Frédéric Hérau
email: herau@univ-reims.fr

Michael Hitrik
email: hitrik@math.ucla.edu

Johannes Sjöstrand
email: johannes@math.polytechnique.fr

Tunnel effect for Kramers-Fokker-Planck type operators: return to equilibrium and applications

Keywords: Kramers, Fokker-Planck, semiclassical limit, return to equilibrium, heat semigroup, eigenvalue splitting, supersymmetry, Witten Laplacian Mathematics Subject Classification 2000: 35P15, 35P20, 47B44, 47D06, 47N55, 81Q60, 82C31 Kramers-Fokker-Planck operators . . . . . . . . . .

In the first part of this work, we consider second order supersymmetric differential operators in the semiclassical limit, including the Kramers-Fokker-Planck operator, such that the exponent of the associated Maxwellian φ is a Morse function with two local minima and one saddle point. Under suitable additional assumptions of dynamical nature, we establish the long time convergence to the equilibrium for the associated heat semigroup, with the rate given by the first non-vanishing, exponentially small, eigenvalue. In the second part of the paper, we consider the case when the function φ has precisely one local minimum and one saddle point. We also discuss further examples of supersymmetric operators, including the Witten Laplacian and the infinitesimal generator for the time evolution of a chain of classical anharmonic oscillators.

Introduction and statement of the main result

The principal purpose of the present paper is to apply the spectral results of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF] to obtain a precise information concerning the large time behavior of the heat semigroup generated by the semiclassical Kramers-Fokker-Planck operator

P = y • h∂ x -V ′ (x) • h∂ y + γ 2 (-h∂ y + y) • (h∂ y + y) , x, y ∈ R n , γ > 0. (1.1)
In fact, as in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], our main result will be valid for a large class of supersymmetric second order differential operators, including (1.1). Physically, the semiclassical limit h → 0 in (1.1) corresponds to the régime of low temperatures. Recall that by supersymmetry, we mean the fact that the Kramers-Fokker-Planck operator P can be viewed as a Witten Laplacian in degree 0 associated to a certain non-semidefinite scalar product in the spaces of differential forms. These relations with the Witten complex [START_REF] Witten | Supersymmetry and Morse theory[END_REF] were exhibited in the works of J. Tailleur, S. Tanase-Nicola, J. Kurchan [START_REF] Tailleur | Kramers equation and supersymmetry[END_REF] and J. M. Bismut [START_REF] Bismut | The hypoelliptic Laplacian on the cotangent bundle[END_REF], using respectively the languages of supersymmetry and differential forms (See also [START_REF] Lebeau | Le bismutien, Séminaire équations aux dérivées partielles[END_REF] for a quick introduction to the differential form version of Bismut and [2]).

The paper [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], which is a natural continuation of [START_REF] Hérau | Semiclassical analysis for Kramers-Fokker-Planck type operators[END_REF], analyzed resolvent estimates and the low lying eigenvalues of P , assuming that the potential V in (1.1) is a smooth real valued Morse function on R n , such that

∂ α V = O(1), |α| ≥ 2, (1.2)
and with |∇V | ≥ 1/C, for |x| ≥ C > 0.

(1.3)

Assuming furthermore that V has precisely three critical points: two local minima,

x ±1 , and one critical point x 0 of index 1, it was established in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF] that for C > 0 large enough and h > 0 sufficiently small, the operator P has precisely two eigenvalues in the disc D(0, h/C) = {z ∈ C; |z| < h C }, µ 0 and µ 1 , such that µ 0 = 0 and µ 1 is real and of the form µ 1 = h a 1 (h)e -2S 1 /h + a -1 (h)e -2S -1 /h , S j = V (x 0 ) -V (x j ).

(1.4)

Here a j are real with a j (h) ∼ a j,0 + ha j,1 + . . . a j,0 > 0.

(

Notice that the eigenfunction corresponding to the eigenvalue µ 0 = 0 is the Maxwellian exp (-φ/h) ∈ L 2 (R 2n x,y ), φ(x, y) = y 2 2 + V (x).

(1.6)

In the case when V → +∞ as x → ∞ and V has precisely one local minimum, it follows from the results of [START_REF] Hérau | Semiclassical analysis for Kramers-Fokker-Planck type operators[END_REF] that in a disc D(0, Ch), C ≥ 1, apart from the eigenvalue µ 0 = 0, the real part of the other eigenvalues is ≥ h/C. In this case, precise results describing the behavior of the semigroup exp (-tP/h) for large t, were obtained in [START_REF] Hérau | Semiclassical analysis for Kramers-Fokker-Planck type operators[END_REF] -the rate of the return to equilibrium, given by the projection onto the ground state (1.6), is then of the order of magnitude 1. In this work, we shall complement this study by analyzing the question of a return to equilibrium in the presence of exponentially small eigenvalues, due to the tunneling between the local minima.

The following is the main result of this work, specialized to the case of the Kramers-Fokker-Planck operator (1.1). Here we shall also write P for the m-accretive realization of the operator (1.1) -see also section 2 and section 3 in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF].

Theorem 1.1 Assume that V in (1.1) is a C ∞ real valued Morse function satisfying (1.2) and (1.3) and having precisely 3 critical points: 2 local minima, x ±1 , and a critical point of index 1, so that the disc D(0, h/C) for C > 0 large enough, contains precisely 2 eigenvalues of P , µ 0 = 0 and µ 1 given in (1.4). Let Π j be the spectral projection associated with the eigenvalue µ j , j = 0, 1. Then we have

Π j = O(1), h → 0.
(1.7)

We have furthermore, uniformly as t ≥ 0 and h → 0, e -tP/h = Π 0 + e -tµ 1 /h Π 1 + O(1)e -t/C , C > 0, in L(L 2 , L 2 ).

(1.8)

The structure of the article is as follows: In section 2, relying upon the results of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], [START_REF] Hérau | Semiclassical analysis for Kramers-Fokker-Planck type operators[END_REF], we establish the basic a priori coercivity estimate for the operator P in a suitable exponentially weighted space, introduced in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF]. In this sense it can be interpreted as an hypocoercive estimate (see e.g. [START_REF] Hérau | Méthodes microlocales pour les équations cinétiques[END_REF], [START_REF] Villani | Hypocoercivity[END_REF]). In section 3 it is then quite straightforward to prove Theorem 1.1 in its general form, by combining the results of section 2 together with the analysis of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF].

The second part of the paper, consisting of sections 4-6 is of a somewhat different nature, complementing and extending the previous analysis. In section 4, we study the case left out in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], when the potential V in (1.1) has precisely two critical points: one local minimum and a critical point of index one. In this case, 0 is not an eigenvalue of P , and the large time behavior of the heat semigroup is governed entirely by the first, exponentially small, eigenvalue. In section 5, we give some examples and describe a probabilistic framework in which the Witten Laplacian and the Kramers-Fokker-Planck operator both arise naturally. Finally, in section 6, we describe another example of a supersymmetric operator, for which the question of a convergence to equilibrium is of interest, namely a chain of classical interacting anharmonic oscillators, coupled to a heat bath. We show how to adapt the analysis of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF] to cover also this case.

An (hypo-)coercive estimate

The purpose of this section is to establish an a priori estimate for P , instrumental in proving Theorem 1.1. This estimate will imply the exponential decay for the heat semigroup exp (-tP/h) in L 2 , when restricted to the kernel of the spectral projection corresponding to the eigenvalues µ 0 = 0 and µ 1 in (1.4). When doing so, as in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], rather than working directly with (1.1), we shall consider a broader class of scalar real second order non-elliptic non-selfadjoint operators on R n . For completeness, we shall now recall, following [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], the definition and the main assumptions concerning this class.

Let us consider

P = n j,k=1 hD x j • b j,k (x) • hD x k + 1 2 n j=1 c j (x)h∂ x j + h∂ x j • c j (x) + p 0 (x) (2.1) =: P 2 + iP 1 + P 0 , D x j = 1 i ∂ ∂ x j .
Here the coefficients b j,k , c j , p 0 all belong to C ∞ (R n ; R), with b j,k = b k,j . Associated to P in (2.1) is the symbol in the semiclassical sense,

p(x, ξ) = p 2 (x, ξ) + ip 1 (x, ξ) + p 0 (x), (2.2)
p 2 (x, ξ) = n j,k=1 b j,k (x)ξ j ξ k , p 1 (x, ξ) = n j=1 c j (x)ξ j , (2.3)
so that p j (x, ξ) is a real-valued polynomial in ξ, positively homogeneous of degree j, 0 ≤ j ≤ 2. We may notice that p(x, ξ) coincides with the Weyl symbol of P modulo O(h 2 ), locally uniformly.

As in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], we shall assume that

p 2 (x, ξ) ≥ 0, p 0 (x) ≥ 0. (2.4)
Furthermore, we shall impose the following growth conditions,

∂ α x b j,k (x) = O(1), |α| ≥ 0, (2.5)
∂ α x c j (x) = O(1), |α| ≥ 1, (2.6) ∂ α x p 0 (x) = O(1), |α| ≥ 2.
(2.7)

From section 3 in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], we recall that under these assumptions, the graph closure of P : S(R n ) → S(R n ), still denoted by P and such that Re P ≥ 0, coincides with the maximal closed realization of P , with the domain given by D(P ) = {u ∈ L 2 ; P u ∈ L 2 }. In particular, this shows that the operator P is m-accretive, and hence, the contraction semigroup e -tP/h :

L 2 → L 2 , t ≥ 0 (2.8)
is well-defined.

We proceed next to recall the additional assumptions of a dynamical nature, introduced in section 4 of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF]. Let

ν(x, ∂ x ) = n j=1 c j (x)∂ x j , (2.9)
and recall the Hypothesis 4.1 of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF]:

The set {x ∈ R n ; p 0 (x) = 0, ν(x, ∂ x ) = 0} is finite = {x 1 , . . . x N }. (2.10)
With ρ j = (x j , 0), 1 ≤ j ≤ N, we define the critical set

C = {ρ 1 , . . . ρ N } ⊂ R 2n . (2.11)
The coefficients p 0 , p 1 , p 2 in (2.2) all vanish to the second order at each ρ j ∈ C. As in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], we define p(x, ξ) = p 0 (x) + ξ -2 p 2 (x, ξ), (2.12)

and consider the time average

p T 0 = 1 T 0 T 0 /2 -T 0 /2 p • exp (tH p 1 ) dt, T 0 > 0. (2.13)
We shall assume that the Hypothesis 4.3 of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF] holds true:

For T 0 > 0 fixed, we have near each ρ j , p T 0 (ρ) ∼ |ρ -ρ j | 2 , (2.14)
and in any set of the form |x| ≤ C, dist(ρ, C) ≥ 1/C, we have

p T 0 (ρ) ≥ 1 C(C) , C(C) > 0. (2.15)
We also need an additional dynamical hypothesis near

∞ in R n , ∀ neighborhood U of π x C, and ∀ x ∈ R n \ U, ∃C > 0, meas {t ∈ [- T 0 2 , T 0 2 ]; p 0 (exp tν(x)) ≥ 1 C } ≥ 1 C . (2.16)
Under the assumptions above, the paper [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF] defines an auxiliary real valued weight function

ψ ε (x, ξ) = O(ε) on T * R n , ε > 0, such that ∂ α x ∂ β ξ ψ ε (x, ξ) = O ε 1-|α+β|/2 ξ -|β| ,
together with an associated canonical transformation

κ(δ) : R 2n → Λ δ := {(x, ξ) + iδH ψε (x, ξ); (x, ξ) ∈ R 2n }, 0 < δ ≪ 1, (2.17)
for which

κ(δ)(x, ξ) = (x, ξ) + iδH ψε (x, ξ) + O(ε 1/2 δ 2 ). (2.18)
Here we let

H f denote the Hamilton field f ′ ξ (x, ξ) • ∂ x -f ′ x (x, ξ) • ∂ ξ of a C 1 -function f (x, ξ).
We refer to section 4 of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF] for the details of the construction of ψ ε and κ δ .

Here we shall merely recall that δ > 0 fixed in (2.18) should be small enough, and ε = Ah, with A arbitrarily large but fixed.

Associated to κ(δ) in (2.17) there is an elliptic Fourier integral operator with a complex phase A δ,ε , constructed in Section 5 of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], such that

A δ,ε : S → S (2.19)
continuously, and

A δ,ε = O A (1) : L 2 → L 2 . (2.20)
Moreover, it is proved in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF] that A δ,ε is invertible, when ε/h ≫ 1, with the inverse A -1 δ,ε also enjoying the mapping properties (2.19), (2.20). When B ≥ A fixed is to be chosen, and A ≫ B is large enough, as in sections 6,7 of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], we shall consider the conjugated operator

P δ, ε = A -1 δ, ε P A δ, ε , ε = Ah, (2.21)
acting on L 2 (R n ). It was then proved in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF] that the real part of the symbol of

P δ, ε is ≥ δ ε C -Ch outside of C + B(0, √ ε), C > 0. In the set C + B(0, √ ε), the symbol of P δ, ε is independent of ε modulo O( ε h ε ∞
) and is of the form

P δ ∼ p δ + h 2 r 2 + . . . (2.22)
where

p δ (ρ) ∼ dist(ρ, C) 2 , Re p δ (ρ) ∼ dist(ρ, C) 2 .
(2.23)

Here we write B(0, r) = {ρ ∈ R 2n ; |ρ| < r}, r > 0.

As has also been recalled in section 8 in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], in the set C + B(0, √ ε), we have

P δ, ε -P δ = O ε h ε ∞ ,
while when away from C + B(0, √ ε), we shall use that

P δ, ε -P δ = O ε + ρ 2 .
(2.24)

Here, following [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], we use the notation

f ε = O(ε) to express that ∂ α x ∂ β ξ f ε (x, ξ) = O ε 1-|α+β|/2 ξ -|β| , for arbitrary multi-indices α, β ∈ N n .
We shall study estimates for the real part of the quadratic form associated to the operator P δ, ε . The starting point here is Proposition 7.1 of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF]

: let 0 ≤ k ε = O(ε) be equal to ε in C + B(0, √ ε) and have its support in C + B(0, √ 2ε). Let K ε = Op h (k ε )
stand for the Weyl quantization of k ε (x, hξ). It is then established in Proposition 7.1 of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF] that

Re (P δ,ε + K ε )u|u ≥ δε C -Ch u 2 , u ∈ S, (2.25)
when ε = Ah, C > 0 is independent of δ, A, and h is small enough depending on these 2 parameters. Rather than working with the estimate (2.25), we shall use that

Re (P δ, ε + K ε )u|u ≥ δε C -Ch u 2 , u ∈ S, (2.26)
which is proved in exactly the same way as in section 7 of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF]. Here we recall that

ε = Ah, ε = Ah, A ≫ A.
In what follows we shall use that the estimate (2.26) holds also for u ∈ D(P δ, ε ) = {u ∈ L

2 ; P δ, ε u ∈ L 2 } = A -1 δ, ε D(P ). Let Π B = Π δ, ε B
be the spectral projection of P δ, ε associated with the spectrum of P δ, ε in the open disc D(0, Bh). From Theorem 8.3 in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF] we recall that the spectrum of P δ, ε in D(0, Bh) is discrete, and the eigenvalues are of the form

λ j,k (h) ∼ h µ j,k + h 1/N j,k µ j,k,1 + h 2/N j,k µ j,k,2 + . . . ,
where the µ j,k are all numbers in D(0, B) of the form

µ j,k = 1 i n ℓ=1 ν j,k,ℓ + 1 2 λ j,ℓ , ν j,k,ℓ ∈ N, (2.27)
for some j ∈ {1, . . . N}. Here λ j,ℓ , 1 ≤ ℓ ≤ n, are the eigenvalues of the Hamilton map of the quadratic part of p at ρ j ∈ C, for which Im λ j,ℓ > 0. Here we also assume that B is chosen such that |µ j,k | = B, for all j, k.

Assume that u ∈ L 2 is such that

u ∈ Ran(1 -Π B ).
(2.28)

We are interested in lower bounds for

Re (P δ, ε u|u), (2.29)
which, in view of (2.26), amounts to estimating K ε u. In doing so, we shall assume, for notational simplicity only, that the critical set C defined in (2.11) consists of a single point, ρ 1 = (0, 0). From (2.23), we know that the leading symbol of P δ , p δ , is such that

p δ (ρ) ∼ |ρ| 2 , ρ ∈ B(0, √ ε). Let p 0 (x, ξ) = p δ 0 (x, ξ) = |α+β|=2 ∂ α x ∂ β ξ p δ (0, 0) α!β! x α ξ β
be the quadratic approximation of p δ , so that

p δ -p 0 = O((x, ξ) 3 ) = O (h + (x, ξ) 2 ) 3/2 , (x, ξ) → (0, 0). ( 2

.30)

Then p 0 is an elliptic quadratic form on R 2n , with a positive definite real part. The quadratic differential operator

P 0 = Op h (p 0 ) : L 2 → L 2 ,
has discrete spectrum, and from [START_REF] Sjöstrand | Parametrices for pseudodifferential operators with multiple characteristics[END_REF] we know that the eigenvalues of P 0 are of the form hµ 1,k , with µ 1,k defined as in (2.27).

When estimating K ε u for u ∈ Ran(1 -Π B ), we also introduce the spectral projection Π 0,B associated to P 0 and the spectrum of P 0 in D(0, Bh). Then, since Π B u = 0,

K ε u = K ε (Π 0,B -Π B )u + K ε (1 -Π 0,B )u. (2.31)
We shall estimate the first term in the right hand side of (2.31), using the following result.

Lemma 2.1 We have

Π B -Π 0,B = O B ( A 3/2 h 1/2 + A -1 ) : L 2 → L 2 .
(2.32)

Proof: Let χ ∈ C ∞ 0 (B(0, 2)), 0 ≤ χ ≤ 1, be such that χ(x, ξ) = 1 for |(x, ξ)| ≤ 1. Set χ √ ε (x, ξ) = χ( ε -1/2 (x, ξ)). We shall first show that Π 0,B 1 -χ √ ε = O B h ε ∞ : L 2 → L 2 , (2.33)
and similarly, that

Π B 1 -χ √ ε = O B h ε ∞ : L 2 → L 2 .
(2.34)

When proving (2.33), we shall use the well-posed Grushin problem for the quadratic operator P 0 , described in [START_REF] Hérau | Semiclassical analysis for Kramers-Fokker-Planck type operators[END_REF], [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF]. Let

Λ = (x, hD) h 1/2 = 1 + x 2 + (hD x ) 2 h 1/2 , (2.35)
so that the quadratic elliptic operator P 0 is equipped with the natural domain

D(P 0 ) = {u ∈ L 2 ; Λ 2 u ∈ L 2 }.
In section 11 in [START_REF] Hérau | Semiclassical analysis for Kramers-Fokker-Planck type operators[END_REF], using the generalized eigenfunctions of P 0 and of the adjoint P * 0 , the authors have constructed the operators

R -: C N 0 → L 2 , R + : L 2 → C N 0 , N 0 ∈ N, (2.36)
such that when z ∈ D(0, Bh), the problem

(P 0 -z) u + R -u -= v, R + u = v + , (2.37)
for v ∈ L 2 , v + ∈ C N 0 has a unique solution u ∈ D(P 0 ), u -∈ C N 0 . Moreover, we have the a priori estimate

h Λ 2 u + |u -| ≤ O(1) ( v + h |v + |) . (2.38)
Associated to (2.37) is the Grushin operator

P 0 (z) = P 0 -z R - R + 0 : D(P ) × C N 0 → L 2 × C N 0 , z ∈ D(0, Bh), (2.39)
with an inverse

E 0 (z) = E(z) E + (z) E -(z) E -+ (z) : L 2 × C N 0 → D(P ) × C N 0 , (2.40)
depending holomorphically on z. From section 11 of [START_REF] Hérau | Semiclassical analysis for Kramers-Fokker-Planck type operators[END_REF], we recall that E 0 (z) enjoys the following localization properties, when k ∈ R,

Λ 2-k E(z)Λ k = O 1 h : L 2 → L 2 , (2.41)
and

Λ k E + (z) = O k (1) : C N 0 → L 2 , E -(z)Λ k = O k (1) : L 2 → C N 0 , (2.42)
Let γ ⊂ D(0, B) be a simple positively oriented closed h-independent contour, such that all eigenvalues of P 0 and P δ, ε in D(0, Bh) are contained in the interior of hγ, so that we have dist(z, Spec(P 0 ) ∪ Spec(P δ, ε )) ≥ h/O(1), z ∈ hγ.

Here we continue to assume that B > 0 is chosen so that there are no numbers of the form µ j,k in (2.27) on the boundary of D(0, B).

Writing Π 0,B = 1 2πi hγ (z -P 0 ) -1 dz (2.43)
and using the well-known formula

(z -P 0 ) -1 = -E(z) + E + (z)E -+ (z) -1 E -(z), (2.44)
we obtain that

Π 0,B = 1 2πi hγ E + (z)E -+ (z) -1 E -(z) dz.
(2.45) Now (2.42) gives that for each k ∈ N,

E -(z)(1 -χ √ ε ) = E -Λ k Λ -k (1 -χ √ ε ) = O h ε k/2 : L 2 → C N 0 . (2.46)
Using also that along hγ, we have

E + (z) = O(1) : C N 0 → L 2 ,
and

E -+ (z) -1 = O(h -1 ) : C N 0 → C N 0 ,
as well as the fact that length of hγ is O B (h), we obtain (2.33).

The proof of (2.34) proceeds along the similar lines, relying upon the well-posed Grushin problem for P δ, ε -z, constructed from the Grushin problem for P 0 and described in detail in section 11 of [START_REF] Hérau | Semiclassical analysis for Kramers-Fokker-Planck type operators[END_REF] and section 8 of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF]. In particular, the analogue of the localization property (2.42) holds true for the inverse of the Grushin operator for P δ, ε , and arguing as above, we get (2.34).

We now come to consider estimates for the difference (Π B -Π 0,B ) χ √ ε , where we claim that

(Π B -Π 0,B ) χ √ ε = O B ε 3/2 h + 1 A : L 2 → L 2 , ε = Ah. (2.47)
In view of (2.33) and (2.34), this will complete the proof of Lemma 2.1.

When proving (2.47), we shall first estimate the difference Π B -Π 0,B χ √ ε , where Π B is the spectral projection of the operator P δ associated with the spectrum of P δ in D(0, Bh). Using 2.43) together with the similar formula for Π B , we get, by an application of the resolvent identity,

Π B -Π 0,B χ √ ε = 1 2πi hγ z -P δ -1 P δ -P 0 (z -P 0 ) -1 χ √ ε dz. (2.48)
Now (2.41), (2.42), (2.44), together with the fact that along hγ we have

E -1 -+ (z) = O(h -1 ), imply that for k ∈ R, Λ k (z -P 0 ) -1 Λ -k = O 1 h : L 2 → L 2 , (2.49)
and even that

Λ 2+k (z -P 0 ) -1 Λ -k = O 1 h : L 2 → L 2 .
(2.50)

Here we shall take k = 3 in (2.49). Writing the integrand in (2.48) as

z -P δ -1 P δ -P 0 (z -P 0 ) -1 χ √ ε (2.51) = z -P δ -1 P δ -P 0 Λ -3 Λ 3 (z -P 0 ) -1 Λ -3 Λ 3 χ √ ε .
we see that we have to estimate the operator norm of ( P δ -P 0 )Λ -3 . Now it follows from (2.22), (2.30) that

( P δ -P 0 )Λ -3 = O(h 3/2 ) : L 2 → L 2 . (2.52) Also, Λ 3 χ √ ε = O ε 3/2 h 3/2 : L 2 → L 2 . ( 2
(z -P δ ) -1 = O(h -1 ) : L 2 → L 2 , z ∈ hγ,
which follows from Theorem 8.4 in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], and that the length of hγ is O B (h), we obtain that

Π B -Π 0,B χ √ ε = O B ε 3/2 h : L 2 → L 2 .
(2.54)

It only remains now to estimate the operator norm of Π B -Π B χ √ ε . To that end, we write, as in (2.48),

Π B -Π B χ √ ε = 1 2πi hγ z -P δ, ε -1 P δ, ε -P δ z -P δ -1 χ √ ε dz (2.55) = 1 2πi hγ z -P δ, ε -1 P δ, ε -P δ χ √ ε z -P δ -1 χ √ ε dz + 1 2πi hγ z -P δ, ε -1 P δ, ε -P δ 1 -χ √ ε z -P δ -1 χ √ ε dz = I + II,
with the natural definitions of I and II. Using, as in section 8 of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], that

P δ, ε -P δ χ √ ε = O h h ε : L 2 → L 2 ,
together with the O(h -1 )-estimates for the resolvents of P δ, ε and P δ along the contour hγ, we get

I = O B 1 h ε = O B 1 A : L 2 → L 2 .
(2.56)

We now come to estimate the term II in (2.55). We have

χ √ ε P δ, ε -P δ = O h h ε : L 2 → L 2 , (2.57)
and using this estimate, as well as the O(h -1 )-resolvent bounds for P δ, ε and P δ , we see that modulo a term whose operator norm on L 2 is

O B h ε = O B A -1 ,
we may replace the integrand in II by the following expression,

z -P δ, ε -1 1 -χ √ ε P δ, ε -P δ 1 -χ √ ε z -P δ -1 χ √ ε . (2.58)
Here following (2.51), we write

P δ, ε -P δ 1 -χ √ ε z -P δ -1 χ √ ε (2.59) as P δ, ε -P δ 1 -χ √ ε Λ -k-2 Λ k+2 z -P δ -1 Λ -k Λ k χ √ ε .
(2.60)

Here using (2.24) we see that

P δ, ε -P δ 1 -χ √ ε Λ -k-2 = O h k/2+1 ε k/2 . ( 2

.61)

On the other hand, as in (2.50),

Λ k+2 z -P δ -1 Λ -k = O 1 h : L 2 → L 2 , (2.62)
and also, We now come to estimate the remaining factor in (2.58). To that end, we let

Λ k χ √ ε = O ε k/2 h k/2 : L 2 → L 2 . ( 2
L ε = O 1 + min((x, ξ) 2 , ε) h
be an elliptic symbol in the class defined by the right hand side, and write

z -P δ, ε -1 1 -χ √ ε = z -P δ, ε -1 L ε L -1 ε 1 -χ √ ε . (2.64)
Here we know that

z -P δ, ε -1 L ε = O 1 h : L 2 → L 2 ,
and since

L -1 ε 1 -χ √ ε = O h ε : L 2 → L 2 ,
it follows that the expression (2.64) is O 1 ε . Using finally that the length of the integration contour in (2.55) is O B (h), we get

II = O B h ε = O B 1 A : L 2 → L 2 .
(2.65)

Combining (2.55), (2.56), (2.65), we conclude that

Π B -Π B χ √ ε = O B 1 A : L 2 → L 2 .
In view of (2.54), the bound (2.47) follows, and this completes the proof of Lemma 2.1. 2

We now come to estimate the second term in the right hand side of (2.31), given by K ε (1 -Π 0,B )u. Here, the difficulty is that in general, due to a pseudospectral phenomenon [START_REF] Dencker | Pseudospectra for semiclassical pseudodifferential operators[END_REF], the operator norm of Π 0,B may exhibit some exponential growth, as h → 0. To circumvent this issue, our fist task will be to establish a more manageable characterization of the vector v = (1 -Π 0,B )u. Specifically, we shall now discuss properties of the range of the projection 1 -Π 0,B on L 2 . In (2.27), following [START_REF] Sjöstrand | Parametrices for pseudodifferential operators with multiple characteristics[END_REF], we have already recalled the form of the eigenvalues of the elliptic quadratic operator P 0 . From [START_REF] Sjöstrand | Parametrices for pseudodifferential operators with multiple characteristics[END_REF], we know furthermore that the generalized eigenfunctions of P 0 are of the form

h -n/4 p x √ h e iΦ(x)/h , (2.66)
where p(x) is a polynomial on R n and Φ(x) is a quadratic form with Im Φ > 0. The degree of the polynomial p(x) in (2.66) tends to ∞ together with the real part of the eigenvalue hµ j,k in (2.27). We may also recall from [START_REF] Sjöstrand | Parametrices for pseudodifferential operators with multiple characteristics[END_REF] that Φ in (2.66) is such that the positive Lagrangian subspace Λ Φ = {(x, Φ ′ (x)), x ∈ C n } is the direct sum of the generalized eigenspaces of the Hamilton map of p 0 , corresponding to the eigenvalues with a positive imaginary part. Correspondingly, the generalized eigenfunctions of the formal L 2 adjoint P * 0 are of the form

h -n/4 q x √ h e iΨ(x)/h , (2.67)
where q is a polynomial and Ψ is a quadratic form such that Im Ψ is positive definite. Let e 1 , . . . e N be a basis for Ran (Π 0,B ) and let e * 1 , . . . e * N be the corresponding dual basis for Ran ((Π 0,B ) * ). If v ∈ L 2 , we have

Π 0,B v = N j=1 (v|e * j )e j , (2.68)
and therefore, v ∈ Ran(1 -Π 0,B ) precisely when v is orthogonal to Ran ((Π 0,B ) * ).

Proposition 2.2 There exists a selfadjoint h-differential operator Q = Op h (q), where q is a positive definite quadratic form on T * R n , such that

Ran E Q, Bh C ⊂ Ran((Π 0,B ) * ) ⊂ Ran(E(Q, CBh)), (2.69
)

for some C > 1 which is independent of Q and B. Here E(Q, λ) = 1 (-∞,λ] (Q) is the finite rank spectral projection associated to Q and the interval (-∞, λ].
Proof: The operator Q will be seen to be essentially the h-Weyl quantization of the classical harmonic oscillator on R n . When constructing Q, recall that the generalized eigenfunctions of P * 0 are of the form (2.67). We shall write Ψ(x) = (Bx, x), where B is a symmetric matrix, B = B 1 + iB 2 , where B j are real, j = 1, 2, and B 2 > 0. The real linear canonical transformation

κ 1 : (x, ξ) → (x, ξ -B 1 x) (2.70)
maps the positive Lagrangian subspace Λ Ψ = {(x, Bx); x ∈ C n } to the positive Lagrangian subspace {(x, iB 2 x); x ∈ C n }. Now since B 2 > 0, there exists an invertible real n × n matrix C such that the real linear canonical transformation

κ 2 : (x, ξ) → (C -1 x, C t ξ) (2.71) maps {(x, iB 2 x); x ∈ C n } = κ 1 (Λ Ψ ) to {(x, ix); x ∈ C n }.
We take the operator

Q = 1 2 n j=1 x 2 j + (hD x j ) 2 = Op h ( q), q(x, ξ) = 1 2 n j=1 x 2 j + ξ 2 j , (2.72) associated to Λ ϕ = {(x, ix); x ∈ C n } = (κ 2 • κ 1 ) (Λ Ψ ), ϕ(x) = ix 2 /2.
To obtain the operator Q it only remains to notice that associated to κ 1 and κ 2 we have the metaplectic operators

U 1 : L 2 → L 2 , U 1 f (x) = e -i(B 1 x,x)/2h f (x), (2.73)
and

U 2 : L 2 → L 2 , U 2 f (x) = f (Cx) |det C| 1/2 , (2.74)
both unitary on L 2 , and hence with

U := U 2 • U 1 , we can take Q := U -1 QU = Op h (( q • (κ 2 • κ 1 ))).
Notice that the eigenfunctions of Q are of the form

e α,h (x) = e α,h=1 x √ h , e α,h=1 (x) = H α (C -1 x)e iΨ(x) , Qe α,h = h |α| + n 2 e α,h , (2.75
) where H α (x) = n j=1 H α j (x j ), α ∈ N n , are the Hermite polynomials. The result follows.

2

Having established a favorable comparison for the linear space Ran ((Π 0,B ) * ) ⊂ L 2 , we return to the problem of estimating

K ε v, for v = (1 -Π 0,B )u. Let ψ ∈ C ∞ (R; [0, 1]) with supp(ψ) ⊂ [0, 1], and set ψ λ (t) = ψ t λ , λ > 0. (2.76) It follows then from Proposition 2.2 that ψBh C (Q)v = 0. (2.77)
To understand the operator occurring in (2.77), it is convenient to perform a suitable dilation in phase space. Assume therefore that λ > 0 in (2.76) is such that h ≪ λ ≪ 1. Let us make the change of variables

x = λ 1/2 x, D x = λ -1/2 D x .
Then, since the operator Q is quadratic,

1 λ Q = 1 λ q w (x, hD x ) = 1 λ q w λ 1/2 x, h λ D x = q w x, h λ D x .
(2.78)

It follows therefore from the functional calculus in the version of [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF] that

ψ(λ -1 Q) = Op h λ , x r x, ξ; h λ = r w λ -1/2 (x, hD x ) ; h λ , (2.79)
where r ∈ S( • -N ) for any N ∈ N, with a complete asymptotic expansion in each of these symbol spaces, and with the leading symbol ψ(q( x, ξ)).

Remark. It is well known [START_REF] Hörmander | Symplectic classification of quadratic forms and general Mehler formulas[END_REF] that when Q = Op w h (q) where q is a positive definite quadratic form, then the Weyl symbol of f (Q), f ∈ C ∞ 0 (R), is of the form f (q; h) = f (q) + O(h). It follows therefore that in (2.79) we have

r x, ξ; h λ = ψ q( x, ξ); h λ ,
where the leading term of χ is ψ(q( x, ξ)).

It is therefore clear that in (2.25) we can take

K ε = εψBh C (Q), ε = Ah, (2.80)
for a suitable choice of B ≥ A fixed, where we can take B as a fixed multiple of A.

With this choice, we get, using (2.77),

K ε (1 -Π 0,B )u = 0. ( 2
∈ Ran(1 -Π B ), we have δε C -Ch u 2 ≤ Re (P δ, ε u, u) + εO B A 3/2 h 1/2 + A -1 u 2 .
(2.82)

Recall that here B ≥ A, B = O(A), is taken fixed, and δ > 0 is sufficiently small but fixed. Choosing first A ≫ B large enough and then taking h sufficiently small depending on these parameters, we absorb the second term in the right hand side of (2.82) into the left hand side.

Proposition 2.3 When A ≤ B ≪ A, let Π B be the spectral projection of P δ, ε , ε = Ah, associated with D(0, Bh). Here B is a fixed multiple of A. Assume that u ∈ D(P δ, ε ) is such that u ∈ Ran(1 -Π B ).
Then for h sufficiently small, we have

Re P δ, ε u|u ≥ Bh O(1) u 2 , ε = Ah. (2.83) Now recall that P δ, ε = A -1 δ, ε P A δ, ε , where A δ, ε , A -1 δ, ε : S → S, L 2 → L 2 , have L 2 norm O A (1). It is therefore clear from Proposition 2.3 that if u is such that u ∈ Ran(1 -Π), where Π = 1 2πi hγ (z -P ) -1 dz (2.84)
is the spectral projection of P associated with the spectrum of P in D(0, Bh), then e -tP/h u ≤ O(1)e -t/C u , C = C(B) > 0.

(2.85)

Therefore, it only remains to consider the restriction of the semigroup e -tP/h to the finite-dimensional subspace Ran(Π), generated by the generalized eigenfunctions of P corresponding to the eigenvalues of P of modulus < Bh. We shall now proceed to do so, in the framework of supersymmetric differential operators.

3 Supersymmetric operators and return to equilibrium in the double well case

The purpose of this section is to establish Theorem 1.1 in its general form, for a class of supersymmetric second order differential operators, including (1.1). Specifically, let

A : R n → R n (3.1)
be an invertible constant matrix. We decompose

A = B + C, t B = B, t C = -C, (3.2)
and assume that

B ≥ 0. (3.3) When φ ∈ C ∞ (R n ; R) is a Morse function such that ∂ α x φ(x) = O(1), ∂ α x ( B∂ x φ, ∂ x φ ) = O(1), |α| ≥ 2, (3.4)
we consider the Witten-Hodge Laplacian associated to A and φ, acting on scalar functions, defined as in section 10 of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF],

P = -∆ (0) A = j,k hD x j B j,k hD x k + j,k ∂ x j φ B j,k (∂ x k φ) -htr(Bφ ′′ ) (3.5) + j,k (∂ x k φ) C j,k h∂ x j + h∂ x j • C j,k (∂ x k φ) .
The principal symbol of P is of the form

p(x, ξ) = Bξ, ξ + 2i Cφ ′ x , ξ + Bφ ′ x , φ ′ x , (3.6)
so that the assumptions (2.5), (2.6), (2.7) are satisfied. Assume that the Morse function φ has finitely many critical points x 1 , . . .

x N ∈ R n and that |φ ′ (x)| ≥ 1 C , |x| ≥ C. (3.7)
The assumption (2.10) holds with C = {ρ j ; j = 1, . . . N}, ρ j = (x j , 0), and we shall also assume that the dynamical assumptions (2.14), (2.15), and (2.16) are valid. We then know that the results of section 2 can be applied to P in (3.5).

As in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], we shall assume now that φ has precisely three critical points, of which (3.8) two are local minima U ±1 , and the third one U 0 is of index one.

Then we know from [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF] that for C > 0 large enough, P in (3.5) has precisely 2 eigenvalues µ 0 = 0 and µ 1 in the disc D(0, h/C) for h small enough. Here µ 1 is real and such that

µ 1 = h a 1 (h)e -2S 1 /h + a -1 (h)e -2S -1 /h , S j = φ(U 0 ) -φ(U j ) > 0, j = ±1, (3.9) 20
where a j (h) are real, a j (h) ∼ a j,0 + ha j,1 + . . ., a j,0 > 0.

The set φ -1 ((-∞, φ(U 0 ))) has precisely two connected components D j , j = ±1, determined by the condition U j ∈ D j . Let 0 ≤ χ j ∈ C ∞ 0 (D j ) be such that χ j = 1 on D j ∩ φ -1 ((-∞, φ(U 0 ) -ε 0 )) for ε 0 > 0 fixed but arbitrarily small. If

f j = h -n/4 c j (h)e -1 h (φ(x)-φ(U j )) χ j (x), j = ±1,
where c j (h) > 0 is a normalization constant such that f j = 1, and

Π = 1 2πi γ (z -P ) -1 dz, γ = ∂D(0, h C ), C > 0,
is the rank 2 spectral projection of P corresponding to the eigenvalues µ 0 = 0 and µ 1 in (3.9), we have the basis

e j = Πf j , j = ±1,
for Ran(Π), introduced in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF]. From section 11 of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF] we recall that

e j = f j + O(h -N 1 e -1 h (S j -ε 0 ) ) in L 2 , N 1 > 0,
and that the restriction of P to the space Ran(Π) has the matrix

λ * -1 λ * 1 λ -1 λ 1 = λ * -1 λ -1 λ * -1 λ 1 λ * 1 λ -1 λ * 1 λ 1 , (3.10)
with respect to the basis (e -1 , e 1 ), with the eigenvalues µ 0 = 0 and

µ 1 = λ * -1 λ -1 + λ * 1 λ 1 .
A simple computation shows that a corresponding basis of the eigenvectors is given by

λ 1 e -1 -λ -1 e 1 (3.11) and λ * -1 e -1 + λ * 1 e 1 . (3.12)
Here we recall from the formulas (11.43), (11.45), and the following discussion in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF] that if

|λ 1 | ≥ 1 C |λ -1 | then |λ * 1 | ≥ 1 2C λ * -1 and λ 1 λ * 1 > 0.
We have the same fact after permuting the indices -1, 1 and the λ j , λ * j . It follows that

µ 1 ∼ max |λ j | 2 ∼ max λ * j 2 .
(3.13)

Rather than using (3.11) and (3.12), we shall make a normalized choice of the eigenfunctions, given by

v 0 = 1 √ µ 1 (λ 1 e -1 -λ -1 e 1 ) , (3.14)
and

v 1 = 1 √ µ 1 λ * -1 e -1 + λ * 1 e 1 . (3.15)
The corresponding matrix of the coefficients is given by

V = 1 √ µ 1 λ 1 -λ -1 λ * -1 λ * 1 . (3.16)
We have det V = 1 and it follows from (3.13) that V = O(1). Hence the inverse matrix V -1 has the same properties, so that v 0 , v 1 is a well-behaved basis of eigenfunctions for P . If (e * -1 , e * 1 ) ∈ Ran(Π * ) is the basis that is dual to (e -1 , e 1 ), then the corresponding basis of eigenfunctions of P * , dual to (v 0 , v 1 ) is given by the matrix t V -1 , so that

v * 0 = 1 √ µ 1 λ * 1 e * -1 -λ * -1 e * 1 , (3.17)
and

v * 1 = 1 √ µ 1 λ -1 e * -1 + λ 1 e * 1 . (3.18)
We summarize the discussion above in the following proposition.

Proposition 3.1 Let v j and v * j be defined as in (3.14),(3.15), (3.17), (3.18). Then the spectral projections Π j = (•|v * j )v j , j = 0, 1 associated to the eigenvalues µ 0 = 0 and µ 1 in (3.9) are uniformly bounded as h → 0.

Combining (2.85) together with Proposition 3.1, as well as with Theorem 8.4 of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], we get the result in Theorem 1.1 in the general case.

Theorem 3.2 Let P = -∆ (0)
A where we assume (3.1-3.4), (3.7), and (3.8). We also assume that P satisfies the dynamical hypotheses (2.10), (2.14), (2.15), so that the disc D(0, h/C) for C > 0 large enough, contains precisely 2 eigenvalues of P , µ 0 = 0 and µ 1 given in (3.9). Let Π j be the spectral projection associated with the eigenvalue µ j , j = 0, 1. Then we have

Π j = O(1), h → 0. (3.19)
We have furthermore, uniformly as t ≥ 0 and h → 0,

e -tP/h = Π 0 + e -tµ 1 /h Π 1 + O(1)e -t/C , C > 0, in L(L 2 , L 2 ). (3.20)
Here we have also used that the eigenvalues of P in D(0, Bh)\D(0, h/C) have real parts ≥ h/O(1).

4 Tunnel effect for a well and the sea

In this section we shall show how to adapt the analysis of section 11 of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF] and that of section 3 of the present work to cover the case of a potential with a single well and a saddle point, rather than a double well and a saddle point as before. Some parts of this section are very close to the corresponding ones of section 8 in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], and rather than repeating the arguments, we shall often merely refer to the discussion there.

As in section 3, we shall consider the supersymmetric case. Assume that we are given the constant matrices A = B + C and a Morse function φ, that satisfy (3.1)-(3.4). We then have the corresponding Witten-Hodge Laplacian in degree 0, given by (3.5), with a principal symbol (3.6) so that the assumptions (2.5)-(2.7) hold. We refer to the formula (11.3) of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF] for the more general expression for the Witten-Hodge Laplacian in degree q ≥ 0, P (q) . As before, we shall assume that φ has finitely many critical points x 1 , ..., x N ∈ R n and that |φ ′ (x)| ≥ 1/C when |x| ≥ C, with C large. The assumptions (2.10) is therefore satisfied with C = {ρ j ; j = 1, ..., N} where ρ j = (x j , 0). We also assume that the dynamical assumptions (2.14), (2.15), (2.16) hold.

As in section 11 of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], an application of Theorem 8.3 of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF] to P (q) shows that the eigenvalues µ j,k there are of the form

µ j,k = 1 i n k=1 ν j,k,ℓ + 1 2 λ ℓ + γ j,k , (4.1)
where γ j,k is any eigenvalue of the subprincipal symbol S P (q) at (x j , 0). From the calculations in subsection 10.3 of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF] we recall that the µ j,k will be confined to a sector {0} ∪ {|arg z| < π/2 -1/C} around [0, +∞). Recall that it is precisely when

x j is of index q (i.e. when the Hessian of φ at x j has precisely q negative eigenvalues) that one of the µ j,k is equal to 0.

We shall now introduce more specific conditions for the case that we study here. Instead of assuming that we are in the double well case, let us shall suppose that we have a single well and a sea, that is φ has precisely two critical points, one local minimum U 1 , and a "saddle point" U 0 of index one.

(4.2)

Notice that this implies that the Maxwellian e -φ/h is no longer an eigenfunction of P (0) , since φ(x) does not go to +∞ with |x|.

Put S 1 = φ(U 0 ) -φ(U 1 ) so that S 1 > 0. The set φ -1 (] -∞, φ(U 0 )[) has precisely two connected components D j , j = ±1, where D 1 is determined by the condition that U 1 ∈ D 1 , while D -1 is unbounded.

Under these assumptions we shall prove the following result:

Theorem 4.1 Let P = -∆ (0)
A be as in (3.5), where we assume (4.2). Then for C > 0 large enough, P has precisely 1 eigenvalue µ 1 in the disc D(0, h/C) when h > 0 is small enough. Here µ 1 is real and of the form

µ 1 = ha 1 (h)e -2S 1 /h , (4.3)
where

a 1 (h) are real, a 1 (h) ∼ a 1,0 + a 1,1 h + ..., a 1,0 > 0, S 1 = φ(U 0 ) -φ(U 1 ).
Remark 4.2. It is clear that Theorem 4.1 implies an analog of Theorem 3.2 in the present metastable case. We shall refrain from formulating it explicitly.

Proof: We first know that P (0) = -∆

A has precisely one eigenvalue µ 1 = o(h) spanning a corresponding 1-dimensional spectral subspace E (0) since there is a unique local minimum for φ. Now while e -φ/h does not belong to L 2 , a truncation of this function can be used as a quasimode near U 1 and it follows therefore as in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], that

µ 1 = O(h ∞ ). Moreover, -∆ (1)
A has precisely one eigenvalue µ 1 = o(h) and -∆ (k)

A has no eigenvalue = o(h) for k ≥ 2 from the discussion in the beginning of the paragraph. Since our operators are real we know that the spectra are symmetric around the real axis, hence µ 1 , µ 1 are real. From the intertwining relations -∆

(1)

A d φ = d φ (-∆ (0) A ), -∆ (0) A d A, * φ = -d A, * φ ∆ (1)
A ,

we then also know that µ 1 = µ 1 (see also the discussion at the end of page 69 of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF]). A is exactly the same as in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF]. We only retain the following from there:

U 1 D 1 D -1 U 0 φ(x) U 0 U 1 level line at φ(U 0 )
We begin with -∆ (0)

A . Let χ 1 ∈ C ∞ 0 (D 1 ) be equal to 1 on D 1 ∩ φ -1 (] -∞, φ(U 0 ) - ǫ 0 ]
) for ǫ 0 > 0 fixed but arbitrarily small. Consider

f 1 = h -n/4 c 1 (h)e -1 h (φ(x)-φ(U 1 )) χ 1 (x), (4.4)
where c 1 (h) ∼ c 1,0 + hc 1,1 + ... > 0 is a normalization constant with c 1,0 > 0, such that f 1 = 1. Then the normalized eigenfunction associated to µ 1 is given by

e 1 := f 1 + O(h -N 1 e -1 h (S 1 -ǫ 0 ) ) in L 2 . (4.5)
We continue with the study of -∆

A . Let E (1) be the one-dimensional eigenspace of P (1) corresponding to µ 1 . From an easy extension of [12, Theorem 9.1] (see also Remark 9.2 there) to the non-scalar case with the presence of the other non-resonant well U 1 , we know that E (1) is generated by an eigenform

e 0 (x; h) = χ 0 (x)e -1 h φ + (x) h -n 4 a 0 (x; h) + O(e -S 0 /h ), (4.6)
where χ 0 ∈ C ∞ 0 (neigh (U 0 )) is equal to one near U 0 , S 0 > 0, and

a 0 (x; h) ∼ ∞ 0 a 0,k (x)h k
is a symbol as in Theorem 9.1 of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], with a 0,0 (U 0 ) = 0. Here the phase

φ + ∈ C ∞ (neigh (U 0 ); [0, ∞)) satisfies φ + (x) ∽ |x -U 0 | 2
and solves the eikonal equation q(x, φ ′ + (x)) = 0, with q = p 2 + p 1 -p 0 .

From [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], let us recall that the phase function φ + arises as the generating function for the stable outgoing manifold through (U 0 , 0) for the H q -flow, Λ Φ + , and recall also that φ ′′ + (U 0 ) > 0 by Proposition [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF]Proposition 8.2]. (Similarly we have a stable incoming manifold Λ φ -.) Let k ± be the number of eigenvalues of the linearization of H q | Λ φ at that point with ± real part > 0, so that k

+ + k -= n. Let K + , K -⊂ Λ φ
be the corresponding stable outgoing and incoming submanifolds of dimension k + and k -respectively. Then K + ⊂ Λ φ + , K -⊂ Λ φ -and φ -φ(U 0 ) -φ ± vanishes to the second order on π x (K ± ). Since φ ′′ (U 0 ) has signature (n -1, 1), we conclude that dim

K + = n -1, dim K -= 1. It is also clear that Λ φ , Λ φ ± intersect cleanly along K ± , so we get φ + -(φ -φ(U 0 )) ∽ dist (x, π x (K + )) 2 , φ -φ(U 0 ) -φ -∽ dist (x, π x (K -)) 2 . (4.7)
We now make some remarks about the adjoint operator -∆t A = (-∆ A ) t A, * . As proved in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF]Subsection 10.4]), we get the corresponding phases and submanifolds in this case, which satisfy

φ * + -(φ -φ(U 0 )) ∽ dist (x, π x (K * + )) 2 , φ -φ(U 0 ) -φ * -∽ dist (x, π x (K * -)) 2 . (4.8)
and for symmetry reason we recall that

φ * -= -φ + , φ * + = -φ -, (4.9)
giving in particular from (4.7), (4.8),

φ -φ(U 0 ) + φ * + ∽ dist (x, π x (K -)) 2 , φ -φ(U 0 ) + φ + ∽ dist (x, π x (K * -)) 2 .
(4.10)

Let µ * 1 be the eigenvalue of

P (0) * := -∆ (0) t A that is o(h).
As before, this is also the eigenvalue o(h) of -∆

e * 0 (x; h) = χ 0 (x)e -1 h φ * + (x) h -n 4 a * 0 (x; h) + O(e -S 0 /h ) and e * 1 (x, h) = h -n/4 c j (h)e -1 h (φ(x)-φ(U 1 )) χ 1 (x) + O(h -N 1 e -1 h (S 1 -ǫ 0 ) ) (4.11)
Now, using that our eigenvalues and operators are real, we know by duality that µ * 1 = µ 1 , and that (E (0) * , E (0) ) and (E

(1) * , E (1) ) are dual pairs for the scalar products (u|v) L 2 and (u|v) A respectively. Following [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF]Subsection 10.3] we know that (a * 0,0 (U 0 )|a 0,0 (U 0 )) A = 0 and that e * 0 can be normalized so that (e * 0 |e 0 ) A = 1. (4.12)

Similarly, denoting by e * 1 the L 2 normalized eigenfuncion spanning E Let (λ 1 ) be the (scalar) matrix of d φ : E (0) → E (1) with respect to the bases (e 1 ) and (e 0 ). Let also (λ * 1 ) be the (scalar valued) matrix of d A, * φ for the same bases. The eigenvalue µ 1 can be viewed as the scalar d A, * φ d φ : E (0) → E (0) . We get

µ 1 = λ * 1 λ 1 , (4.14)
and

λ 1 = (e * 0 |d φ e 1 ) A , λ * 1 = (g 1 |d A, * φ e 0 ) A , j = ±1, (4.15)
where

g 1 = e * 1 (1 + O(e -1 Ch )) is the vector in E (0) * that is dual to e 1 .
Here the complex conjugate signs are superfluous since we work with real operators, eigenvalues and functions.

We skip the computation of λ 1 , which is exactly the same as in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], just recalling that the main term is equal to

-c 1 (h)h 1-n 2 χ 1 (x) A(x)a * 0 (x; h)|dχ(x) e -1 h (φ * + (x)+φ(x)-φ(U 1 )) dx. (4.16)
and can be evaluated thanks to the stationary phase using (4.10). We get

λ 1 = h 1 2 ℓ 1 (h)e -1 h S 1 (1 + O(e -1 Ch )), ℓ 1 ∼ ℓ 1,0 + hℓ 1,1 + ..., ℓ 1,0 = 0. (4.17)
similarly, λ * 1 ca be evaluated in a dual point of view as in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF] and we also get

λ * 1 = h 1 2 ℓ * 1 (h)e -1 h S 1 (1 + O(e -1 Ch )), ℓ * 1 (h) ∼ ℓ * 1,0 + hℓ * 1,1 + ..., ℓ * 1,0 = 0. (4.18)
We eventually claim that ℓ 1,0 ℓ * 1,0 > 0. Indeed, this number is real and different form zero and if we deform our matrices to reach the selfadjoint case (with A > 0) we see that we have a positive sign). Combining this with (4.14), the proof of Theorem 4.1 is complete. 2

5 Some models of KFP type operators

Probabilistic description

Here we shall give some examples of Kramers-Fokker-Planck type operators. We begin with a very short review of stochastic calculus in order to explain their probabilistic origin, and refer to the books [START_REF] Nelson | Dynamical theories of Brownian motion[END_REF], [START_REF]Stochastic Differential equations[END_REF] for more details. Part of this material can be also found in [START_REF] Eckmann | Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures[END_REF], [START_REF] Eckmann | Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators[END_REF] and [START_REF] Eckmann | Spectral properties of hypoelliptic operators[END_REF], from where the example of the chain of anharmonic oscillators is taken. Let x(t) ∈ R n be a stochastic process satisfying the following stochastic differential equation

dx(t) = b(x(t))dt + σdw, (5.1)
where w is the m-dimensional Wiener process, σ is a linear map from R m to R n , and b is a C ∞ -vector field on R n , all of whose derivatives are bounded. Under these assumptions, there exists a unique global solution x(t) of (5.1), for a given initial data x(0) = x, independent of w, in an adapted stochastic L 2 setting -see the references already mentioned. Then we can define a semigroup of operators T t , t ≥ 0, by

E (ϕ(x(t))|F s ) = T t-s ϕ(x(s)), a.s. (5.2)
when 0 ≤ s ≤ t. Here F t is the filtration associated to {w(s) -w(0); 0 ≤ s ≤ t} and x, and ϕ ∈ C (0) (R n ), where C (0) (R n ) is the Banach space of continuous functions vanishing at infinity, with the topology of the uniform convergence. Then T t is a strongly continuous positivity preserving contraction semigroup, whose infinitesimal generator is given on

C ∞ 0 (R n ) by L = ∇ • D∇ + b(x) • ∇,
where D = 1 2 σσ t . The idea now is to extend T t to a larger class of test functions, and then to study the evolution of the adjoint (T t ) * on the dual space of bounded measures. To be precise, let us denote by dµ t (x) the probability distribution for x(t), defined for all t ≥ 0. We then have

E (ϕ(x(t))) = ϕ(x)dµ t (x), ϕ ∈ C (0) (R n ),
and we get by (5.2) that µ t = (T t ) * µ 0 , where (T t ) * is the adjoint of the operator T t acting on the Banach space of bounded measures on R n .

We shall now extend the space of test functions. When doing so, we introduce the Hilbert space

H = L 2 (R n , e -Φ(x) dx) (5.3)
where Φ ∈ C ∞ (R n ). We shall make the following assumptions concerning Φ :

∂ α x Φ(x) = O(1), |α| ≥ 2, (5.4) 1 2 b(x) • ∇Φ(x) + 1 4 D∇ x Φ, ∇ x Φ - 1 2 div b ≤ O(1) (5.5)
and

∂ α x 1 2 b(x) • ∇Φ(x) + 1 4 D∇ x Φ, ∇ x Φ - 1 2 div b = O(1), |α| ≥ 2. ( 5.6)
These conditions will be fulfilled in the case that we shall study in what follows, since in the supersymmetric case it is straightforward to verify that they are equivalent to (3.4). Now we can identify the dual H ′ of H with the space of densities

H * = L 2 (R n , e Φ(x) dx).
Assume that the measures dµ t are absolutely continuous with density in H * , and write dµ t = f (t, .)dx, identifying the measure dµ t with the corresponding density f t . We denote again by (T t ) * acting on H * the adjoint of T t acting on H. We introduce the formal adjoint operator L * on H * of L on H, with the domain C ∞ 0 (R n ), which is given by

L * = ∇ • D∇ -∇ • b(x).
We have the following result.

Lemma 5.1 Assume that Φ ∈ C ∞ (R n ) satisfies (5.4), (5.6). Then operator -L on H (resp. -L * on H * ) is m-accretive, and T t (resp (T t ) * ) is a strongly continuous semigroups on H (resp. H * ), with infinitesimal generators given by L (resp. L * ).

Proof.

It will be more convenient to work in the unweighted space

L 2 (R n ). To this end, if φ ∈ H, we write φ(x) = e Φ(x)/2 ψ(x), ψ ∈ L 2 . If ∂ t φ = Lφ,
then the equation satisfied by ψ is

∂ t ψ = e -Φ/2 Le Φ/2 ψ, so that ∂ t ψ = (∂ x + ∂ x Φ/2) • D(∂ x + ∂ x Φ/2)ψ + b(x) • (∂ x + ∂ x Φ/2)ψ.
Let C be a sufficiently large constant. According to (5.4-5.6), the operator

-((∂ x + ∂ x Φ/2) • D(∂ x + ∂ x Φ/2) + b(x) • (∂ x + ∂ x Φ/2)) + C
has a symbol satisfying the hypotheses (2.4)-(2.7). Here we may recall that the vector field b is bounded on R n together with all of its derivatives. An application of Corollary 3.2 in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF] shows that its maximal closed realization in L coincides with the graph closure on S. Coming back to H and denoting by L again its maximal closed extension, we get that L + C is maximal accretive, and that T t is a strongly continuous semigroup, thanks to the Hille-Yosida Theorem. As for the dual semigroup on H * , we also get (for example, using Corollary 10.6 in [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]), that the same occurs for (T t ) * and L * . The proof is complete.

2

From the preceding discussion, we get the equation satisfied by the density f for an initial data f 0 ∈ H * ,

∂ t f + (-L * )f = 0 f | t=0 = f 0 i.e. ∂ t f + (-∇ • D∇ + ∇ • b)f = 0 f | t=0 = f 0 (5.7)
where we recall that D = σσ t /2. In particular we have dµ t = f (t, .)dx = (T t ) * µ 0 in the space of bounded measures. If there exists an invariant probability measure µ ∞ , then its density M is in H * . In our present study we shall essentially make the choice C -1 e -Φ = M, but there are cases (see e.g. [START_REF] Eckmann | Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures[END_REF], [START_REF] Eckmann | Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators[END_REF]), where it may happen that no invariant measure is known, and that another choice of the function Φ is necessary. Such a function M will be called a Maxwellian of the process. Notice that if it exists, it is a 0-eigenfunction of L * and positive. Remark 5.2. Notice that it may also happen that there exists an invariant measure, which fails to be finite. We also associated to it a function that we will call again Maxwellian (and denote again by M). In that case of course it cannot be normalized. Equation (5.7) is nearly the Kramers-Fokker-Planck type equation that we studied in the first part of the paper. In the following sections we shall also do the following two things: first we shall exhibit the semiclassical scaling, which corresponds to the low temperature limit in the models we are going to study later.

Second, we shall change our unknown by posing f = e -Φ(x)/2 u (forgetting for a while the semiclassical scaling), in order to work in the flat space L 2 rather than in H. Finally, in the three models that we present in the next subsections (Witten, Kramers-Fokker-Planck, and the chain of anharmonic oscillators), we shall recognize the supersymmetric structure.

Witten and Kramers-Fokker-Planck operators

We begin with the Witten case. It corresponds to an evolution equation with a gradient field -γ∇V (x) and a diffusion force coming from a heat bath at a temperature T . We have dx = -γ∂ x V dt + 2γT dw.

Here x ∈ R n is the spatial variable, the parameter γ is a friction coefficient, and w is an n-dimensional Wiener process of mean 0 and variance 1. With the notation of the preceding subsection, we recover an equation of type (5.1) with D = σσ t /2 = γT I d and b(x) = -γ∂ x V . Equation (5.7) for the density in this case is then

∂ t f -γT ∂ 2 x f -γ∂ x (∂ x V f ) = 0 f | t=0 = f 0 i.e. ∂ t f -γ∂ x (T ∂ x + ∂ x V )f = 0 f | t=0 = f 0 . (5.8)
Posing T = h/2 and multiplying by h gives the semiclassical equation

h∂ t f - γ 2 h∂ x (h∂ x + 2∂ x V )f = 0. ( 5.9)
It is then clear that an associated Maxwellian of the process is

M(x) = e -2V (x)/h .
Writing f = M 1/2 u, we obtain from (5.9) that

h∂ t u + γ 2 (-h∂ x + ∂ x V )(h∂ x + ∂ x V )u = 0.
(5.10)

Here we recognize the Witten operator W = (-h∂ x + ∂ x V )(h∂ x + ∂ x V ). In the notation of (3.1)- (3.5), it corresponds to a supersymmetric operator with

A = γ 2 I d , φ(x) = V (x)
Assumptions of type (3.4) on V are then fulfilled if ) are satisfied (we skip the proof here, which will be given later in the more complex case of the chain of oscillators). In particular, in this case we get Theorem 3.2. Of course, the corresponding result follows also in the case of a single well and the sea, i.e. when V has precisely one local minimum and a saddle point (Theorem 4.1 and the remark following it).

∂ α V (x) = O (
We proceed now to discuss the Kramers-Fokker-Planck case, and follow the same method. The stochastic equation of type (5.1) comes here from the Newton law

dx = ydt dy = -γydt -∂ x V (x)dt + √ 2γT dw .
(5.12)

The parameter γ is a friction coefficient, and the particle of position x ∈ R n and velocity y ∈ R n is submitted to an external force field derived from a potential V , with w being an n-dimensional Brownian process of mean 0 and variance 1. With the notation of the preceding subsection, we therefore have

D = σσ t /2 = 0 0 0 γT I d and b(x, y) = y -γy -∂ x V .
The corresponding equation for the density (5.7) is then

∂ t f -γT ∆ y f + ∂ x (yf ) + ∂ y (-γyf -∂ x V f ) = 0 f | t=0 = f 0 (5.13) i.e. ∂ t f -γ∂ y .(T ∂ y + y)f + y∂ x f -∂ x V ∂ y f = 0 f | t=0 = f 0 . (5.14)
Posing T = h/2 and multiplying by h gives the semiclassical formulation

h∂ t f - γ 2 h∂ y .(h∂ y + 2y)f + yh∂ x f -∂ x V h∂ y f = 0. (5.15)
A Maxwellian of the process is then

M(x, y) = C -1 e -2(V (x)+y 2 /2)/h
where C is a normalization constant. If we write f = M 1/2 u, then (5.15) gives

h∂ t u + γ 2 (-h∂ y + y) • (h∂ y + y)u + γy • h∂ x u -∂ x V • h∂ y u = 0 (5.16)
This is the semiclassical Kramers-Fokker-Planck operator (1.1) that we studied in [START_REF] Hérau | Semiclassical analysis for Kramers-Fokker-Planck type operators[END_REF], [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], and the first part of the present paper. In the notation of section 3, it corresponds to a supersymmetric operator with

A = 1 2 0 I d -I d γ and φ(x, y) = V (x) + y 2 /2.
The assumptions (3.4) are fulfilled if

∂ α V (x) = O(1)

Chains of anharmonic oscillators

The last example that we give comes from the series of papers [START_REF] Eckmann | Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures[END_REF], [5] [6]. It is a model describing a chain of two anharmonic oscillators coupled with two heat baths at each side. The particles are described by their respective position and velocity (x j , y j ) ∈ R 2d . We suppose that for each oscillator j ∈ {1, 2}, the particles are submitted to an external force derived from a potential V j (x j ), and that there is a coupling between the two oscillators derived from a potential V c (x 2 -x 1 ). We denote by V the sum

V (x) = V 1 (x 1 ) + V 2 (x 2 ) + V c (x 2 -x 1 )
, where x = (x 1 , x 2 ), and we also write y = (y 1 , y 2 ). By z j , j ∈ {1, 2} we shall denote the variables describing the state of the particles in each of the heat baths, and set z = (z 1 , z 2 ). We suppose that the particles in each bath are submitted to a coupling

(x 1 , y 1 ) (x 2 , y 2 ) z 1 α 1 h z 2 α 2 h
Figure 2: Oscillators coupled to heat baths with the nearest oscillator, a friction force and a thermal diffusion at temperature T j , (j = 1, 2). We denote by w j , j ∈ {1, 2}, two d-dimensional brownian motions of mean 0 and variance 1, and set w = (w 1 , w 2 ).

The fundamental system of equations of type (5.1) is then written as follows, (see [START_REF] Eckmann | Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures[END_REF] for more detail concerning the physical constants)

               dx 1 = y 1 dt dy 1 = -∂ x 1 V (x)dt + z 1 dt dz 1 = -γz 1 dt + γx 1 dt - √ 2γT 1 dw 1 dz 2 = -γz 1 dt + γx 2 dt - √ 2γT 2 dw 2 dy 2 = -∂ x 2 V (x)dt + z 2 dt dx 2 = y 2 dt. (6.1)
The parameter γ is the friction coefficient in the baths. In the (x, y, z) variables, the diffusion matrix and the drift appearing in (5.1) are therefore

D = σσ t /2 =   0 0 0 0 0 0 0 0 γT I d   and b(x) =   y -∂ x V + z γ(x -z)   .
(for simplicity we identified T with the 2d times 2d diagonal matrix with coefficients T 1 I and T 2 I. The corresponding equation (5.7) for the density is then

∂ t f -γT ∂ 2 z f + ∂ x (yf ) + ∂ y (-∂ x V f + zf ) + ∂ z (γ(x -z)) = 0 f t=0 = f 0 (6.2) where T ∂ 2 z stands for T 1 ∂ 2 z 1 + T 2 ∂ 2 z 2 . We get ∂ t f -γ∂ z (T ∂ z + (z -x))f + y∂ x f -(∂ x V -z)∂ y f = 0 f t=0 = f 0 . (6.3)
Notice that it is very close to the Witten and Kramers-Fokker-Planck operator. For a semiclassical formulation, we pose T 1 = α 1 h/2 and T 2 = α 2 h/2, and we multiply (6.3) by h. This gives

h∂ t f + γ 2 α 1 (-h∂ z 1 )(h∂ z 1 + 2(z 1 -x 1 )/α 1 )f + γ 2 α 2 (-h∂ z 2 )(h∂ z 2 + 2(z 2 -x 2 )/α 2 )f + (yh∂ x f -(∂ x V -z)h∂ y )f = 0. (6.4)
At this stage it is difficult to exhibit a Maxwellian. Indeed the existence of an invariant measure is a difficult problem solved in some particular case in [START_REF] Eckmann | Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures[END_REF]. Anyway, it is clear that the function Φ(x, y, z) = V (x) + y 2 /2 + z 2 /2 -zx plays a special role, in fact it is the classical energy at temperature 1. We can also check that in the case of same temperatures (α 1 = α 2 def = α), the function

M α = C -1 e -2Φ/αh
is a Maxwellian of the process. We use this function to define the weighted space H def = L 2 (e -2Φ/αh dxdydz) as in (5.3), and in order to work in the flat space L 2 we make the change of unknown f = M 1/2 α u. Equation (5.14) becomes

h∂ t u + γ 2 α 1 -h∂ z 1 + 1 α (z 1 -x 1 ) h∂ z 1 + 2 α 1 - 1 α (z 1 -x 1 ) u + γ 2 α 2 -h∂ z 2 + 1 α (z 2 -x 2 ) h∂ z 2 + 2 α 2 - 1 α (z 2 -x 2 ) u + (yh∂ x -(∂ x V -z)h∂ y ) u = 0. (6.5)
We impose the following condition on the parameter α :

α ≥ max {α 1 , α 2 } /2.
which corresponds to a semiclassical study at "reference" temperature αh/2 not too low.

Unfortunately we are not able to find any supersymmetric structure in the case of different temperatures, since a Maxwellian is not known in this case. From now on we therefore stick to the case of identical temperatures T = h/2 so that

α = α 1 = α 2 = 1. Equation (6.5) becomes h∂ t u + γ 2 (-h∂ z 1 + (z 1 -x 1 )) (h∂ z 1 + (z 1 -x 1 )) u + γ 2 (-h∂ z 2 + (z 2 -x 2 )) (h∂ z 2 + (z 2 -x 2 )) u + (yh∂ x -(∂ x V -z)h∂ y ) u = 0 (6.6)
and the Maxwellian was already exhibited M 1 = C -1 e -2Φ/h . This equation can be written h∂ y u + P u = 0 where

P = γ 2 (-h∂ z 1 + (z 1 -x 1 )) (h∂ z 1 + (z 1 -x 1 )) + γ 2 (-h∂ z 2 + (z 2 -x 2 )) (h∂ z 2 + (z 2 -x 2 )) + (yh∂ x -(∂ x V -z)h∂ y ) . (6.7)
In the notations of Section 3 (3.1-3.5), we can write P as a Witten-Hodge laplacian P = -∆

A with a supersymmetric phase φ given by

φ def = Φ = V (x) + y 2 /2 + z 2 /2 -zx, (6.8)
and the non-degenerate matrix A = B + C given by

A = 1 2   0 I d 0 -I d 0 0 0 0 γI d   with B = 1 2   0 0 0 0 0 0 0 0 γI d   , C = 1 2   0 I d 0 -I d 0 0 0 0 0   .
In order to complete the semiclassical study as in [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF], we only need additional conditions on the potentials V 1 , V 2 and V c . It is clear that the conditions

∂ α V ε (x) = O(1)
when |α| ≥ 2, with ε = 1, 2 and c (6.9) imply (3.4). In view of the definition (6.8), it is straightforward that φ has exactly the same number of critical points than V (x) -x 2 /2 with same index. For this it is sufficient to notice that there is a natural splitting of the variables for Φ given by

Φ = (V (x) -x 2 /2) + y 2 /2 + (z -x) 2 /2.
We postpone to the end of this section the proof of the following lemma: Lemma 6.1 Suppose that V satisfies (6.9). If in addition V (x) -x 2 /2 is a Morse function and there exists C such that

|∂V (x) -x| ≥ 1/C when |x| ≥ C, (6.10)
then (3.7) and the dynamical conditions (2.14-2.16) are fulfilled.

the corresponding Hodge Laplacian on 0-forms (minus the constant γhd/2) where denoting (ξ, η, ζ) the dual variables of (x, y, z) we have

p 2 = γ 2 ζ 2 , p 1 = yξ -(∂ x V -z)η, p 0 = γ 2 (z -x) 2 .
In particular, with the notations of Section 2, we have

ν(x, y, z, ∂ x , ∂ y , ∂ z ) = y∂ x -(∂ x V -z)∂ y . (6.11)
We denote by x 1 , ..., x N the critical points of V (x) -x 2 /2, and notice that the critical points of Φ are (x j , 0, x j ) for j = 1, ..., N. According to definitions (2.11) and (6.11), the critical set C of p is made of the points ρ j = (x j , 0, x j , 0, 0, 0) for j = 1, ..., N. We also introduce π x (resp. π xyz ) the orthogonal projections on R d x (resp. R 3d x,y,z ) from R 6d .

Let now ε > 0 be a fixed constant. Since V (x) -x 2 /2 is a Morse function, it is non-degenerate, so that with (6.10) we get the following: there exists C ε such that

∀x ∈ π x (C + B(0, ε)) , |∂V (x) -x| ≥ 1/C ε . (6.12)
From this result we get easily the following one: there exists C ′ ε such that In order to complete the proof of (2.16) we study in detail the characteristics of the flow generated by the vector field ν(x, y, z, ∂ x , ∂ y , ∂ z ). We first notice that the flow is complete since the gradient of Φ is Lipschitz. Let (x 0 , y 0 , z 0 ) ∈ π x,y,z (C + B(0, ε)). We denote by (x(t), y(t), z(t)) the integral curves of ν for t ∈ R with x(0) = x 0 , y(0) = y 0 , z(0) = z This implies that on [0, t V ] with t V ≤ 1/C V ≤ 1/C |y(s) - Let us now take again 0 < θ < t V . We consider three cases:

y 0 + s(∂ x V (x 0 ) -z 0 )| ≤ t 2 2 C V (|y 0 | + t|∂ x V (x 0 ) -z 0 |) ≤ C V t 2 |∂ x V (x 0 ) -z 0 |, (6.21
|x(t)-z 0 | 2 ≥ |∂ x V (x 0 )-z 0 |( 1 4 t 2 -18tθ-18θ 2 ) ≥ t 2 V 16 |∂ x V (x 0 )-z 0 | on [t V /2, t V ].
if 18θ 2 ≤ t 2 V /16. In all cases we get that there exists a positive constant c ′ V only depending on C V such that |x(t) -z 0 | ≥ c ′ V max {|y 0 |, |x 0 -z 0 |, |∂ x V (x 0 ) -z 0 |} (6.24) on an interval of length at least θ.

Conclusion From (6.18), (6.24) and using (6.14), we get that in all cases there exists an interval of length at least min(θ, t V /2) on which p 0 (exp (tν)(x 0 , y 0 , z

0 )) = γ 2 |x(t) -z 0 | 2 ≥ γ 2 (min (c ′ V , c V ) max {|y 0 |, |x 0 -z 0 |, |∂ x V (x 0 ) -z 0 |}) 2 .
(6.25)

Since (x 0 , y 0 , z 0 ) ∈ π xyz (C + B(0, ε)) and using (6.13) we get that the dynamical condition (2.16) is fulfilled. The proof is complete. 2

Figure 1 :

 1 Figure 1: A well and the sea

1 )

 1 when |α| = 2 O( x -1 ) when |α| ≥ 3 If we also suppose that V is a Morse function with two local minima and a saddle point of index one, such that |∇V | ≥ 1/C, for |x| ≥ C > 0. (5.11) then the dynamical asumptions (2.14)-(2.16

∀

  (x, y, z) ∈ π x,y,z (C + B(0, ε)) , |∂V (x) -x| + |z -x| + |y| ≥ 1/C ′ ε .

0 V 0 V

 00 ẏ = -(∂ x V (x) -z) ż = 0,and we have to studyp 0 (exp (tν)(x 0 , y 0 , z 0 )) = γ 2 |x(t) -z(t)| 2 = γ 2 |x(t) -z 0 | 2 . (6.14)We split the study into two cases. Let t ∈ [0, 1].First case: Suppose |y 0 | ≥ |∂ x V (x 0 ) -z 0 |. We write that y(t) = y 0 + V (x(s)) -z 0 )ds = y 0 -t(∂ x V (x 0 ) -z 0 ) -′′ (x(r)) ẋ(r)dr ds = y 0 -t(∂ x V (x 0 ) -z 0 ) -′′ (x(r))y(r)dr ds. (6.15) Since V ′′ is uniformly bounded, and denoting by C V ≥ 1 a corresponding bound we get sup s∈[0,t] |y(s) -y 0 | ≤ t|∂ x V (x 0 ) -z 0 | + t 2 2 C V sup s∈[0,t] |y(s)|.

= x 0 + ty 0 - t 2 2 ( 3 C V t 3

 233 ) since by assumption |y0 | ≤ |∂ x V (x 0 ) -z 0 |. We therefore get sup s∈[0,t] |y(s) -y 0 + s(∂ x V (x 0 ) -z 0 )| ≤ 2C V t 2 |∂ x V (x 0 ) -z 0 |.We can then write thatx(t) = x 0 + ∂ x V (x 0 ) -z 0 ) + 2 |∂ x V (x 0 ) -z 0 |ψ(t)(6.22)with |ψ(t)| ≤ 1 on [0, t V ]. Recalling (6.14) and using the triangular inequality, weget for t ∈ [0, t V ] that if t V ≤ 3/(8C V ), |x(t) -z 0 | 2 ≥ max {|x 0 -z 0 | -t|y 0 | -t 2 |∂ x V (x 0 ) -z 0 |, t|y 0 | -|x 0 -z 0 | -t 2 |∂ x V (x 0 ) -z 0 |, t 2 4 |∂ x V (x 0 ) -z 0 | -|x 0 -z 0 | -t|y 0 | } (6.23)

1 . 2 .

 12 If |x 0 -z 0 | ≥ max {θ|y 0 |, 18θ 2 |∂ x V (x 0 ) -z 0 |},we use the first expression in (6.23) and we get that|x(t) -z 0 | 2 ≥ |x 0 -z 0 |(1 -t/θ -18t 2 /θ 2 ) ≥ 1 4 |x 0 -z 0 | on [0, θ/6]. If θ|y 0 | ≥ max {|x 0 -z 0 |, 18θ 2 |∂ x V (x 0) -z 0 |}, we use the second expression in (6.23) and we get that|x(t) -z 0 | 2 ≥ |y 0 | t -θ -t 2 /(18θ) ≥ θ 2 |y 0 | on [2θ, 3θ]. 3. If 18θ 2 |∂ x V (x 0 ) -z 0 | ≥ max (|x 0 -z 0 |, θ|y0|), we use the third expression in (6.23) and we get that

  .81) Combining (2.26), (2.31), Lemma 2.1, and (2.81), we see that for u ∈ L 2 with u

  when |α| ≥ 2As in the Witten case, if we also suppose that V is a Morse function with precisely two local minima and a saddle point of index one, and that|∇V | ≥ 1/C, for |x| ≥ C > 0.(5.17)then the dynamical asumptions (2.14)-(2.16) are satisfied, and Theorem 3.2 is applicable. The corresponding result in the case of a single well and the sea, i.e. when V has one local minimum and a saddle point, is also valid (Theorem 4.1 and the following remark).

  Since by assumption |y0 | ≥ |∂ x V (x 0 ) -z 0 | we get sup s∈[0,t] |y(s) -y 0 | ≤ 2t|y 0 | + t 2 C V |y 0 | ≤ 3t|y 0 |. V ]. Recalling(6.14) and using the triangular inequality, weget for t ∈ [0, t V ] that if t V ≤ 1/4 |x(t) -z 0 | 2 ≥ max |x 0 -z 0 | -2t|y 0 |, t 2 |y 0 | -|x 0 -z 0 | (6.17)Let now 0 < θ < t V . We split again the study into two parts:1. If |x 0 -z 0 | ≥ θ|y 0 |, we use the first expression in (6.17) and we get that|x(t) -z 0 | 2 ≥ |x 0 -z 0 |(1 -2t/θ) ≥ 1 2 |x 0 -z 0 | on [0, θ/4].2. If θ|y 0 | ≥ |x 0 -z 0 |, we use the second expression in (6.17) and we get that|x(t) -z 0 | 2 ≥ |y 0 |(t/2 -θ) ≥ t V 8 |y 0 | on [t V /2, t V ]. if θ ≤ t V /8.In all cases we get that there exists a constant c V > 0 depending only onC V such that |x(t) -z 0 | 2 ≥ c V max {|y 0 |, |x 0 -z 0 |, |∂ x V (x 0 ) -z 0 |} (6.18)on an interval of length at least θ.Second case: Suppose |y 0 | ≤ |∂ x V (x 0 ) -z 0 |. As in (6.15) we can write that This implies that on [0, t V ] with again t V ≤ 1/C V ≤ 1/C

			1/2 V	we have
		1 2	sup s∈[0,t]
	We can then write that x(t) = x 0 + = x 0 + ty 0 + t 0 ẋ(s)ds = x 0 + 3 2 t 2 |y 0 |φ(t) so that sup s∈[0,t] ≤ t 2 2 C V sup s∈[0,t] with |φ(t)| ≤ 1 on [0, t t 2 0 2	t C V sup y(s)ds s∈[0,t] 1/2 V we have |y(s)|,	(6.16) (6.20)
	1 2	sup s∈[0,t]

|y(s) -y 0 | ≤ t|∂ x V (x 0 ) -z 0 | + t 2 2 C V |y 0 |. y(t) = y 0 -t(∂ x V (x 0 ) -z 0 ) -t 0 s 0 V ′′ (x(r))y(r)dr ds.

(6.19)

Since V ′′ is uniformly bounded, and denoting again by

C V ≥ 1 a corresponding bound we get sup s∈[0,t] |y(s) -y 0 + s(∂ x V (x 0 ) -z 0 )| ≤ |y(s) -y 0 + s(∂ x V (x 0 ) -z 0 )| |y(s) -y 0 + s(∂ x V (x 0 ) -z 0 )| + t 2 2 C V (|y 0 | + t|∂ x V (x 0 ) -z 0 |) .

It is a great pleasure for him to thank its Département de Mathématiques, and in particular Alain Grigis, for the extraordinary hospitality and excellent working conditions. His best thanks are also due to the Centre de Mathématiques of École Polytechnique for the generous hospitality in June of 2007. The partial support of his research by the National Science Foundation under grant DMS-0653275 and by an Alfred P. Sloan Research Fellowship is also gratefully acknowledged.

As a consequence we can apply Theorem 3.2 to operator P : Proposition 6.2 Consider P given by (6.7) and suppose that V satisfies (6.9) and (6.10). Then if the effective potential V (x) -x 2 /2 is of double well type, (two local minima and a saddle point of index 1), the hypotheses of Theorem 3.2 are fulfilled and as a consequence its conclusions apply to operator P .

Proof.

It is straightforward. From the construction of P , hypotheses (3.1-3.3) are fulfilled. From (6.9), hypothesis (3.4) is satisfied. Since the effective potential V (x) -x 2 /2 is a morse function of double well type, then (3.8) is also satisfied since, as already noticed, V (x) -x 2 /2 and φ have the same number of critical point with same index. Eventually using Lemma 6.1 and (6.10) we get that hypothese (3.7) and the dynamical conditions (2.14-2.16) are fulfilled. The proof is complete and the conclusions of Theorem 3.2 apply to P . 2 Remark 6.3. Of course the corresponding result follows in the case of a well and the sea, ie when V (x) -x 2 /2 has one minimum and a saddle point (Theorem 4.1 and the remark after).

A simple family of such potentials is given for example by the ones for which

1 /2 of double well type, V 2 (x 2 )-x 2 2 /2 of single well type, and V c sufficiently small. Here is an example of such potentials in 1d:

Here x 1 , x 2 , x ′ ∈ R.

Remark 6.4. We did all the computations in the case of 2 oscillators. It is clear that the preceding supersymmetric construction works as well in the case of N oscillators coupled and with two heat bathes at each side as in [START_REF] Eckmann | Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures[END_REF]. We did not try to verify the dynamical conditions in these cases. Eventually recall that the complete study at different temperatures seems difficult to treat (see e.g. the recent work by Hairer and Mattingly [START_REF] Hairer | Slow energy dissipation in systems of anharmonic oscillators[END_REF] in the case of 3 oscillators).

Proof of Lemma 6.1.

We only prove here condition (3.7) and (2.16) since the proof of other ones follow the same kind of arguments. The proof uses strongly the fact that the symbols are with quadratic growth at most. We write p = p 2 + ip 1 + p 0 for the symbol of


Charly Caredda

Laurent Mahieu-Williame

Rapha El Sablong

Micha El Sdika

Jacques Guyotat

Bruno Montcel

Pixel-wise modied Beer-Lambert model for intraoperative functional brain mapping

Keywords:

e qf mer nd ontinuous wve white light illumintion is suitle pproh to intropertively lolize the sensory nd motor res of the ptient rinF he nlysis of the reetne spetr through the modied feerEvmert lw enles us to mesure the onentrtion hnges of oxygented nd deoxygented hemogloin during the video quisitionF rowever these onentrtion hnges depend on the wvelength dependent optil men pth lengthF e mnul imge segmenttion nd wonteEgrlo simultions llow us to preisely hoose the men pth length in pixelEwise mnnerF

INTRODUCTION

xonEinvsive funtionl rin mpping is n imging tehnique used to lolize the funtionl res of the ptient rinF his tehnique is used efore rin tumor resetion surgery to indite to the neurosurgeon the ortil tissues whih should not e removed without ognitive impirmentF wsEf is widely used to lolize ptient sensory or motor resF roweverD fter ptient rniotomyD rin shift invlidtes the relevne of neuroE nvigtion to intropertively lolize the funtionl res of the ptient rinF por this resonD the loliztion of ortil motor or sensory res is urrently hieved through eletril stimultion ut this tehnique suers from limittionsF Chance et al. hve demonstrted tht optil imging n e used to monitor rin tivity with the determintion of onentrtion hnges of oxygented hemogloin ¡HbO 2 D deoxygented hemogloin ¡HbD lood volume ¡Hb C ¡HbO 2 nd oxygented ytohromeEoxidseF 1 st lso hs een demonstrted tht devie onsisting of ontinuous wve white light illumintion nd qf 2 or hyperspetrl mer 3 is suitle pprohF he motivtion for this work is derived from the neessity to preisely ssess ¡HbO 2 nd ¡Hb for eh mer pixelF his requires preise determintion of the optil men pth length for the omputtion of the modied feerEvmert lwF 4 e propose to estimte wvelength dependent optil men pth length for eh mer pixel with mnul imge segmenttion nd wonteEgrlo simultionsF 2. MATERIAL AND METHODS 2.1 Experimental setup he instrumenttion onsisted of digitl qf gwy mer @fevi ePHHHE ITSuAD ontinuous wve white light soure @yew glssi IIT PQH vight ulA nd lptop @proessorX sntel gore iS UPHHUD P:SHGHz¢RD rmX IS:QGiBAF he qf mer ws used with n idmund yptis mer lens @f a ITmmf5 a I X I X RA nd dt were diretly quired y the lptop vi n f linkF he sensor provided V its qf imges every QQ msF purther uthor informtionX @end orrespondene to ghrly greddA ghrly greddX iEmilX hrlyFredddretisFinsElyonFfr 2.2 Patient inclusion and experimental paradigm he video quisition ws relized efore rin tumor resetion surgeryF he ptient ws wkened nd funtionl tests hve een relizedF timultion of the motor ortex ws hieved through suessive st lenhing nd relxingF he prdigm onsisted of Q stepsX QHs of restD followed y QHs of stimultion nd QHs of restF he neurosurgeon relized eletril rin stimultion to lolize ptient rin motor nd sensory resF 2.3 Data pre-processing he rst imge of the video sequene ws segmented with semi utomted proedure into three lssesX grey mtterD surfe lood vessel nd uried lood vesselF ixels were lustered into four lusters using the uEwens lgorithm from the gCC lirry ypeng @vQ:P:HAF 5 hen these lusters were mnully sorted nd ttriuted to the three lssesF xon seleted lusters orresponded to sturted res @speulr reetionA were disrdedF he lgorithm developed y Sdika et al. 6,7 ws implemented to ensure tht eh mer pixel orresponds to the sme ortil re ll long dt quisitionF es the rin repetitive motion is ompenstedD the segmenttion of the rst imge is vlid during the whole video sequeneF he qf intensities of the quired imge re then independently ltered with lowEpss ss fessel temporl lterF 1 he slow drift of the olleted intensity due to ortil tissue desition during the video quisition 8 ws orreted through liner regression sutrted to the originl dtF 2.4 Model he ssessment of H b nd H bO 2 onentrtion hnges ws relized through the modied feerEvmert lwF 4 hese mesurements depend on the determintion of the wvelength dependent men pth length of the deteted photonsF he men pth length vlues were omputed with wonteEgrlo simultionsF 9 sn pigF IEe nd fD two ortil tissues were modeled under homogeneous white light illumintionF ih voxel of the modeled tissues inluded the vlues of optil prmeters @sorptionD redued stteringD nisotropy oeients nd refrtive indexAF 3. RESULTS AND DISCUSSION sn pigF IEhD the mnul segmenttion of the pigF IEg ims to ssoite eh pixel of the quired frme with spei tissueF sn pigF IEiD severl wvelength dependent men pth lengths mesured on the surfe of the modeled tissues @see pigF IEe nd fA re plottedF hese men pth lengths re used in the modied feerElmert lw to ompute dierent ¡Hb nd ¡HbO 2 time ourses @see pigF IEpAF he grey mtter point lolized y red ross in pigF IEg is situted t the surfe of the tissueD I:QRmm wy from lood vesselF o the most pproprite men pth length is represented y the lk urve in the pigF IEi nd the most pproprite ¡Hb nd ¡HbO 2 time ourses re represented y the lk nd green urves in pigF IEpF he inuene of the hoie of the men pth length in the modied feerEvmert lw n e oserved in the pigF IEpF yur method n e improved with n utomti men pth length estimtion y tting the quired reetne spetr with the simulted reetne spetrF his omprison ppers not to e n esy tsk sine only three spetrl vlues re quired whih re integrted over the rod wvelength rnges overed y eh detetorF en hyperspetrl mer seems to e more pproprite solution to nswer this prolemF e similr method hs een implemented y Pichette et al. 3 using n hyperspetrl mer nd two lss imge segmenttion ut this tehnique does not tke ount of the distne etween ortil tissues nd lood vesselF 4. CONCLUSION e qf mer nd white light illumintion re suitle pproh to intropertively monitor the sensory nd motor res of ptient rinF e semi utomted segmenttion nd wonteEgrlo simultions llow us to preisely ssess the H b nd H bO 2 onentrtion hnges with pixelEwise seletion of the men pth lengthF ACKNOWLEDGMENTS his work ws supported y the vefi swi @exEIIEvefEHHTQA of niversit e de vyonD within the progrm snvestissements devenir @exE IIEshiEHHHUA operted y the prenh xtionl eserh egeny nd prne vife smgingF pigure IF A and B E epresenttion of modeled ortil tissuesF olumes re mde up of UH ¢ UH ¢ UH voxels with Imm 3 resolutionF he model I represents P mm dimeter lood vessel on the surfe of the ortil tissueD the model PD P mm dimeter lood vessel uried under I mm of grey mtterF he red voxels represent the lood vessel nd the grey voxels the grey mtterF e flk rrow symolizes wonteEgrlo simultion for single position @IH 6 pkets of photonAF C E pirst frme of the ptient rinF Mi nd Si designte the ptient motor nd sensory res i vlidted y eletril stimultionF D E wnul imge segmenttion @red X surfe lood vesselD lue X uried lood vesselD grey X grey mtter nd yn X unproessed dtAF E E wen pth length of the models represented in pigF e nd fF F E ¡Hb nd ¡HbO2 time ourses of the ortil tissue lolized y the red ross in pigF gF REFERENCES I ghneD fFD hungD FD nehD gFD elterD gFD nd viptonD vFD gognitionEtivted lowEfrequeny modultion of light sorption in humn rinFD4 Proceedings of the National Academy of Sciences 90D QUUH{QUUR @eprF IWWQAF P wlonekD hF nd qrinvldD eFD sntertions etween eletril tivity nd ortil miroirultion reveled y imging spetrosopyX smplitions for funtionl rin mppingD4 272D SSI{R @HS IWWTAF Q ihetteD tFD vureneD eFD enguloD vFD vesgeD pFD fouthillierD eFD xguyenD hF uFD nd velondD pFD sntropE ertive videoErte hemodynmi response ssessment in humn ortex using snpshot hyperspetrl optil imgingD4 Neurophotonics 3D HRSHHQ @ytF PHITAF R uohlEfreisD wFD iertD fFD hreierD tF FD veithnerD gFD vinduerD FD nd oylD qFD epprtus for mesuring lood prmetersD4 @II PHIPAF S frdskiD qFD he ypeng virryD4 Dr. Dobb's Journal of Software Tools @PHHHAF T dikD wFD elstonD vFD ousseuD hFD quyottD tFD whieuEillimeD vFD nd wontelD fFD epetitive motion ompenstion for rel time intropertive video proessingD4 Medical Image Analysis 53D I { IH @PHIWAF U dikD wFD elstonD vFD whieuEillimeD vFD quyottD tFD ousseuD hFD nd wontelD fFD oust rel time motion ompenstion for intropertive video proessing during neurosurgeryD4 in 2016 IEEE 13th International Symposium on Biomedical Imaging D IHRT{IHRWD siiiD rgueD gzeh epuli @eprF PHITAF V yelshl gelD wFD weyerD FD hlD rFD oottkD F fFD uirshD wFD hkertD qFD nd worgensternD FD ivlution of intropertive optil imging nlysis methods y phntom nd ptient mesurementsD4 Biomedizinische Technik/Biomedical Engineering 58 @tnF PHIQAF W pngD F nd fosD hF eFD wonte grlo simultion of photon migrtion in Qd turid medi elerted y grphis proessing unitsD4 Optics express 17@PPAD PHIUV{PHIWH @PHHWAF