ α,0 (m i ) = α(m i ) and γ V,0 (m i ) = 2 V ′′ (m i ) π . Proof. Since (Ψ 1 , Ψ 2 ) is an orthonormal family, one has W h = (⟨W Ψ i , Ψ j ⟩ L 2 (T) ) i,j=1,2 . In addition, Ψ 1 = O(e -C h ) in L 2 (supp(ϕ 2 )). Hence, using (3.7), for all i ≠ j, ⟨W Ψ i , Ψ j ⟩ L 2 (T) = O(e -C h ). Suppose now that i ∈ {1, 2} is fixed. By definition of W , one has ⟨W Ψ i , Ψ i ⟩ L 2 (T) = ⟨αΨ i , Ψ i ⟩ L 2 (T) + 2⟨ ∂ x V Ψ i , Ψ i ⟩ L 2 (T)
. By definition of Ψ 1 and Ψ 2 above, using Laplace's method, one has as h → 0:

⟨αΨ i , Ψ i ⟩ L 2 (T) ∼ ∑ k≥0 h k γ α,k (m i ) with γ α,0 (m i ) = α(m i ).
On the other hand, for i ∈ {1, 2}, one has for δ > 0 small enough, 1 4 ), this ends the proof of the lemma. ◻

⟨ ∂ x V Ψ i , Ψ i ⟩ L 2 (T) = Z -2 i,h x-m i <δ ∂ x V (x) e -2(V (x)-V (m i )) h dx + O(e -c h ) = Z -2 i,h m i +δ m i ∂ x V (x)e -2(V (x)-V (m i )) h dx - m i m i -δ ∂ x V (x)e -2(V (x)-V (m i )) h dx + O(e -c h ) = hZ -2 i,h + O(e -c h ). Using the fact that Z i,h ∈ E cl ((πh V ′′ (m i ))
We are now in position to prove Theorem 2.

3.2.

End of the proof of Theorem 2. Recall that the strategy consists in localizing the singularities of

E -1 -+ (λ) in D(0, 0 h) (see (2.25)). First of all, λ = 0 is always a singularity of λ ∈ D(0, 0 h) ↦ E -1 -+ (λ), since T h (0)Ψ 1 = 0 (see Proposition 2.

5). Let us now look for the other singularities of

λ ∈ D(0, 0 h) ↦ E -1 -+ (λ). Since dim E 0 = 2, ∆ V,h Ψ 1 = 0, and Ψ 2 ∈ E 0 is orthogonal to Ψ 1 , it holds: ∆ V,h Ψ 2 = µ ∆ 2,h Ψ 2 and therefore, M V,h = diag(0, µ ∆ 2,h
). By Lemma 3.1, (2.23), and (2.24), one then has

-E -+ (λ) = 0 0 0 µ ∆ 2,h -λ γ 1,h 0 0 γ 2,h + r h (λ), where λ ∈ D(0, 0 h) ↦ r h (λ) is holomorphic and r h (λ) = O(λ 2 h -1 + λe -C h ) for all λ ∈ D(0, 0 h) and h small enough. Set Γ h = diag(γ 1,h , γ 2,h ).
Then, according to Lemma 3.1 and without extra assumption on α(m 1 ) and α(m 2 )1 ,

(3.8) Γ -1 h = O(h -1 2
). Therefore, one deduces that (3.9)

-E -+ (λ) = Γ h F h (λ) + R h (λ) with F h (λ) = -λ 0 0 µ ∆ 2,h γ 2,h -λ , where R h (λ) is holomorphic with respect to λ ∈ D(0, 0 h) and R h (λ) = O(λ 2 h -3 2 +λh -1 2 e -C h
) for all λ ∈ D(0, 0 h) and h small enough. Set η 2,h ∶= µ ∆ 2,h γ 2,h . According to Lemma 3.1, to (3.1), and to the relation α(m 2 ) > 0 implied by (1.4) (see indeed (1.5)), one has in the limit h → 0:

(3.10) η 2,h = ζ h (m 2 ) h e -2 h (V (s 2 )-V (m 2 )) ,
where

ζ h (m 2 ) ∼ ∑ k≥0 h k 2 ζ k (m 2 ) and ζ 0 (m 2 ) = 1 π V ′′ (m 2 )V ′′ (s 2 ) α(m 2 ). Let K ≥ 2 be fixed in what follows. Set D K = λ ∈ C, λ -η 2,h < h K e -2 h (V (s 2 )-V (m 2 )) ,
whose closure is by (3.10) included in D(0, 0 h) for h small enough. Hence, for any λ ∈ D K , since K ≥ 2, one has, for all h small enough,

(3.11) C -1 he -2 h (V (s 2 )-V (m 2 )) ≤ λ ≤ Che -2 h (V (s 2 )-V (m 2 ))
. Consequently, for any λ ∈ ∂D K , the matrix F h (λ) is invertible and

(3.12) F h (λ) -1 = O(h -K e 2 h (V (s 2 )-V (m 2 ))
) on ∂D K . On the other hand, it follows from the estimate on R h below (3.9) that for any λ ∈ ∂D K ,

R h (λ) = O( √ h e -4(V (s 2 )-V (m 2 )) h + √ h e -(2(V (s 2 )-V (m 2 ))+C) h ) = O(e -(2(V (s 2 )-V (m 2 ))+C) h ).
Hence, by (3.9), E -+ (λ) is invertible on ∂D K for all h small enough and

-E -1 -+ (λ) = 1 + O(e -C 2h ) F h (λ) -1 Γ -1 h on ∂D K ,
where we have used (3.12) and the invertibility of Γ h . Using in addition F h (λ) -1 ∂D K = O(1) and (3.8), this implies that for all h small enough:

1 2iπ ∂D K E -1 -+ (λ)dλ = - 1 2iπ ∂D K -λ 0 0 η 2,h -λ -1 dλ Γ -1 h + Γ -1 h O(e -C 2h ) = 0 0 0 γ -1 2,h + O(e -C 4h
) is non trivial.

(3.13)

Hence, for all K ≥ 2, λ ↦ E -1 -+ (λ) admits at least a singularity α h,K in the disk D K . In particular, one has for all h small enough,

α h,K = η 2,h + O(h K e -2 h (V (s 2 )-V (m 2 ))
), where η 2,h satisfies (3.10). But, since (1.4) holds, Theorem 1 implies that for all h small enough, σ(P h )∩{Re(z) ≤ 0 h 2 } is composed of two elements 0 and λ 2,h . According to (2.25), the nonzero eigenvalue λ 2,h then necessarily satisfies λ 2,h = α h,K for all K ≥ 2. This ends the proof of Theorem 2.

The multiple well case

In this section, we prove Theorem 3. We assume that U is a Morse function and that (1.10) is satisfied. Recall that the local minima m 1 , . . . , m n 0 of V are labeled such that (S(m j )) j∈{1,...,n 0 } is decreasing (see (1.12)). We then label {Ψ 1 , . . . , Ψ n 0 }, the basis of E 0 , accordingly.

4.1. An adapted basis of quasimodes. In order to compute accurately the matrices M V,h and W h , we will build the basis (Ψ j ) j=1,...,n 0 of E 0 from a family of quasi-modes (ϕ j ) j=1,...,n 0 constructed in [START_REF] Bony | Eyring-Kramers law for Fokker-Planck type differential operators[END_REF]. First, as in the previous section, set, for h > 0,

ϕ 1 = 1 Z 1,h e -(V -V (m 1 )) h with Z 1,h = e -(V -V (m 1 )) h L 2 (T) ∈ E cl ((πh V ′′ (m 1 )) 1 4 ),
and Ψ 1 = ϕ 1 . For any j = 2, . . . , n 0 , let ϕ j be the L 2 (T)-normalized quasi-mode associated with m j given in [START_REF] Bony | Eyring-Kramers law for Fokker-Planck type differential operators[END_REF]Definition 4.3] applied to the case of the Witten Laplacian ∆ V,h . Since ϕ i is by definition supported in a neighborhood of C(m i ) (see [START_REF] Bony | Eyring-Kramers law for Fokker-Planck type differential operators[END_REF]Section 4]), it follows from the second item of Assumption (1.10) that, for all i ∈ {1, . . . , n 0 } and r > 0 small enough, one has for all h small enough:

(4.1)

ϕ i = 1 D(m i ,r) Z i,h e -(V -V (m i )) h + O(e -c h ) in L 2 (T),
where Z i,h ∈ E cl ((πh V ′′ (m i )) 1 4 ). For j ∈ {1, . . . , n 0 }, we set κ j,h ∶= ⟨∆ V,h ϕ j , ϕ j ⟩.

We have obviously κ 1,h = 0 and, from [6, Proposition 5.1], we have for all j ∈ {2, . . . , n 0 },

(4.2) κ j,h ∈ E cl he -2S(m j ) h V ′′ (m j )V ′′ (j(m j )) 1 2
π .

The basis (Ψ j ) j=1,...,n 0 of E 0 = Ran Π is then constructed from the family (ϕ j ) j=1,...,n 0 by the following procedure. Set, for j = 1, . . . , n 0 : v j = Πϕ j . For h small enough, the family (v j ) j=1,...,n 0 is then a basis of E 0 (since, according to [6, Proposition 5.3], ⟨v j , v i ⟩ L 2 (T) = δ i,j +O(e -c h ), for all i, j = 1, . . . , n 0 ). We then consider the family (Ψ j ) j=1,...,n 0 obtained from (v j ) j=1,...,n 0 by a Gram-Schmidt procedure as in [6, page 30]. According to [6, Proposition 5.3 and Eq. (5.15)], this orthonormal basis of E 0 satisfies the following properties:

(4.3) for all j, k = 1, . . . , n 0 , ⟨∆ V,h Ψ j , Ψ k ⟩ = δ j,k κ j,h + O h ∞ κ j (h)κ k (h)
and (4.4) for all j = 1, . . . , n 0 , Ψ j = ϕ j + O(e -c h ) in L 2 (T).

Working in the basis (Ψ j ) j=1,...,n 0 of E 0 , we obtain the following asymptotic equivalent of W h (see (2.22)).

Lemma 4.1. There exists C > 0 such that the matrix W h satisfies, for h small enough,

W h = Γ h + O(e -C h ),
where Γ h = diag(γ 1,h , . . . , γ n 0 ,h ) and, for j = 1, . . . , n 0 :

γ j,h ∼ k≥0 h k γ α,k (m j ) + √ h k≥0 h k γ V,k (m j )
with γ α,0 (m j ) = α(m j ), and γ V,0 (m j ) = 2

V ′′ (m j ) π .

Proof. The asymptotic computation of ⟨W Ψ i , Ψ i ⟩ as h → 0 is exactly the same as in Lemma 3.1, using (4.1) and (4.4). The only point to be checked is that for any i ≠ j, one has ⟨W Ψ i , Ψ j ⟩ = O(e -C h ). Thanks to (4.4), this is equivalent to say that for all i ≠ j, one has ⟨W ϕ i , ϕ j ⟩ = O(e -C h ), which follows directly from (4.1). ◻ 4.2. Proof of Theorem 3. Let Γ h be defined by Lemma 4.1 and let M V,h be given by (2.22). Notice that in the case n 0 ≥ 3, the matrix M V,h has not to be diagonal and we vill use the following result.

Lemma 4.2. Introduce the symmetric positive semi-definite matrix

M Γ V,h = Γ -1 2 h M V,h Γ -1 2
h . Then, there exists 0 > 0 such that, for all h small enough, the n 0 eigenvalues β

1,h ≤ ⋅ ⋅ ⋅ ≤ β n 0 ,h of M Γ V,h satisfy: β 1,h = 0 and, for all k = 2, . . . , n 0 , β k,h ∈ E cl (µ ∆ k,h γ k,h ), where µ ∆ k,h is given by (2.

11).

Proof. First, observe that M V,h admits 0 as a simple eigenvalue (since 0 is a simple eigenvalue of ∆ V,h ), so it is also a simple eigenvalue of M Γ V,h . For A = (a i,j ) 1≤i,j≤n 0 a matrix, we define à ∶= (a i,j ) 2≤i,j≤n 0 . We then have:

MΓ V,h = ( Γh ) -1 2 MV,h ( Γh ) -1 2 .
Moreover, by (4.2) and (4.3), the matrix MV,h writes

MV,h = Ω h D V,h + O(h ∞ ) Ω h ,
with Ω h = diag(e -S(m 2 ) h , . . . , e -S(mn 0 ) h ) and D V,h = diag(a h (m 2 ), . . . , a h (m n 0 )). Since Γh is diagonal, one has Ω h Γh = Γh Ω h and it follows that MΓ V,h writes MΓ

V,h = Ω h D Γ V,h + O(h ∞ ) Ω h with D Γ V,h = diag(µ ∆ j,h γ j,h , j = 2
, . . . , n 0 ). Hence MΓ V,h admits a graded structure in the sense of [START_REF] Peutrec | Sharp spectral asymptotics for non-reversible diffusion processes[END_REF]Definition A.1]. We can thus apply [START_REF] Peutrec | Sharp spectral asymptotics for non-reversible diffusion processes[END_REF]Theorem A.4] which yields the result. ◻

We are now in position to prove Theorem 3. Let U be a Morse function and assume that (1.10) is satisfied. Recall that we look for the singularities of E -1 -+ (λ) and that λ = 0 is always a singularity of E -1 -+ (λ) (since dim Ker(P h ) = 1). Let us now look for the remaining singularities of E -1 -+ (λ) in D(0, 0 h). From Lemma 4.2, there exists a unitary change of basis

B h such that M Γ V,h = B * h diag(β 1,h , . . . , β n 0 ,h )B h (since M Γ V,
(λ) = Γ 1 2 h B * h diag(β 1,h , . . . , β n 0 ,h ) -λI C n 0 -R h (λ) B h Γ 1 2 h ,
where, for h small enough, R h (λ) is holomorphic with respect to λ ∈ D(0, 0 h) and

(4.6) R h (λ) = O(λ 2 h -3 2 + λh -1 2 e -C h ),
and where we have used that Γ -1 h = O(h -1 2 ) by Lemma 4.1. Hence, the singularities of E -1 -+ (λ) in D(0, 0 h) are exactly those of L -1 h , where (4.7)

L h (λ) = F h (λ) -R h (λ), with F h (λ) = diag(β 1,h -λ, . . . , β n 0 ,h -λ).
Recall that β 1,h = 0 and β j,h ∈ E cl (µ ∆ j,h γ j,h ) for all j ≥ 2. For all j = 2, . . . , n 0 , using the asymptotic equivalents of µ ∆ j,h and γ j,h given in Theorem 5 and in Lemma 4.1, one has when h → 0, using (1.4) which implies α(m j ) > 0:

(4.8) β j,h = ζ h (m j ) h e -2 h S(m j ) , where ζ h (m j ) ∼ ∑ k≥0 h k 2 ζ k (m j
) and ζ 0 (m j ) is given by (1.14). Let us now consider j ∈ {2, . . . , n 0 } and K ≥ 2. Denote D j,K = D(β j,h , h K e -2 h S(m j ) ) and let λ ∈ ∂D j,K . Since S(m ) < S(m k ) when ≥ k, for h > 0 small enough, the D j,K are pairwise disjoint, their closures are included in D(0, 0 h), and for all i ∈ {2, . . . , n 0 }:

∀i > j, λ -β i,h ≥ β j,h -β i,h -λ -β j,h ≥ Che -2 h S(m i ) ≥ Che -2 h S(m j )
∀i < j, λ -β i,h ≥ β j,h -β i,h -λ -β j,h ≥ Che -2 h S(m j ) .

(4.9)

Moreover, for h small enough, (4.10) λ ≥ Che -2 h S(m j ) .

Consequently, for λ ∈ ∂D j,K , the matrix F h (λ) is invertible and

F h (λ) -1 = O(h -K e
Combining this estimate with (4.6) and reasoning as around (3.13), we prove that for all j = 2, . . . , n 0 , K ≥ 2, and h small enough, (4.11) L -1 h (λ) (and thus E -1 -+ (λ)) admits a singularity α j,h,K in D j,K , so in particular α j,h,K ≠ α i,h,K when i ≠ j ∈ {2, . . . , n 0 } (the D j,K being pairwise disjoint).

In addition, since (1.4) holds, Theorem 1 implies that for all h small enough, σ(P h ) ∩ {Re(z) ≤ 0 h 2 } ∖ {0} is made of n 0 -1 real eigenvalues 0 < λ 2,h ≤ . . . ≤ λ n 0 ,h (counted with algebraic multiplicity). It then follows from (2.25) that, for each j = 2, . . . , n 0 , the eigenvalue λ j,h satisfies λ j,h = α j,h,K for all K ≥ 2. Since α j,h,K = β j,h +O(h K e -2 h S(m j ) ) for all K ≥ 2 and β j,h ∈ E cl (µ ∆ j,h γ j,h ), this completes the proof of Theorem 3, using the asymptotic equivalent of µ ∆ j,h γ j,h as h → 0 (see Theorem 5 and Lemma 4.1).

4.3. The case when (1.4) is not satisfied. In this section, we prove Theorem 4, where, compared to Theorem 3, we do no longer assume that (1.4) holds but that (1.15) holds. We can thus no longer use Theorem 1 as we did at the very end of the proof of Theorem 3 to say that P h admits n 0 -1 nonzero eigenvalues (counted with multiplicity) in {Re(z) ≤ 0 h 2 }. We can however make use of all the intermediate results in the proof of Theorem 3 until (4.11) (included), except that one has the following minor scaling changes:

(1) Equation (4.8) must be changed into (4.12) Let j ∈ {2, . . . , n 0 } and note that, for h small enough, the sequence (α j,h,K ) K≥2 defined in the proof of Theorem 3 is stationary by analyticity of the nontrivial map λ ↦ E -+ (λ). We denote by α j,h its limit and recall that, according to (2.25), {0, α 2,h , . . . , α n 0 ,h } ⊂ σ(P h ). In addition, the α j,h 's are exponentially small and thus belong to D(0, ch 2 ) for any c > 0 and all h small enough. Let us now prove that, for h small enough, 0 and α j,h , j = 2, . . . , n 0 , are the only singularities of λ ↦ E -1 -+ (λ) in D(0, ch 2 ), that they are all real, and have geometric multiplicity 1 as eigenvalues of P h . Recall that by (4.5), λ ∈ D(0, 0 h) is a singularity of E -1 -+ (λ) if and only if it is a singularity of L h (λ) -1 (see (4.7)). Let us denote, for λ ∈ C,

β j,h = ζ h (m j ) √ h e -
d h (λ) = det L h (λ) = det(F h (λ) -R h (λ)),
which is holomorphic on D(0, 0 h). For all λ ∈ ∂D(0, ch 2 ), using (4.6), we get R h (λ) = O(h 4 h -3 2 ) = O(h 5 2 ), and by (4.12), det F h (λ) = λ × Π n 0 j=2 (λ -β j,h ) = c n 0 h 2n 0 (1 + o h (1)). Thus, for h small enough, it holds uniformly on ∂D(0, ch 2 ): (4.13) d h (λ) = λ × Π n 0 j=2 (λ -β j,h ) + O(h 5 2 h 2(n 0 -1) ) = det F h (λ)(1 + o h (1)). In particular, for any λ ∈ ∂D(0, ch 2 ), one has d h (λ) -det F h (λ) < det F h (λ) , which implies by Rouché's theorem that d h (λ) and det F h (λ) have the same number n 0 of zeros (counted with multiplicity) in D(0, ch 2 ). These n 0 zeros are thus 0 and α j,h , j = 2, . . . , n 0 . Let us recall that the α j,h are pairwise disctinct and satisfy α j,h = β j,h + O(h K e -2 h S(m j ) ) for all K ≥ 2 and j = 2, . . . , n 0 . They are then all simple zeros of d h (λ) (and thus dim Ker E -+ (α j,h ) = 1 for j = 2, . . . , n 0 ). Using also (2.10) and [25, Equation (2.7)], dim Ker(P h -α j,h ) = dim Ker T h (α j,h ) = dim Ker E -+ (α j,h ) = 1, for all j = {2, . . . , n 0 }. Lastly, since the operator P h has real coefficients, its spectrum is stable by complex conjugation. Hence, since moreover 0 and the α j,h , j = 2, . . . , n 0 , have different asymptotic equivalents as h → 0, one has α j,h = α j,h for every j = 2, . . . , n 0 . To conclude the proof of Theorem 4, it then just remains to use that, for every j = 2, . . . , n 0 , β j,h ∈ E cl (µ ∆ j,h γ j,h ), and the asymptotic equivalents of µ ∆ j,h γ j,h as h → 0 given by Theorem 5 and Lemma 4.1.

Theorem 4 .

 4 Let U be a Morse function. Assume that (1.15) and (1.10) are satisfied.

( 2 .

 2 14) Π = π [0,ε * h] (∆ V,h ) and E 0 = Ran(Π), where π [a,b] (∆ V,h ) is the spectral projector associated with ∆ V,h and the interval [a, b].

Lemma 2 . 4 .

 24 Let U be a Morse function. For any λ ∈ C, the operator Th (λ) ∶= ΠT h (λ) Π acting on ΠL 2 ∶= Π(L 2 (T)) with domain D( Th (λ)) = {u ∈ ΠL 2 , u ∈ H 2 (T)} is closed.

  2 h S(m j ) ,where ζ h (m j ) ∼ ∑ k≥0 h k 2 ζ k (m j ) with ζ 0 (m j ) satisfying (1.17).(2) In Equations (4.9) and (4.10), all the Ch must be replaced by C √ h.

  see(2.22)), where W , R + , and R -are respectively defined in (2.6), (2.15), and (2.16). The operators R + and R -are defined with the help of an orthonormal basis (Ψ 1 , Ψ 2 ) of E 0 (see(2.14)).

  h is symmetric and thus diagonalizable in an orthonormal basis). Combined with (2.23),(2.24), and Lemma 4.1,

	this yields
	(4.5)	-E -+

Note that when α(m 1 )α(m

) ≠ 0, it actually holds Γ -1 h = O(1).

h S(m j ) ) on D j,K .

Acknowledgement

This work was supported by the ANR-19-CE40-0010, Analyse Quantitative de Processus Métastables (QuAMProcs). B.N. is supported by the grant IA20Nectoux from the Projet I-SITE Clermont CAP 20-25. The authors are grateful to Pierre Monmarché for fruitful discussions and to the anonymous referees for valuable comments.


Denis E Corpet
email: <d.corpet@envt.fr

Claire Jacquinet

Ginette Peiffer

Sylviane Taché

X Nobiotiques

Insulin Injections Promote the Growth of Aberrant Crypt Foci in the Colon of Rats

Keywords: insulin, colon : carcinogenesis, cancer promotion, rat, azoxymethane, ACF, aberrant crypt foci abdominal fat, plasma triglycerides, blood glucose, sialomucin, sulfomucin, colorectal cancer, prevention, diet

The main objective of the present study was to test the hypothesis that exogenous insulin would enhance colon carcinogenesis. Thirty-six female F344 rats, fed ad libitum a low fat rodent chow, received a single azoxymethane injection (20 mg/kg), and were randomized a week later to two groups. Control rats were given 5 days a week a s.c. saline injection, and experimental rats were given ultralente bovine insulin, 20 U/kg. The promoting effect of insulin injections was assessed by the multiplicity (number of crypts) of aberrant crypt foci after 100 d of treatment (72 injections). The rats given insulin ate more and were heavier than controls (215 11 ± vs. 182 7 g, p<0.001). Insulin injections also increased the amount of abdominal fat, the plasma triglycerides, and the insulinemia, ± and decreased blood glucose (all p<0.05). The number of aberrant crypt foci was the same in both groups, but their multiplicity was significantly increased by the insulin injections (2.8 0.3 vs. 2.5 0.2 crypt/focus in controls, p 0.007). Besides, the proportion of

sialomucin producing foci was higher in insulin injected rats than in controls (p 0.04). These data show that exogenous insulin can = promote colon carcinogenesis in rats, and suggest that lifestyle and diets leading to low blood insulin might protect humans against colorectal cancer.

Introduction

The current hypothesis linking diet and colon cancer is that a high fat, low fiber and low calcium diet increases the cancer risk, by increasing the concentration of promoting bile acids in the lumen of the gut ( ). However randomized intervention trials assessing the 1 recurrence of polyps in volunteers given highfiber low-fat diets do not support this hypothesis ( , ). Although most studies focus on 2 3 luminal factors in the fecal stream, it is also possible that the risk factor reaches the colonic cells by way of the general circulation ( ). The 4 major risk factors for colorectal cancer are the followings: sedentary lifestyle, high caloric intake, and diets high in fat, sucrose and ethanol, and low in fiber, resistant starch and n-3 polyunsaturated fatty acids ( , ). These factors are associated with elevated serum 1 4 triglycerides and plasma glucose ( ). High blood glucose and lipids influence circulating hormones, such as insulin, and can lead to 5 insulin resistance, abdominal obesity, and eventually, non-insulin dependant diabetes ( ). McKeown-Eyssen speculated that blood 5 triglycerides, glucose, or insulin could promote cancer development ( ). The idea was later supported by other reviews on insulin and 5 colon cancer ( ), or insulin and breast cancer ( , ). We thus decided to study the direct effect of repeated insulin injections on rat colon 6 7 8

carcinogenesis.

Promotion of cancer was assessed with the aberrant crypt focus (ACF) assay ( ). ACF are putative precursors of colon cancer ( ), 9 10 specifically induced by colon carcinogens ( ), promoted by promoting diets ( ), inhibited by inhibitors of carcinogenesis ( ). In 11 12 13 rodents and humans, ACF display mutations and histologic changes observed in colonic tumors ( , , ), and the growth of ACF 14 15 16 correlates with the adenocarcinoma yield ( , , ). The number of ACF per animal is an assay for initiators of colon cancer, although 17 18 19 the number of crypts per focus (multiplicity) is a measure of promotion effect ( , , ). 17 [START_REF] Magnuson | Ability of Aberrant Crypt Foci Characteristics to Predict Colonic Tumor Incidence in Rats Fed Cholic Acid[END_REF] 20

Materials and Methods

Animals and Treatments

Thirty-six five-week-old female F344 rats were obtained from Iffa-Credo (Lyon, France). They were acclimatized to the colony for one week, housed two rats per stainless steel wire drop-bottom cage, at 22 C with light-dark cycle 12h-12h, day starting at 7h30 a.m.,

°" "

and fed a laboratory chow (6 fat, UAR, Villemoisson, France) and water ad libitum. The rats were initiated between 9 and 10 a.m. with a % single i.p. injection of azoxymethane (Sigma, St. Quentin, France) at a dose of 20 mg/kg in NaCl 9 g/l. They were randomly allocated to the treatments 7 d later. Rats in the control group were given subcutaneous injection of sterile saline (NaCl 9 g/l, 2.5 ml/kg, id est 0.21 to 0.52 ml/rat). In the experimental group they were given s.c. injection of insulin (20 U/kg/d, UltralenteMC Novo Nordish Pharm., Boulogne, France, containing per ml 40 U bovine insulin and 0.08 mg zinc) diluted with saline (8 U/ml in NaCl 9 g/l). Injections were given five times per week, between 8h30 and 9 a.m.. A preliminary experiment showed that, provided they have free access to food, the rats may be given insulin up to 48 U/kg/d without major problems. Food intake, water intake and body weight were measured weekly. The animals were sacrificed 106 d after the carcinogen injection, 99 d after the first insulin injection, the day of the 72nd insulin injection, by cervical dislocation between 2 and 3h15 p.m.. Blood was taken from the heart, the abdominal fat was excised and weighed, and the colon was measured, and kept in formalin for ACF scoring.

Assay of Blood Insulin, Glucose and Triglycerides

Blood samples (1.5 ml) were collected by cardiac puncture in a syringe containing 5 l heparin within 1 min after death, within 5h30 μ and 6h15 after the last insulin injection. After centrifugation for 10 min at 1300 g the plasma was divided in coded microtubes kept at 80

-°C
for further analyses. Insulin was measured by radioimmunoassay, using a kit designed for human plasma, giving a 100 cross-reaction % with bovine and rat insulin (Insik-5, Sorin Biomedica, Antony France). Glucose and triglycerides were measured by enzymatic-spectrophotometric methods (kits PAP-500 and -150, BioM rieux, Marcy France).

é

Assay of Aberrant Crypt Foci

ACF were scored 106 d after the carcinogen injection using the procedure described by Bird ( ). Immediately after sacrifice, colons 10 were removed and flushed with Kreb s Ringer solution (Sigma), then opened longitudinally and fixed flat between coded filter paper in 10 ' buffered formalin (Sigma). The colons were stained with methylene blue (0.1 ) for 10 15 min, then the mucosal side was observed at % % -32 magnification. ACF were distinguished by their slit-like opening, increased staining, size and pericryptal zone. The multiplicity × (number of crypts per ACF) was recorded for each ACF in each colon. Those were scored blindly by a single observer (C.J.).

After ACF methylene-blue scoring, the colons were rinsed then stained with high-iron diamine Alcian blue procedure to determine the type of mucin secretion ( ). As already described ( , ), the normal crypts in the proximal part of rat colon were stained blue 21 22 23 (sialomucin). The normal crypts were stained dark brown (sulphomucin) in the distal colon, like in the normal human colorectal mucosa ( ). ACF mucus staining was examined in this distal part (length 6.8 0.3 cm in controls vs 7.1 0.9 cm insulin rats, p 0.1). The 22

± ± =
sialomucin secreting blue ACF were easily identified against the background of normal brown crypts. It has been suggested that a shift from sulphomucin to sialomucin production may represent an early feature of malignancy ( , ). Colons were scored again blindly by a 22 [START_REF] Caderni | Characterisation of Aberrant Crypt Foci in Carcinogen-treated Rats: Association with Intestinal Carcinogenesis[END_REF] single observer (G.P.), to determine the type of mucin secreted by ACF.

Statistics

Data are given as mean standard deviations. The statistical analysis of ACF assay for colon cancer promotion was based on ACF ± multiplicity as defined by the mean number of aberrant crypts per focus ( ). Data were examined by two-sided Student s t-test, or Welsh 19

' '
s test when variances were unequal.

Results

Weights

Rats given insulin grew faster than controls as shown on . The difference reached significance on day 28 (p 0.015) and after. fig. 1 = The food intake also was higher in insulin injected rats, even before they become heavier than control, e.g., between days 20 and 23, they ate (mean SD) 16.6 0.9 vs. 13.7 0.4 g/d respectively (p<0.001). Differences in food intake, water intake, fecal output, and body

± ± ±
weight remained significant at all time points (data no shown), except during weekends: the injections were not done Saturday and Sunday, and rats in the insulin group tended to eat less than controls those days, e.g., 13.0 0.7 vs. 14.1 0.7 (p 0.05) between days 17

± ± =
and 20. As shown on the body weight difference between groups increased particularly during the last week, but weights were taken fig. 1 after 2 p.m. instead of 8h30 a.m. usually. We guess that control rats do not eat during daytime (and can loose weight via fecal excretion), while insulin injected rats eat mostly between 9 a.m. and 2 p.m., i.e., just after the insulin injection.

On the day of sacrifice, the rats given insulin had more abdominal fat (10.4 1.8 vs. 7.7 1.3 g), and a longer colon (13.8 1.1 vs.

± ± ±

12.5 0.7 cm) than controls. These differences were significant (both p<0.005), but only the difference between abdominal fat remained ± so after adjustment for body weight (p<0.05).

Blood Values

On the day of sacrifice, about 6h after the last insulin injections, the blood insulin was higher in rats given insulin than in controls (440 230 vs. 80 25 U/ml, p<0.0001). The blood glucose was lower in rats given insulin than in controls (4.2 1.2 vs. 10.8 1.6 mmol/l,

± ± μ ± ±
p<0.0001). The blood triglycerides were higher in rats given insulin than in controls (1.19 0.25 vs. 0.97 0.21 mmol/L, p 0.03).

± ± =

Promotion of ACF

A total of 9355 ACF was detected in the 36 colons. The number of ACF per colon, which is not associated with tumor promotion, was the same in both groups (see ). By contrast, the crypt multiplicity (or number of crypts/focus), a predictor of the tumor incidence, Table 1 3 7 was higher in insulin treated rats than in controls (an 11 increase, p 0.007). The number of large ACF, containing 7 crypts or more,

% =
which is another measure for ACF promotion, was higher in rats given insulin than in controls, with borderline significance (p 0.056).

=

Thus data show that ACF grew faster in rats given insulin.

Nature of Mucus

The proportion of ACF excreting sialomucin was higher in insulin injected rats than in controls (43 11 vs. 34 8 , p 0.038), which

± ± % =
suggests enhanced carcinogenesis. The proportion of large ACF excreting sialomucin was not changed by insulin. As shown on , the fig. 2 proportion of sialomucin excreting ACF in the distal colon (mean 7 cm from anus) increased with ACF multiplicity. The correlation between the proportion of sialomucin marked ACF and the number of crypt per ACF was very high (r 0.99, p<0.0001). Thus, the = sialomucin might just be another marker for large ACF.

Discussion

The results of this study show that exogenous insulin leads to increased food intake, higher body weight and more abdominal fat, increased levels of plasma insulin and triglycerides, and reduced levels of plasma glucose. Insulin also increased the growth of aberrant crypt foci, with more crypts per ACF, and more large ACF per colon in insulin treated rats than in controls. Thus, insulin injections promoted colon carcinogenesis in this model. We also observed more sialomucin producing ACF in insulin treated rats than in controls.

This might either suggest a more advanced phenotype ( , ), or simply reflect the increased size of ACF. 22 23 A recent report by Tran et al. also shows that insulin is a colon tumor promoter in rats ( ). After azoxymethane initiation and daily 24 injections of insulin, the fraction of rats with colon adenomatous polyps was greater in rats receiving insulin than in rats receiving saline (79 vs. 50 respectively, p<0.05). The number of tumors per rat was also greater in the insulin group (2.0 vs. 0.7, p<0.05). Insulin did % % not increase body weight gain ( ). Design differences between Tran s study and this one may explain the discrepancies in body weight 24

'
gain and tumor outcome. Tran et al. used male rats (vs. female here), starting at 240 g body weight (vs. 85 g), fed an AIN93M diet (vs.

rodent chow), given medium-acting porcine insulin (vs. long-action bovine) at 15 U/kg for 132 days (vs. 20 U/kg for 100 d), after two 15 mg/kg azoxymethane injections (vs. a single dose 20 mg/kg), with a light cycle shifted 6 h earlier than daylight ( ). It is surprising that in 24 Tran et al. study the insulin did not increase body weight, but it may be explained by differences in starting weight, gender, diet and insulin quickness. Although both studies show a significant promotion of carcinogenesis by insulin, at first glance the magnitude of effect appears smaller in the present study. However the crypt multiplicity increased here by a 1.11 factor, which may be compared to the 1. ). This increase is higher than the 1.6 increased tumor incidence 17 [START_REF] Magnuson | Ability of Aberrant Crypt Foci Characteristics to Predict Colonic Tumor Incidence in Rats Fed Cholic Acid[END_REF] observed by Tran ( ). Thus, the promotion potency of insulin in both studies may be similar. [START_REF] Tran | Insulin Promotion of Colon Tumors in Rats[END_REF] Previous reports show that insulin is a growth factor for rats and for tumors. Daily insulin injections (25 U/kg for 28 d) increase body weight, fat tissue weight, and fat cell size in female rats ( ). Daily insulin injections to hypophysectomized rats enable the growth of 25 liver tumors, initiated by dietary methyldimethyl-aminobenzene ( ). Conversely, diabetes induced by alloxan in female rats previously 26 initiated with dimethylbenzenthra-dimethylbenzenthracene produces the rapid regression of 90 of established mammary tumors. These % tumors were shown to be insulin dependent in vitro ( ). 27

Exogenous insulin may promote tumor growth by at least three mechanisms.

-First, exogenous insulin increases food intake and weight gain. Overfeeding and overweight increase, and caloric restriction decreases, carcinogenesis in many studies ( ). Overfeeding modifies the hormonal balance and increases the energy available through the 28 circulation for neoplastic cells.

-Second, exogenous insulin leads to hyperinsulinemia, which may directly promote tumor growth ( ), for instance by easing the 27 uptake of nutrients by tumor cells.

-Last, insulin may modify other blood-borne factor(s), for instance the insulin-like growth factor IGF1. Most colon cell lines have receptors for IGF1, and high fat diets induce both insulin and IGF1 resistance in rats ( ). 29

A study in pair-fed rats receiving exogenous insulin, but as many calories as controls, might tell if hyperinsulinemia by itself can promote tumors. In human beings, however, hyperinsulinemia is always associated with hyperphagia.

Diets high in calories, fat, sucrose, fructose, amylopectin starch, and low in fibers, resistant starch and n-3 unsaturated lipids, favor the insulin resistance syndrome in rodents. These diets increase blood insulin, and lead to abdominal fat store, obesity, and non-insulinodependant diabetes in many animal models ( , , , ), and probably in humans too ( , ). These same diets are 30 31 32 33 34 35 also known to promote experimental carcinogenesis in rodents, and are considered risk factors for colorectal cancer in human populations ( , ). It is usually considered that bile acids in the fecal stream can explain the promoting effect of these diets on colon cancer, but blood insulin resistance may explain the high risk of colorectal cancer associated with sedentary lifestyle, overfeeding, and nutriments cited above ( , , ). 4 5 6

This study is the second to show a promotion of colon carcinogenesis by exogenous insulin. If nutritional studies, in progress in our laboratory, also show the promoting effect of endogenous insulin, it will be necessary to study the protection afforded by interventions that reduce insulinemia in humans.

  20 increase published by Zhang et al. ( ) and Magnuson et al. ( ). According to their data, the diets that increase the crypt multiplicity by 1.2 at 17 19 100 d also increase the adenocarcinoma incidence by 3 later ( ,

  34 . Frayn KN , Klingman SM . Dietary Sugars and Lipid Metabolism in Humans . Am J Clin Nutr . 62 : S250 -S263 1995 ; 35 . Zemel MB . Insulin Resistance, Obesity and Hypertension: an Overview . J Nutr . 125 : S1715 -S1717 1995 ;

Fig. 1

 1 Fig. 1Mean body weights of F344 female rats during daily injections of saline or insulin, starting 0 day.

Fig. 2

 2 Fig. 2Type of mucus produced by the aberrant crypt foci in the distal colon of rats according to their multiplicity. Percentage of ACF/total ACF, secreting sialomucins, sulphomucins, or both types of mucins together. Data are based on 2444 ACF.

Table 1

 1 Effect of repeated insulin injections on aberrant crypt foci (ACF) in F344 rats after a single azoxymethane injection a 72 insulin injections (20 U/kg) were given during 106 days after a single azoxymethane injection (20 mg/kg) b Large ACF: > 6 crypts c Values are means SD; , no. of rats;

	Group n	ACF/Rat	Multiplicity crypts/ACF

± n

[START_REF] Stoll | Nutrition and Breast Cancer Risk: Can an Effect via Insulin Resistance be Demonstrated?[END_REF] insulin may also explain the promotion by these diets ( ). The present results support the idea that in humans, hyperinsulinemia and/or

Ackowledgements:

We thank Gail McKeown-Eyssen, Giovanna Caderni, and John Cummings for their comments on the study protocol, Raymond Gazel for taking care of the rats, Jacques Alary for insulin supply, and W. Robert Bruce for stimulating discussions.

The study was supported in part by the Direction G n rale de l Enseignement et de la Recherche du Minist re de l Agriculture (France), and

by a grant of the Association de la Recherche contre le Cancer.


Amal Guezmil

Vadim Iosif

Gabriel Velu

Daniel Roger

High temperature induction motor preliminary sizing

High temperature induction motor preliminary sizing

Amal Guezmil, Vadim Iosif, Gabriel Velu and Daniel Roger Univ. Artois, EA4024 LSEE, Technoparc Futura, F-62400 Bethune, France Department guezmil.amal@gmail.com, vadim.iosif@univ-artois.fr, gabriel.velu@univ-artois.fr, daniel.roger@univ-artois.fr Abstract-This paper presents the preliminary sizing of a High Temperature (HT°) induction motor based on a copper cage rotor and a distributed winding built with an inorganic Electric Insulation System (EIS) able to withstand 500°C. The specific part of the design derives of the poor mechanical and electrical characteristics of the inorganic HT°insulation materials. HT°w ires are protected by a cement and form rigid coils. The laminated magnetic core has a specific design for suiting with the rigid coils. An Electrical and thermal models are validated.

Index Terms-High Temperature, Induction Motor, Thermal Model

I. INTRODUCTION

Up to now, electrical machines cannot be used in very hot environments because of the thermal limits of the organic materials of their electrical insulation system (EIS). Standard electrical machines cannot be used permanently in environments over more or less 300°C because of the thermal limits of the polymers [START_REF] Schanler | High temperature breakdown strength and voltage endurance characterization of nanofilled polyamideimide[END_REF]. However, several applications could use the electrical machines beyond of these temperatures for high temperature environment, for example in the fields of aviation, spatial [START_REF] Fingers | Application of high temperature magnetic materials[END_REF] or nuclear industry [START_REF] Mitsui | Investigation of the Voltage Life of Mica-Alumina Composite Insulation at High Temperatures[END_REF].

Research works have been performed for replacing polymers by inorganic EIS able to withstand 500°C [START_REF] Iosif | Assessment and improvements of inorganic insulation for high temperature low voltage motors[END_REF]. They show that HT°EISs have breakdown voltages lower than organic ones and poor mechanical characteristics. The proposed works are carried out in the continuity of previous ones in order to design a HT°induction motor and to analyze the limits of this structure [START_REF] Cozonac | Conception d'une machine asynchrone haute temprature[END_REF].

II. EQUIVALENT CIRCUIT OF HT°INDUCTION MOTOR

For getting the operating parameters of the motor, it is necessary to know the elements of the equivalent circuit as presented in Fig. 1. If we take into account that the HT°i nduction motor is built with the rotor of a conventional motor, we can take the parameters of the equivalent diagram of the conventional motor. The only change is at the stator coil and rotor bar resistances R 1 and R 2 , respectively (see Fig. 1). The critical situation for a HT°induction motor will be the heating of the stator winding and rotor bars. This situation will increase the resistance of the copper nickel wire also the resistance of the copper bars. Increasing the current in winding turns will increase the power of the motor, but this power will be limited by the section of the copper rotor bar sized for the conventional motor. Eqs. ( 1) and ( 2) are derived from the equivalent circuit of Fig. 1.

I 2 = V μ R 2 + jX 2 ; I 1 = I 2 + V μ R μ + jX μ (1) V = V μ + I 1 × (R 1 + jX 1 ) (2)
The expression of V μ is derived from Eqs. ( 1) and (2). Then, the I 1 and I 2 are computed from Eq. [START_REF] Schanler | High temperature breakdown strength and voltage endurance characterization of nanofilled polyamideimide[END_REF].

V μ = V 1 + R1+jX1 R2+jX2 + R1+jX1 Rμ+jXμ (3)
Let's define the power expression as

P F = 3 V 2 μ Rμ ; P R = 3 R2 I 2 2 ; P B = 3 R1 I 2 1
with P R , P B and P F represent respectively the losses in the rotor, in the coils and in the iron of the stator.

III. HT°INDUCTION MOTOR THERMAL MODEL

Machines with a high power density are often fast and therefore quite long with a small diameter, their thermal equilibrium can be determined by considering that all the heat produced in the rotor passes through the stator and the carcass before being vented to the atmosphere by the ventilation system. 2 presents a simplified thermal model that allows to estimate the internal temperature of the main elements of the machine: the rotor, the winding and the carcass; these elements correspond to the points R, B and C on the equivalent diagram of Fig. 3; the point A corresponds to the ambient temperature assumed to be constant. Thermal resistors characterize the main heat exchanges in the machine in the equilibrium regime. So a small part of the rotor losses are evacuated directly by the shaft (R RA ) the other must cross the air gap and evacuate outside by the stator. It exists also a coupling between the coils and the rotor (R RB ). This simplified thermal model does not take into account the complex exchanges that exist at the level of the end-windings connections.

The rotor bars are not insulated so they can withstand higher temperatures than the active conductors of the stator. When machines are isolated by polymers whose thermal limits are critical point for sizing is the temperature of the T B coils which must remain permanently under the value maximum which depends on the nature of the insulators. The stator teeth are semi closed, so G RB is neglected.

Considering the thermal conductances G = 1 R , the temperature determination is represented in matrix form:

⎡ ⎣ T R T B T C ⎤ ⎦ = A -1 ⎛ ⎝ ⎡ ⎣ P R P B P F ⎤ ⎦ -BT A ⎞ ⎠ (4)
with

A = ⎡ ⎣ G RC + G RA 0 -G RC 0 G BC -G BC -G RC -G BC G CA + G BC + G RC ⎤ ⎦ and B = ⎡ ⎣ G RA 0 G CA ⎤ ⎦
The heat flow that crosses the air-gap can be estimated roughly considering a static thin air layer. This very approximate approximation gives a thermal resistance higher than the actual one; therefore the real rotor temperature be law.

R RC = 1 C T hair e S 1 = 1 C T hair e 2π R r + e 2 L
(5)

Depending on the axial direction, the conduction thermal resistance R CA for a surface S is calculated with :

R CA = 1 C T hacier e S 2 = 1 C T hacier 1 2πR r (6) with R r = 4.510 -2 m, C T hair = 0.0258W m -1 K -1 , C T hacier = 50.2W m -1 K -1 .
With the expressions ( 5) and ( 6) and using (4), it is now possible to calculate the resistances of the thermal model based on some experimental tests which have been performed by feeding thermocouples into the different points of our motor.

IV. COUPLED ANALYSIS OF THERMAL AND ELECTRICAL EQUIVALENT SCHEME Fig. 4 presents the motor operating characteristics by means of slip. It's clear that we have a nominal torque equal to 9N/m and the coils temperature is higher than 600°C. Also, the coils losses are about 1000 W which is higher compared to the motor power. So, we have a low efficiency.

V. CONSTRUCTION OF THE HT°INDUCTION MOTOR

The HT°motor is equipped with a copper cage rotor that has the same shape as the conventional one except that the shaft of HT°motor will be lengthened because of the length of the end-connections of the stator. This length is explained that the heads of the coils are larger because the insulation of the ceramic wire is very fine in radius of curvature. The air gap of the HT°motor is expected to be 0.6 mm, 2 times larger compared to the conventional one. The stator core is made of F e S i electrical sheets able to withstand high temperature [START_REF] Ababsa | High-Temperature Magnetic Characterization Using an Adapted Epstein Frame[END_REF]. A general view of the HT°induction motor is shown in figure 5. This motor have a stainless steel carcass and the bearing will be ceramic to resist the heat evacuated by rotor shaft.

Fig. 1 :

 1 Fig. 1: Equivalent schema of HT°induction motor.

Fig. 2 :

 2 Fig. 2: HT°induction motor thermal mode.

Fig. 3 :

 3 Fig. 3: The cross section of the active part of a machine.

Fig.

  Fig.2presents a simplified thermal model that allows to estimate the internal temperature of the main elements of the machine: the rotor, the winding and the carcass; these elements correspond to the points R, B and C on the equivalent diagram of Fig.3; the point A corresponds to the ambient temperature assumed to be constant. Thermal resistors characterize the main heat exchanges in the machine in the equilibrium regime. So a small part of the rotor losses are evacuated directly by the shaft (R RA ) the other must cross the air gap and evacuate outside by the stator. It exists also a coupling between the coils and the rotor (R RB ). This simplified thermal model does not take into account the complex exchanges that exist at the level of the end-windings connections.The rotor bars are not insulated so they can withstand higher temperatures than the active conductors of the stator. When machines are isolated by polymers whose thermal limits are critical point for sizing is the temperature of the T B coils which must remain permanently under the value maximum which depends on the nature of the insulators. The stator teeth are semi closed, so G RB is neglected.Considering the thermal conductances G = 1 R , the temperature determination is represented in matrix form: ⎡

Fig. 4 :

 4 Fig. 4: HT°induction motor operating characteristics by means of slip. Torque, temperature, power losses.

Fig. 5 :

 5 Fig. 5: General view of the HT°induction motor.

ACKNOWLEDGMENT

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This work was supported by the north region of France. The authors would like to thank MEDEE Regional Research Center for the financial support that has been contributed in the paper of the CE2I State Region Contract.