Dorian Le Peutrec
email: dorian.lepeutrec@math.u-psud.fr

Boris Nectoux
email: boris.nectoux@asc.tuwien.ac.at

Repartition of the quasi-stationary distribution and first exit point density for a double-well potential

Keywords: overdamped Langevin process, double-well, metastability, tunneling effect, semiclassical analysis, quasi-stationary distribution AMS classification (2010): 35P15, 35P20, 47F05, 35Q82

Let f : R d → R be a smooth function and (Xt) t≥0 be the stochastic process solution to the overdamped Langevin dynamics

Let Ω ⊂ R d be a smooth bounded domain and assume that f | Ω is a doublewell potential with degenerate barriers. In this work, we study in the small temperature regime, i.e. when h → 0 + , the asymptotic repartition of the quasi-stationary distribution of (Xt) t≥0 in Ω within the two wells of f | Ω . We show that this distribution generically concentrates in precisely one well of f | Ω when h → 0 + but can nevertheless concentrate in both wells when f | Ω admits sufficient symmetries. This phenomenon corresponds to the so-called tunneling effect in semiclassical analysis. We also investigate in this setting the asymptotic behaviour when h → 0 + of the first exit point distribution from Ω of (Xt) t≥0 when X0 is distributed according to the quasi-stationary distribution.

1 Setting and results

Quasi-stationary distribution and purpose of this work

Let (X t ) t≥0 be the stochastic process solution to the overdamped Langevin dynamics in R d :

dX t = -∇f (X t )dt + √ h dB t , (1)
where f : R d → R is the potential (chosen C ∞ in all this work), h > 0 is the temperature and (B t ) t≥0 is a standard d-dimensional Brownian motion. Let Ω be a C ∞ bounded open and connected subset of R d and introduce

τ Ω = inf{t ≥ 0 | X t / ∈ Ω}
the first exit time from Ω. A quasi-stationary distribution for the process [START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF] on Ω is a probability measure µ h on Ω such that, when X 0 is distributed according to µ h , what we will denote in the following by X 0 ∼ µ h , it holds for any time t > 0 and any Borel set A ⊂ Ω,

P(X t ∈ A | t < τ Ω ) = µ h (A).
From [START_REF] Champagnat | General criteria for the study of quasistationarity[END_REF][START_REF] Gong | Killed diffusions and their conditioning[END_REF][START_REF] Bris | A mathematical formalization of the parallel replica dynamics[END_REF][START_REF] Pinsky | On the convergence of diffusion processes conditioned to remain in a bounded region for large time to limiting positive recurrent diffusion processes[END_REF], there exists a probability measure ν h supported in Ω such that for any probability measure µ 0 on Ω: when X 0 ∼ µ 0 , one has for any borel set A ⊂ Ω, lim

t→+∞ P(X t ∈ A | t < τ Ω ) = ν h (A). (2)
It follows from ( 2) that ν h is the unique quasi-stationary distribution for the process [START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF] on Ω.

In molecular dynamics, the quasi-stationary distribution ν h is used to quantify the metastability of the subdomain Ω of R d as follows: for a probability measure µ 0 supported in Ω, the domain Ω is said to be metastable for the initial condition µ 0 if, when X 0 ∼ µ 0 , the convergence in ( 2) is much quicker than the average exit time from Ω. When Ω is metastable, it is thus relevant to study the exit event (τ Ω , X τΩ ) of the process [START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF] from Ω starting from ν h , i.e. when X 0 ∼ ν h . This is used in several algorithms aiming at accelarating the sampling of the exit even from a metastable domain, see for instance [START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF][START_REF] Bris | A mathematical formalization of the parallel replica dynamics[END_REF][START_REF] Perez | Longtime dynamics through parallel trajectory splicing[END_REF][START_REF] Sorensen | Temperature-accelerated dynamics for simulation of infrequent events[END_REF]. The study of the metastability is a very active field of science research which is at the heart of the numerical challenges observed in molecular dynamics. We refer in particular to [START_REF] Lelièvre | Partial differential equations and stochastic methods in molecular dynamics[END_REF] for an overview on this topic.

In this work, we study the repartition when h → 0 of the quasi-stationary distribution ν h within the wells of a double-well Morse potential f with degenerate barriers (see the assumption [H-Well] below). We show in particular that ν h generically concentrates in one well (see Theorem 1 below) but can also concentrate in both wells when the function f is (nearly) even (see Theorems 2 and 3 below). According to the analysis led in [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 1[END_REF] (see also the preprint [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points[END_REF] which concatenates the results of [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 1[END_REF] and of [START_REF] Lelièvre | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 2[END_REF]), the second phenomenon can only appear when the potential function f admits degenerate deepest barriers. It is particularly unstable (see Remark 4 below) and arises from a strong tunneling effect between the wells. The asymptotic behaviour of the law of X τΩ when h → 0 is also investigated in order to discuss the metastability of Ω for deterministic initial conditions within the wells.

Connections with the existing literature

As it will be clearly stated below in the first part of Section 1.4, the quasidistribution ν h is completely characterized by the ground state of the Dirichlet realization of the infinitesimal generator L (0) f,h of the diffusion [START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF],

L (0) f,h = - h 2 ∆ + ∇f • ∇ = 1 2h e -f h ∆ (0) f,h e f h ,
where ∆

f,h = -h 2 ∆ + |∇f | 2 -h∆f is the usual Witten Laplacian acting on functions. In this respect, the techniques used in this work originate from the semiclassical literature dealing with the obtention of sharp asymptotics on the 2 low spectrum of ∆ (0) f,h in the limit h → 0 and we refer in particular in this direction to [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach[END_REF] in the case without boundary and to [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary[END_REF] in the case of Dirichlet boundary conditions (see also the prior related works [START_REF] Bovier | Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times[END_REF][START_REF] Bovier | Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues[END_REF] using potentialtheoretic methods and which motivated [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach[END_REF][START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary[END_REF]). However, these references focus on the low spectrum of ∆ (0) f,h and not really on the concentration of the corresponding eigenfunctions. In addition, though they consider multiple-well Morse potentials, they do not consider the case of degenerate barriers. The case of general Morse potentials f , allowing in particular degenerate barriers, has nevertheless been recently treated in the case without boundary in [START_REF] Michel | About small eigenvalues of the Witten Laplacian[END_REF] (see also [START_REF] Berglund | The Eyring-Kramers law for Markovian jump processes with symmetries[END_REF] for related results) using the techniques of [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach[END_REF][START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary[END_REF].

More closely related to the present work, the already mentioned paper [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 1[END_REF] involving both authors generalizes in particular the results of [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary[END_REF] to more general multiple-well Morse potentials but actually focuses on where the quasistationary distribution (or equivalently the ground state) concentrates in Ω and where the exit point distribution concentrates on ∂Ω. Moreover, our results heavily rely on intermediate results proven in [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 1[END_REF] (see Propositions 2 and 4 in Section 2.1.2). However, the degenerate situation considered in the present paper is excluded in [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 1[END_REF], where the principal barrier of f is assumed to be non degenerate (see indeed [7, Assumption (A1)]).

Double-well potential

We assume more generally from now on that Ω = Ω ∪ ∂Ω is a C ∞ oriented compact and connected Riemannian manifold of dimension d with boundary ∂Ω. The basic assumption in this work is the following:

Ω f = arg min Ω f = {x 1 , x 2 }.
Finally, the open set {x ∈ Ω , f (x) < min ∂Ω f } has precisely two connected components, denoted by C 1 and C 2 , such that for all j ∈ {1, 2},

x j ∈ C j and ∂C j ∩ ∂Ω = ∅.
Under the assumption [H-Well], the potential function f has precisely two wells, namely the open sets C 1 and C 2 . This double-well potential is moreover said to have degenerate barriers since the depths of C 1 and C 2 are the same and equal (see Figure 1)

H := min ∂Ω f -min Ω f = min ∂Ω f -min Ω f > 0. ( 3
)
Let us also recall that a function g : Ω → R is a Morse function if all its critical points are non degenerate. This implies in particular that g has a finite number of critical points. When replacing the assumption arg min

Ω f = {x 1 , x 2 } by arg min Ω f = {x 1 } in [H-Well] (i.e. when the barriers are not degenerate), it is proved in [7, {f = min ∂Ω f } {f = min Ω f } H C 1 C 2 x 1 x 2 Figure 1: A one dimensional example where [H-Well] is satisfied.
Proposition 9] that the quasi-stationary distribution ν h concentrates in C 1 when h → 0. This work aims precisely at studying the degenerate case arg min Ω f = {x 1 , x 2 } which introduces some additional technical difficulties, see the next section for some explanation.

Let us assume from now on that the assumption [H-Well] is satisfed. The set of saddle points of f of index 1 in Ω is denoted by U Ω 1 . Let us also define

U ∂Ω 1 : = {z ∈ ∂Ω, z is a local minimum of f | ∂Ω } ∩ {z ∈ ∂Ω, ∂ n f (z) > 0} and U Ω 1 := U ∂Ω 1 ∪ U Ω 1 and m Ω 1 := Card(U Ω 1 ).
According to the terminology of [13, Section 5.2], we call the elements of U Ω 1 the generalized saddle points for the Witten Laplacian acting on 1-forms with tangential Dirichlet boundary conditions on ∂Ω. Note that f does not have any saddle point on ∂Ω (since ∇f = 0 there) but that extending f by -∞ outside Ω (which is consistent with zero boundary Dirichlet conditions), the elements of U Ω 1 are geometrically saddle points (since for such an element z, z is a local minimum of f | ∂Ω and a local maximum of f | D , where D is the straight line passing through z and orthogonal to ∂Ω at z).

Notice that from the assumption [H-Well], one has for all i ∈ {1, 2}:

∂C i ∩ ∂Ω ⊂ U ∂Ω 1 ∩ arg min ∂Ω f = (∂C 1 ∪ ∂C 2 ) ∩ ∂Ω .
Let us define, for i ∈ {1, 2}, z i,1 , . . . , z i,ni by

∂C i ∩ ∂Ω = {z i,1 , . . . , z i,ni }, where n i ≥ 1 according to [H-Well]. (4)
One defines furthermore z 3,1 , . . . , z 3,n3 by

{z 3,1 , . . . , z 3,n3 } = U Ω 1 \ ∪ 2 j=1 ∂C j ∩ ∂Ω ,
where

n 3 ∈ N (n 3 = 0 meaning U Ω 1 \ ∪ 2 j=1 ∂C j ∩ ∂Ω = ∅). From [7, Proposi- tion 15], it holds ∂C 1 ∩ ∂C 2 ⊂ U Ω 1 ∩ {f = min ∂Ω f }
and one orders z 3,1 , . . . , z 3,n3 so that

∂C 1 ∩ ∂C 2 = {z 3,1 , . . . , z 3,m3 } ,
where m 3 ∈ {0, . . . , n 3 }. Note finally the relation

m Ω 1 = n 1 + n 2 + n 3 . (5)
See Figures 2 and3 for a schematic representation of the potential f under

[H-Well] when ∂C 1 ∩ ∂C 2 = ∅ and when ∂C 1 ∩ ∂C 2 = ∅. C 1 C 2 Ω ∂Ω z 1,2 x 1 x 2 z 1,1 z 2,1 Figure 2: Schematic representation of the connected components C1 and C2 of {f < min ∂Ω f } when the assumption [H-Well] is satisfied. In this representa- tion, ∂C1 ∩ ∂C2 = ∅, U ∂Ω 1 = {z1,1, z1,2, z2,1}, ∂C1 ∩ ∂Ω = {z1,1, z1,2}, ∂C2 ∩ ∂Ω = {z2,1}, U Ω 1 = ∅ and arg min Ω f = {x1, x2}. Thus, n1 = 2, n2 = 1 and n3 = m3 = 0. C 1 C 2 Ω ∂Ω z 3,1 z 3,2 x 1 x 2 z 1,1 z 2,1
∂C1 ∩ ∂C2 = {z3,1}, U ∂Ω 1 = {z1,1, z2,1, z3,2}, ∂C1 ∩ ∂Ω = {z1,1}, ∂C2 ∩ ∂Ω = {z2,1}, U Ω 1 = {z3,1} and arg min Ω f = {x1, x2}. Thus, n1 = 1, n2 = 1, m3 = 1 and n3 = 2.

Results

Preliminary spectral analysis

Let L (0) f,h be the infinitesimal generator of the diffusion (1),

L (0) f,h = h 2 ∆ (0) H + ∇f • ∇, where ∆ (0)
H is the Hodge Laplacian on Ω and ∇ the gradient associated with the metric tensor on Ω. Let moreover L D,(0) f,h be the differential operator

L (0) f,h on L 2 (Ω, e -2 h f (x) dx) with domain D L D,(0) f,h = w ∈ H 2 (Ω, e -2 h f (x) dx), w = 0 on ∂Ω .
The operator L D,(0) f,h is self-adjoint, positive, and has compact resolvent. Moreover, its smallest eigenvalue λ 1 (h) is positive, non degenerate, and any eigenfunction associated with λ 1 (h) has a sign on Ω (see for instance [START_REF] Evans | Partial differential equations[END_REF]Section 6]). Let u h be an eigenfunction associated with λ 1 (h). According to [START_REF] Bris | A mathematical formalization of the parallel replica dynamics[END_REF], the quasistationary distribution ν h is then given by

dν h := u h (x) e -2 h f (x) Ω u h e -2 h f dx, ( 6
)
where dx is the Lebesgue measure on Ω. We assume furthermore from now on that u h > 0 on Ω and

Ω u 2 h e -2 h f = 1. ( 7
)
In view of [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points[END_REF], in order to study the asymptotic behaviour of ν h when h → 0, we look for an accurate approximation of u h . This is delicate since exponentially small eigenvalues of the same order are into play. Indeed, according to [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 1[END_REF]Theorem 4], under [H-Well], it holds

lim h→0 h ln λ 1 (h) = -2(min ∂Ω f -min Ω f ) = -2H
and there exists C > 1 such that for every h > 0 small enough,

1 < λ 2 (h) λ 1 (h) ≤ C,
where λ 2 (h) denotes the second smallest eigenvalue of L D,(0) f,h . This makes in particular difficult to properly estimate u h by simply projecting a well chosen quasi-mode on Span(u h ) since the quality of such an approximation is typically bounded from above by the quotient λ1(h) λ2(h) which does not tend to 0 when h → 0. To overcome this difficulty, the key point relies on the fact that we are able to precisely analyse the restriction of L D,(0) f,h to the eigenspace associated with λ 1 (h) and λ 2 (h). Indeed, this eigenspace has dimension two and the remaining eigenvalues of L D,(0) f,h are bounded from below by

√ h 2
1 . More precisely, we have according to [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary[END_REF]Theorem 3.2.3] the Lemma 1. Let us assume that the hypothesis [H-Well] is satisfied. Then, there exists h 0 > 0 such that for all h ∈ (0, h 0 ),

dim Ran π [0, √ h 2 ) L D,(0) f,h = 2,
where π [0,

√ h 2 ) L D,(0) f,h
is the orthogonal projector on the vector space associated with the eigenvalues of

L D,(0) f,h in [0, √ h
2 ).

Remark 1. As a consequence of Lemma 1, there exists h 0 > 0 such that for every h ∈ (0, h 0 ), the second smallest eigenvalue λ 2 (h) of L D,(0) f,h is non degenerate.

Moreover, it follows from the general analysis led in [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 1[END_REF] that the matrix

L of L D,(0) f,h Ran π [0, √ h 2 ) L D,(0) f,h
satisfies Proposition 1 below. Before stating it, let us introduce the following notation. For (γ(h)) h>0 ∈ R R * + , one writes γ(h) √ h if there exist C > 0 and h 0 > 0 such that for all h ∈ (0, h 0 ),

1 C √ h ≤ γ(h) ≤ C √ h. (8)
In addition, for α > 0, one says that (r(h)) h>0 admits a full asymptotic expansion in h α , and one writes r(h) ∼ +∞ k=0 a k h αk , if there exists a sequence (a k ) k≥0 ∈ R N such that for any N ∈ N, it holds in the limit h → 0:

r(h) = N k=0 a k h αk + O h α(N +1) . (9)
Proposition 1. Let us assume that the hypothesis [H-Well] is satisfied. Then, there exists h 0 > 0 such that for every h ∈ (0, h 0 ), there exists an orthonormal basis

B 0 = (ϕ 1 , ϕ 2 ) of Ran π [0, √ h 2 ) L D,(0) f,h
such that the matrix L of the re-

striction of L D,(0) f,h to Ran π [0, √ h 2 ) L D,(0) f,h
in B 0 has the form:

L = 1 2 α 1 (h) ε(h) ε(h) α 2 (h) h -1 2 e -2 h H , ( 10
)
where H is defined in (3),

• ε(h) satisfies in the limit h → 0:

ε(h) = O e -c h if ∂C 1 ∩ ∂C 2 = ∅ √ h if ∂C 1 ∩ ∂C 2 = ∅, (11)
for some c > 0 independent of h and where the symbol is defined in (8),

• there exist two sequences

(κ 1,k ) k≥0 ∈ R N and (κ 2,k ) k≥0 ∈ R N such that for i ∈ {1, 2}, in the limit h → 0: α i (h) ∼        +∞ k=0 κ i,k h k if ∂C 1 ∩ ∂C 2 = ∅ +∞ k=0 κ i,k h k 2 if ∂C 1 ∩ ∂C 2 = ∅, (12)
where the symbol ∼ is defined in (9) and

κ i,0 = ni j=1 2 ∂ n f (z i,j ) π 1 2 det Hessf (x i ) det Hessf | ∂Ω (z i,j ) . ( 13
)
Moreover, when ∂C 1 ∩ ∂C 2 = ∅, one has for every i ∈ {1, 2},

κ i,1 = m3 j=1 |λ -(z 3,j )| det Hess f (x i ) 1 2 π det Hess f (z 3,j ) 1 2 , ( 14
)
where λ -(z) is the negative eigenvalue of Hess f (z). Finally, the sequence (κ 1,k ) k≥1 (resp. (κ 2,k ) k≥1 ) only depends on the values of the derivatives of f at x 1 and on ∂C 1 ∩ ∂Ω ∪ ∂C 2 (resp. of the derivatives of f at x 2 and on ∂C 2 ∩ ∂Ω ∪ ∂C 1 ).

Proposition 1 will be proven in Section 2.1. It permits to reduce the study of the asymptotic repartition of ν h within the wells C 1 and C 2 to linear algebra considerations in dimension two. Then, when X 0 ∼ ν h , the study of the asymptotic concentration of the law of X τΩ (which occurs on a subset of arg min ∂Ω f , see [7, Definition 1] for a precise definition) follows from the analysis made in [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 1[END_REF] and based on the following formula [START_REF] Bris | A mathematical formalization of the parallel replica dynamics[END_REF]: for any F ∈ L ∞ (∂Ω, R), it holds

E ν h [F (X τΩ )] = - h 2λ 1 (h) ∂Ω F ∂ n u h e -2 h f Ω u h e -2 h f , (15)
where the notation E ν h stands for the expectation when X 0 ∼ ν h .

Results when ν h concentrates in precisely one well when h → 0 Let us define here the following assumption:

[H1]:

The assumption [H-Well] is satisfied, there exists h 0 > 0 such that either for all h ∈ (0, h 0 ), α 1 (h) < α 2 (h), [START_REF] Helffer | Puits multiples en limite semi-classique. II. Interaction moléculaire. Symétries. Perturbation[END_REF] or for all h ∈ (0, h 0 ),

α 2 (h) < α 1 (h), (17)
and it holds

lim h→0 ε(h) α 1 (h) -α 2 (h) = 0.
Note that the assumption [H1] is generic (given an arbitrary function f satisfying [H-Well]) according to the following:

• when ∂C 1 ∩∂C 2 = ∅ and the asymptotic expansion in h of α 1 (h) and α 2 (h) in ( 12) differ (i.e. when (κ 1,k ) k≥0 = (κ 2,k ) k≥0 ), the assumption [H1] is satisfied and there exists c > 0 such that when h → 0 (see indeed [START_REF] Helffer | Semiclassical analysis for the Schrödinger operator and applications[END_REF]),

ε(h) α 1 (h) -α 2 (h) = O e -c h , (18)
• when ∂C 1 ∩ ∂C 2 = ∅ the assumption [H1] is, according to ( 11) and ( 12), equivalent to κ 1,0 = κ 2,0 , where κ 1,0 and κ 2,0 are defined in [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary[END_REF]. In this case, when h → 0:

ε(h) α 1 (h) -α 2 (h) O √ h . (19)
Our main result under the generic assumption [H1] is the following. It implies in particular that when [H1] holds together with ( 16), ν h concentrates in any neighborhood of x 1 (i.e. lim h→0 ν h (O 1 ) = 1 for any open subset O 1 of Ω containing x 1 , see more precisely (21) below). This can be roughly explained as follows: when [H1] holds, the term ε(h) can be neglected in the expression of the matrix L given in [START_REF] Gong | Killed diffusions and their conditioning[END_REF], and ( 16) breaks the symmetry between the two wells C 1 and C 2 , ensuring more precisely the concentration of ν h in C 1 .

Theorem 1. Let us assume that the hypotheses [H-Well] and [H1] together with (16) are satisfied. Let ν h be the quasi-stationary distribution of the process (1) on Ω (see

(6)). Let O 1 ⊂ Ω be an open neighborhood of x 1 and O 2 ⊂ Ω be an open neighborhood of x 2 such that O 1 ∩ O 2 = ∅.
Then, there exists c > 0 such that in the limit h → 0:

ν h (O 1 ) + ν h (O 2 ) = 1 + O e -c h , (20)
where for k ∈ {1, 2},

ν h (O k ) = δ 1,k + O |ε(h)| |α 2 (h) -α 1 (h)| + O e -c h . (21)
Moreover, for any F ∈ L ∞ (∂Ω, R) and for any family

(Σ i,j ) (i,j)∈ 2 p=1 {p}×{1,...,np} of disjoint open neighborhoods of (z i,j ) (i,j)∈ 2
p=1 {p}×{1,...,np} in ∂Ω, there exists c > 0 such that in the limit h → 0:

E ν h [F (X τΩ )] = (i,j)∈ 2 p=1 {p}×{1,...,np} E ν h 1 Σi,j F (X τΩ ) + O e -c h (22)
and

n2 j=1 E ν h 1 Σ2,j F (X τΩ ) = O |ε(h)| |α 2 (h) -α 1 (h)| + O e -c h . ( 23
)
In addition, when, for some j ∈ {1, . . . , n 1 }, F is C ∞ around z 1,j , one has when h → 0:

E ν h 1 Σ1,j F (X τΩ ) = F (z 1,j ) a 1,j + O |ε(h)| |α 2 (h) -α 1 (h)| + O h , (24)
where, for i ∈ {1, 2} and j ∈ {1, . . . , n i }, the constant a i,j is defined by

a i,j := ∂ n f (z i,j ) det Hess f ∂Ω (z i,j )   ni k=1 ∂ n f (z i,k ) det Hess f ∂Ω (z i,k )   -1 . ( 25
)
Remark 2. When [H-Well] and [H1] are satisfied, one also obtains from Proposition 1 sharp asymptotic estimates on the two smallest eigenvalues

0 < λ 1 (h) < λ 2 (h) of L D,(0) f,h
when h → 0, see indeed (54) and (55).

From Theorem 1, when [H-Well] holds and [H1] is satisfied with ( 16), the quasi-stationary distribution ν h concentrates when h → 0 in C 1 and more precisely around any arbitrary small neighborhood of x 1 . Moreover, when X 0 ∼ ν h , the law of X τΩ concentrates when h → 0 on {z 1,1 , . . . , z 1,n1 } = ∂C 1 ∩ ∂Ω with an explicit repartition given by (25). Adapting the proof of [6, Proposition 11] (see also [START_REF] Lelièvre | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 2[END_REF]) by using ( 20) and ( 21), one can also show that when X 0 = x ∈ C 1 , the law of X τΩ concentrates when h → 0 on {z 1,1 , . . . , z 1,n1 } = ∂C 1 ∩ ∂Ω with the same repartition as when X 0 ∼ ν h . This exhibits a metastable behavior for such initial conditions. Moreover, when |∇f | = 0 on ∂C 2 , it follows from [6, Theorem 2] that when X 0 = x ∈ C 2 , the law of X τΩ concentrates when h → 0 on {z 1,2 , . . . , z 1,n2 } = ∂C 2 ∩ ∂Ω with the repartition given by (25) (with i = 2). This exhibits a non metastable behavior for such initial conditions.

To connect with the literature dealing with semiclassical Schrödinger operators of the form h 2 ∆ (0)

H + V on manifolds without boundary (where V is a potential function independent of h), one can say in this situation that the tunneling effect between the two wells is too weak to mix their respective properties and that these two wells are hence somehow independent, that is, in the terminology of [START_REF] Helffer | Multiple wells in the semi-classical limit III -Interaction through non-resonant wells[END_REF][START_REF] Helffer | Puits multiples en limite semi-classique. II. Interaction moléculaire. Symétries. Perturbation[END_REF], weakly resonant or non resonant. We also refer to [START_REF] Helffer | Semiclassical analysis for the Schrödinger operator and applications[END_REF] for an overview on this topic for semiclassical Schrödinger operators (see in particular pp. 41-42 there). Notice lastly that [START_REF] Perez | Longtime dynamics through parallel trajectory splicing[END_REF] shows that some tunneling effect of order √ h appears nevertheless when ∂C 1 ∩ ∂C 2 = ∅ (see indeed [START_REF] Lelièvre | Partial differential equations and stochastic methods in molecular dynamics[END_REF]), contrary to the case ∂C 1 ∩ ∂C 2 = ∅ when α 1 (h) and α 2 (h) do not have the same asymptotic expansion, see [START_REF] Lelièvre | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 2[END_REF]. As expected, when ∂C 1 ∩ ∂C 2 = ∅, the independence between the two wells in this case is hence generically weaker.

Results when ν h concentrates in both wells when h → 0 Let us define here the following assumption:

[H2]:

The assumption [H-Well] is satisfied. Moreover, there exists h 0 > 0 such that for all h ∈ (0, h 0 ), it holds

ε(h) = 0 and lim h→0 α 1 (h) -α 2 (h) ε(h) = 0.
Let us exhibit situations where the assumption [H2] is satisfied.

• When ∂C 1 ∩ ∂C 2 = ∅, the assumption [H2] is satisfied if and only if κ 1,0 = κ 2,0 and κ 1,1 = κ 2,1 . This equivalence follows from [START_REF] Helffer | Semiclassical analysis for the Schrödinger operator and applications[END_REF] and [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach[END_REF]. Therefore, when ∂C 1 ∩ ∂C 2 = ∅, using ( 13) and ( 14), the assumption [H2] is satisfied if and only if

n1 j=1 ∂ n f (z 1,j ) det Hessf | ∂Ω (z 1,j ) = n2 j=1 ∂ n f (z 2,j ) det Hessf | ∂Ω (z 2,j ) , ( 26
)
and det Hessf (x 1 ) = det Hessf (x 2 ). ( 27
)
Moreover, it holds in this case:

α 1 (h) -α 2 (h) ε(h) = O √ h .
• Let us assume that f is an even function as defined by (34) below. Then, from Theorem 3 below, the assumption [H2] is satisfied (see indeed Remark 7).

Remark 3. When ∂C 1 ∩ ∂C 2 = ∅, we are not able to explicit assumptions on f which imply [H2] except in the symmetric situation described in Theorem 3. Note in particular that when ∂C 1 ∩ ∂C 2 = ∅ and [H2] holds, one has when h → 0:

α 1 (h) = α 2 (h) 1 + O e -c h
(which follows from [H2], [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach[END_REF] and the fact that ε(h) = O e -c h , see [START_REF] Helffer | Semiclassical analysis for the Schrödinger operator and applications[END_REF]) and thus:

κ 1,k = κ 2,k for all k ∈ N. (28)
Moreover, it also holds in this case λ

1 (h) = λ 2 (h) 1 + O e -c h ) (see (67)).
Remark 4.

The assumption [H2] is non generic, that is unstable with respect to perturbations of the potential f in the following sense. For any f satisfying [H2], it follows from (26)-( 28) that there exists an arbitrary small perturbation δf : Ω → R such that f + δf satisfies [H1]. Then, according to Theorem 1, the quasi-stationary distribution for the potential f + δf concentrates when h → 0 in precisely one of the wells

C 1 or C 2 .
The following result shows that when [H2] is satisfied, the quasi-stationary distribution ν h concentrates when h → 0 in the two wells C 1 and C 2 .

Theorem 2. Let us assume that the hypotheses [H-Well] and [H2] are satisfied. Let ν h be the quasi-stationary distribution of the process (1) on Ω (see [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points[END_REF]).

Let O 1 ⊂ Ω be an open neighborhood of x 1 and O 2 ⊂ Ω be an open neighborhood of x 2 such that O 1 ∩ O 2 = ∅.
Then, there exists c > 0 such that in the limit h → 0:

ν h (O 1 ) + ν h (O 2 ) = 1 + O e -c h , (29)
where, for k ∈ {1, 2},

ν h (O k ) = b k + O |α 2 (h) -α 1 (h)| |ε(h)| + O h , (30)
where, defining q by {q} = {1, 2} \ {k},

b k = det Hess f (x q ) 1 4
det Hess f (x 1 )

1 4 + det Hess f (x 2 ) 1 4
.

Moreover, for any F ∈ L ∞ (∂Ω, R) and for any family

(Σ i,j ) (i,j)∈ 2 p=1 {p}×{1,...,np}
of disjoint open neighborhoods of (z i,j ) (i,j)∈ 2 p=1 {p}×{1,...,np} in ∂Ω, there exists c > 0 such that in the limit h → 0:

E ν h [F (X τΩ )] = (i,j)∈ 2 p=1 {p}×{1,...,np} E ν h 1 Σi,j F (X τΩ ) + O e -c h . ( 32
)
Lastly, when, for some (i, j)

∈ 2 p=1 {p} × {1, . . . , n p }, F is C ∞ around z i,j
, one has when h → 0:

E ν h 1 Σi,j F (X τΩ ) = F (z i,j ) a i,j b i + O |α 2 (h) -α 1 (h)| |ε(h)| + O h , (33)
where b i is defined in (31) and a i,j is defined in (25).

Remark 5. When [H-Well] and [H2] are satisfied, one also gives sharp asymptotic estimates on the two smallest eigenvalues

0 < λ 1 (h) < λ 2 (h) of L D,(0) f,h
when h → 0, see indeed (67), ( 68) and (69).

When [H-Well] and [H1] hold, Theorem 2 implies that the quasi-stationary distribution ν h concentrates when h → 0 in C 1 and C 2 , and more precisely around any arbitrary small neighborhood of x 1 and x 2 . Note also that when ∂C 1 ∩ ∂C 2 = ∅, the coefficient (31) specifying the repartition of ν h within the wells equals 1 2 according to (27). Moreover, when X 0 ∼ ν h the law of X τΩ concentrates when h → 0 on {z 1,1 , . . . , z 1,n1 }∪{z 2,1 , . . . , z 2,n2 } = (∂C 1 ∪∂C 2 )∩∂Ω with an explicit repartition given by (25). In addition, when |∇f | = 0 on ∂C 1 ∪ ∂C 2 , it follows from [6, Theorem 2] that when X 0 = x ∈ C k , k ∈ {1, 2}, the law of X τΩ concentrates when h → 0 on {z k,1 , . . . , z k,n k } = ∂C k ∩ ∂Ω with the repartition given by (25). This shows that in this case the domain Ω is not metastable for deterministic initial conditions within

C 1 ∪ C 2 .
Connecting again with the literature dealing with semiclassical Schrödinger operators of the form h 2 ∆ (0) H +V on manifolds without boundary, when the assumptions [H-Well] and [H2] are satisfied, a strong tunneling effect appears when h → 0 and mixes the respective properties of both wells. We refer to [11, pp. 45-46] for a symmetric case with two wells and to [START_REF] Helffer | Puits multiples en limite semi-classique. II. Interaction moléculaire. Symétries. Perturbation[END_REF] for more general symmetric situations.

Let us conclude this section by specifying the statement of Theorem 2 in a completely symmetric situation. To this end, we recall that an isometry Φ :

Ω → Ω is a C ∞ diffeomorphism which satisfies, for all x ∈ Ω and all v, w ∈ T x Ω: v • w = DΦ x (v) • DΦ x (w)
, where • is the scalar product associated with the metric of Ω on the tangent bundle T Ω. One says moreover that f : Ω → Ω is even if there exists an isometry Φ such that Φ(x 1 ) = x 2 , Φ 2 = I, and f

• Φ = f, ( 34
)
where I is the identity map on Ω. When f is even, the following improvement of Theorem 2 holds.

Theorem 3. Let us assume that the hypothesis [H-Well] is satisfied. Let ν h be the quasi-stationary distribution of the process (X t ) t≥0 on Ω (see [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points[END_REF]). Assume that f is an even function as defined by (34). Then, the assumption [H2] is satisfied with in particular, for all h small enough:

α 1 (h) = α 2 (h)
, where α 1 (h) and α 2 (h) are defined by [START_REF] Gong | Killed diffusions and their conditioning[END_REF].

Furthermore, let O 1 ⊂ Ω be an open neighborhood of x 1 and O 2 ⊂ Ω be an open neighborhood of x 2 such that O 1 ∩ O 2 = ∅.
Then, for k ∈ {1, 2}, there exists c > 0 such that in the limit h → 0:

ν h (O k ) = 1 2 + O e -c h .
Moreover, one has n 1 = n 2 (see (4)) and the asymptotic estimates (32) and (33).

2 Proof of our main results For p ∈ {0, . . . , d}, one denotes by Λ p C ∞ (Ω) the space of C ∞ p-forms on Ω and by Λ p C ∞ T (Ω) the subset of Λ p C ∞ (Ω) made of the p-forms v such that tv = 0 on ∂Ω, where t denotes the tangential trace on forms. We recall that tv = 0 on ∂Ω means that the restriction to ∂Ω of the p-form v vanishes when applied to tangential vector fields, and we refer e.g. to [23, Equation (2.25)] for a rigorous definition of the tangential trace. For q ∈ N, one denotes by Λ p H q w (Ω) the weighted Sobolev spaces of p-forms with regularity index q, for the weight e -2 h f on Ω (where the subscript w refers to the fact that the weight function appears in the inner product), and we refer again to [START_REF] Schwarz | Hodge decomposition -a method for solving boundary value problems[END_REF] for an introduction to weighted Sobolev spaces on manifolds with boundaries. The set Λ p H 1 w,T (Ω) is then defined by

Λ p H 1 w,T (Ω) := v ∈ Λ p H 1 w (Ω) , tv = 0 on ∂Ω .
We will denote by . H q w the norm on the weighted space Λ p H q w (Ω) and by

•, • L 2 w the scalar product on Λ p L 2 w (Ω). Notice that Λ 0 L 2 w (Ω) is the space L 2 (Ω, e -2 h f (x) dx) and Λ 0 H 2 w (Ω) is the space H 2 (Ω, e -2 h f (x) dx) introduced in the definition of L D,(0) f,h in Section 1.4.
In the following, one denotes respectively by

d : Λ p C ∞ (Ω) → Λ p+1 C ∞ (Ω) and d * : Λ p+1 C ∞ (Ω) → Λ p C ∞ (Ω)
the exterior and the co-differential derivatives on Ω. Let us introduce the differential operator

L (1) f,h = h 2 ∆ (1) H + L ∇f = 1 2h e f h h 2 ∆
(1)

H + |∇f | 2 + h(L ∇f + L * ∇f ) e -f h acting on Λ 1 C ∞ (Ω), where ∆ (1)
H = (d + d * ) 2 is the Hodge Laplacian on Ω, L ∇f is the Lie derivative with respect to the vector field ∇f , and L * ∇f its formal adjoint in L 2 (Ω). The (tangential) Dirichlet realization of L (1)

f,h is denoted by L D,(1) f,h
and its domain is

D L D,(1) f,h = v ∈ Λ 1 H 2 (Ω), tv = 0 and td * e -2 h f v = 0 on ∂Ω .
From [13, Section 2.4], the operator L D,(1) f,h is self-adjoint, positive and has compact resolvent. One has moreover the following result from [13, Theorem 3.2.3].

Lemma 2. Under the assumption [H-Well], there exists h

0 > 0 such that for all h ∈ (0, h 0 ), dim Ran π [0, √ h 2 ) L D,(1) f,h = m Ω 1 ,
where m Ω 1 is defined in (5) and π [0,

√ h 2 ) L D,(1) f,h
is the orthogonal projector on the vector space associated with the eigenvalues of

L D,(1) f,h in [0, √ h
2 ).

In the following, the exterior differential d will be denoted, with a slight abuse of notation, by ∇. For ease of notation, one also denotes, for p ∈ {0, 1},

π (p) h = π [0, √ h 2 ) L D,(p) f,h
.

From [13, Corollary 2.4.4], the following relation holds on Λ 0 H 1 w,T (Ω):

∇π (0) h = π (1) h ∇ . ( 35
)
This implies in particular that

∇ : Ran π (0) h → Ran π (1) h (36)
and then, when [H-Well] holds, according to Lemma 1, that for every h small enough,

∇u h ∈ Ran π (1)
h .

We refer to [7, Section 3.1.2] for more details concerning this section.

Proof of Proposition 1

In the following, we assume that [H-Well] holds.

The finite dimensional vector spaces Ran π (0) h and Ran π

(1) h are endowed with the scalar product ., . L 2 w of L 2 w (Ω) introduced in Section 2.1.1. Moreover, the set {i j } (i,j)∈ 3 p=1 {p}×{1,...,np} is ordered using the lexicographical order, i.e.

{i j } (i,j)∈ 3 p=1 {p}×{1,...,np} = {1 1 , . . . .1 n1 , 2 1 , . . . , 2 n2 , 3 1 . . . , 3 m3 , 3 m3+1 , . . . , 3 n3 },

where we recall

n 1 = Card (∂C 1 ∩∂Ω), n 2 = Card ∂C 2 ∩∂Ω , m 3 = Card ∂C 1 ∩ ∂C 2 and n 3 = Card U Ω 1 \ (∪ 2 k=1 ∂C k ∩ ∂Ω) = m Ω 1 -n 1 -n 2 are defined in Sec- tion 1.3.

Let us now define

u 1 := χ 1 χ 1 L 2 w ∈ Λ 0 H 1 w,T (Ω) and u 2 := χ 2 χ 2 L 2 w ∈ Λ 0 H 1 w,T (Ω) (38)
where, for i ∈ {1, 2}, 0 ≡ χ i ∈ C ∞ (Ω, R + ) is compactly supported in Ω, χ 1 and χ 2 have disjoint supports, and for some small α > 0 and β > 0,

supp χ i ⊂ C i + B(0, α) ∩ Ω and χ i = 1 on C i ∩ {f < min ∂Ω f -β}.
Let us also consider a family of L 2 w -unitary 1-forms

( ψ ij ) (i,j)∈ 3 p=1 {p}×{1,...,np} (39)
such that, for (i, j)

∈ 3 p=1 {p} × {1, . . . , n p }, ψ ij ∈ Λ 1 H 1 w,T (Ω) ∩ Λ 1 C
∞ Ω , and for some small δ > 0, supp ψ ij ⊂ B(z i,j , δ) ∩ Ω.

It then holds, for every (k, q) ∈ {1, 2}, (i, j) ∈ 3 p=1 {p} × {1, . . . , n p }, and (i , j ) ∈ 3 p=1 {p} × {1, . . . , n p } (for δ > 0 small enough):

u k , u q L 2 w = δ k,q and ψ ij , ψ i j L 2 w = δ i,i δ j,j . (40)
Taking, for every (i, j) ∈ 3 p=1 {p} × {1, . . . , n p }, ψ ij as a (normalized) truncated principal eigen-1-form of a local Witten Laplacian defined around z i,j with Dirichlet boundary conditions2 , we obtain the following proposition (see [7, Section 3.2.2 and Definition 42] and references therein for details). It gathers the statements of [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 1[END_REF]Propositions 43 and 47] which are the starting points of our analysis.

Proposition 2. Let us assume that the function f satisfies [H-Well]. Then, the families ( u 1 , u 2 ) and ( ψ ij ) (i,j)∈ 3 p=1 {p}×{1,...,np} defined in (38), (39) can be chosen so that the following estimates hold when h → 0 (where H is defined in (3)):

1. There exists c > 0 such that: a) for every k ∈ {1, 2}, it holds

(1 -π (0) h ) u k 2 L 2 w ≤ h 1 2 ∇ u k 2 L 2 w ≤ e -2 h (H-c 2 ) , b) for every i ∈ {1, 2, 3} and j ∈ {1, . . . , n i }, it holds (1 -π (1) h ) ψ ij 2 H 1 w = O e -2c h .
2. For every k ∈ {1, 2} and (i, j) ∈ 3 p=1 {p} × {1, . . . , n p }, there exists a real constant ε i,j,k ∈ {-1, 1} independent of h such that it holds

∇ u k , ψ ij L 2 w =      -C i,j,k h -3 4 e -H h 1 + O(h) when z i,j ∈ ∂C k ∩ ∂Ω ε i,j,k C i,j,k h -1 2 e -H h 1 + O(h) when z i,j ∈ ∂C 1 ∩ ∂C 2 0 else,
where the remainder terms O(h) admit a full asymptotic expansion in h, and

C i,j,k =            π -1 4 2 ∂ n f (z i,j ) det Hess f (x k ) 1 4 det Hess f | ∂Ω (z i,j ) 1 4 if z i,j ∈ ∂C k ∩ ∂Ω π -1 2 |λ -(z i,j )| det Hess f (x k ) 1 4
det Hess f (z i,j )

1 4 if z i,j ∈ ∂C 1 ∩ ∂C 2 ,
(41) where λ -(z i,j ) denotes the negative eigenvalue of Hess f (z i,j ). Remark 6. In the second item in Proposition 2, notice that it follows from the notation introduced in Section 1.3 that for every k ∈ {1, 2} and (i, j) ∈ 3 p=1 {p} × {1, . . . , n p }, one has:

• z i,j ∈ ∂C k ∩ ∂Ω if and only if i = k (and thus j ∈ {1, . . . , n k }),

• z i,j ∈ ∂C 1 ∩ ∂C 2 if and only if i = 3 (and thus j ∈ {1, . . . , m 3 }).

As a consequence of (40) and the first item in Proposition 2, there exists c > 0 such that it holds in the limit h → 0:

G 0 := π (0) h u k , π (0) h u q L 2 w k,q∈{1,2} = I 2 + O e -c h (42)
and

G 1 := π (1) h ψ ij , π (1)
h ψ i j L 2 w (i,j)∈ 3 p=1 {p}×{1,...,np} (i ,j )∈ 3 p=1 {p}×{1,...,np} = I m Ω 1 + O(e -c h ). ( 43
)
It then follows from Lemmata 1 and 2 that, for every h > 0 small enough, the family π

(0) h u k k∈{1,2} is a basis of Ran π (0)
h and that π

(1)

h ψ ij (i,j)∈ 3 p=1 {p}×{1,...,np}
is a basis of Ran π

h . Let us now define the m Ω 1 × 2 matrix

S := ∇π (0) h u k , π (1)
h ψ ij L 2 w (i,j)∈ 3 p=1 {p}×{1,...,np}, k∈{1,2} . (44)
According to the two items in Propositions 2, and using the identity

∇π (0) h u k , π (1)
h ψ ij L 2 w = ∇ u k , ψ ij L 2 w -∇ u k , 1 -π (1) h ψ ij L 2 w
which follows from (35), there exists c > 0 such that the coefficients of S satisfy when h → 0:

S ij ,k =      ∇ u k , ψ ij L 2 w 1 + O e -c h if z i,j ∈ ∂C k ∩ ∂Ω ∇ u k , ψ ij L 2 w 1 + O e -c h if z i,j ∈ ∂C 1 ∩ ∂C 2 O e -1 h (H+c)
else.

(

) 45
Let us denote by Υ and Ψ the following families written as row vectors,

Υ := π (0) h u 1 , π (0)
h u 2 and Ψ := π

h ψ ij (i,j)∈ 3 p=1 {p}×{1,...,np} , and define

B 0 = (ϕ 1 , ϕ 2 ) := Υ G -1 2 0 and B 1 = ψ ij (i,j)∈ 3 p=1 {p}×{1,...,np} := Ψ G -1 2 1 , ( 46
)
where G 0 and G 1 are defined in ( 42) and (43). For every h > 0 small enough, the families B 0 and B 1 are then respectively orthonormal bases of Ran π (0) h and of Ran π

L = G -1 2 0 L f,h π (0) h u k , π (0)
h u q L 2 w 1≤k,q≤2 G -1 2 0 . ( 47
)
This matrix is sometimes called the interaction matrix in the literature dealing with the study of semiclassical Schrödinger operators (see e.g. [START_REF] Helffer | Multiple wells in the semi-classical limit[END_REF] or [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF]).

Moreover, the matrix M of ∇ : Ran π

(0) h → Ran π (1)
h (see ( 36)) in the bases B 0 and B 1 is given by

M = G -1 2 1 S G -1 2 0 , (48)
where S is defined in (44). Since L

D,(0) f,h Ran π (0) h = h 2 ∇ * ∇, the matrix M satisfies L = h 2 M * M. (49)
In order to prove Proposition 1, it is then sufficient to get asymptotic estimates on the coefficients of the matrix M . This is the purpose of the next proposition.

Proposition 3. Let us assume that the hypothesis [H-Well] is satisfied. Let ( u k ) k∈{1,2} be defined by (38). Let (ϕ k ) k∈{1,2} and ψ ij (i,j)∈ 3 p=1 {p}×{1,...,np} be defined by (46). Then, for all k ∈ {1, 2}, there exists c > 0 such that when h → 0:

i) for every j ∈ {1, . . . , n k }, ∇ϕ k , ψ kj L 2 w = ∇ u k , ψ kj L 2 w 1 + O(e -c h ) = -C k,j,k h -3 4 e -H h 1 + O(h) , ii) for every j ∈ {1, . . . , n p } with p ∈ {1, 2} \ {k}, ∇ϕ k , ψ pj L 2 w = O e -1 h (H+c) , iii) for every j ∈ {1, . . . , m 3 }, ∇ϕ k , ψ 3j L 2 w = ∇ u k , ψ 3j L 2 w 1 + O(e -c h ) = ε 3,j,k C 3,j,k h -1 2 e -H h 1 + O(h) ,
iv) and for all j ∈ {m 3 + 1, . . . ,

n 3 }, ∇ϕ k , ψ 3j L 2 w = O e -1 h (H+c) ,
where we recall that H = min ∂Ω f -min Ω f (see (3)), the coefficients C i,j,k are defined in (41), and the terms O(h) admit a full asymptotic expansion in h.

Proof. The results of Proposition 3 follow from (42)-( 45), (48), and item 2 in Proposition 2 (see also Remark 6).

Proposition 1 is a consequence of Proposition 3 and of (49). They indeed imply the existence of some c > 0 such that when h → 0, the coefficients ε(h), α 1 (h), and α 2 (h) defined by [START_REF] Gong | Killed diffusions and their conditioning[END_REF] satisfy

ε(h) =    O e -c h if ∂C 1 ∩ ∂C 2 = ∅ m3 j=1 ε 3,j,1 ε 3,j,2 C 3,j,1 C 3,j,2 √ h 1 + O(h) if ∂C 1 ∩ ∂C 2 = ∅, (50)
and, for k ∈ {1, 2},

α k (h) =        n k j=1 C 2 k,j,k 1 + O(h) if ∂C 1 ∩ ∂C 2 = ∅ n k j=1 C 2 k,j,k 1 + O(h) + m3 j=1 C 2 3,j,k √ h 1 + O(h) if ∂C 1 ∩ ∂C 2 = ∅,
where the C i,j,k 's are defined in (41) and the remainder terms O(h) admit a full asymptotic expansion in h. The relations ( 11)-( 14) follow.

Let us conclude this section by noticing the following consequences of Proposition 1 which will needed in upcoming computations.

1. From [START_REF] Gong | Killed diffusions and their conditioning[END_REF], it holds for i ∈ {1, 2} and every h small enough:

λ i (h) = α 1 (h) + α 2 (h) + (-1) i α 2 (h) -α 1 (h) 2 + 4ε(h) 2 4 √ h e -2 H h , (51)
where 0 < λ 1 (h) < λ 2 (h) denote the two smallest eigenvalues of L D,(0) f,h . It then follows from (51), (50), and ( 12) that 4 √ h λ 1 (h) e 2 H h and 4

√ h λ 2 (h) e 2 H h admit a full asymptotic expansion in h when ∂C 1 ∩ ∂C 2 = ∅ and in √ h when ∂C 1 ∩ ∂C 2 = ∅.
2. From [START_REF] Gong | Killed diffusions and their conditioning[END_REF], since u h is the principal eigenfunction of L D,(0) f,h satisfying ( 7), one has for any h > 0 small enough:

-either ε(h) = 0, in which case one has necessarily

α 1 (h) = α 2 (h) (since 0 < λ 1 (h) < λ 2 (h)) and then u h = ±ϕ i ,
where the functions ϕ 1 and ϕ 2 are defined by ( 46) and i ∈ {1, 2} is such that

α i (h) = min α 1 (h), α 2 (h) ,
-or ε(h) = 0, in which case (51) and an elementary computation lead to

u h = ± 1 1 + β(h) 2 ϕ 1 + β(h) 1 + β(h) 2 ϕ 2 , (52)
where β(h) is defined by

β(h) = - 2 ε(h) α 2 (h) -α 1 (h) + α 2 (h) -α 1 (h) 2 + 4ε(h) 2 . ( 53
)
We conclude this section by stating the following proposition which will also be needed to study the asymptotic behaviour when h → 0 of the law of X τΩ when X 0 ∼ ν h . It is the statement of [7, Proposition 65] in our specific setting.

Proposition 4. Let us assume that the hypothesis [H-Well] is satisfied. Let ψ ij (i,j)∈ 3 p=1 {p}×{1,...,np} be defined by (46). Let Σ be an open subset of ∂Ω and F ∈ L ∞ (∂Ω, R). One then has for every (i, j)

∈ 3 p=1 {p} × {1, . . . , n p }, when h → 0, Σ F ψ ij • n e -2 h f = O h d-3 4 e -1 h min ∂Ω f if i ∈ {1, 2} and z i,j ∈ Σ O e -1 h (min ∂Ω f +c) if i = 3 or z i,j / ∈ Σ,
where the constant c > 0 is independent of h. Moreover, when (i, j) ∈

2 p=1 {p}× {1, . . . , n p }, z i,j ∈ Σ, and F is C ∞ around z i,j , it holds Σ F ψ ij •n e -2 h f = π d-1 4 2 ∂ n f (z i,j ) det Hess f | ∂Ω (z i,j ) 1 4 h d-3 4 e -1 h min ∂Ω f F (z i,j )+O(h) ,
where the above remainder term O(h) admits a full asymptotic expansion in h.

Proof of Theorem 1

In this Section, one proves Theorem 1. To this end, let us assume that the hypotheses [H-Well] and [H1], with [START_REF] Helffer | Puits multiples en limite semi-classique. II. Interaction moléculaire. Symétries. Perturbation[END_REF], are satisfied. Then, from (51), [START_REF] Helffer | Semiclassical analysis for the Schrödinger operator and applications[END_REF], and ( 12), one has in the limit h → 0:

• when ∂C 1 ∩ ∂C 2 = ∅, it holds ε(h) = O e -c h
) for some c > 0 and then, for

every i ∈ {1, 2}, 2 √ h e 2 h H λ i (h) = α i (h) + O e -c h ) ∼ +∞ k=0 κ i,k h k , (54)
• when ∂C 1 ∩ ∂C 2 = ∅, it holds ε(h) √ h and then, for every i ∈ {1, 2},

2 √ h e 2 h H λ i (h) = α i (h) + O h) = κ i,0 1 + O( √ h) , (55)
where, the remainder term O √ h in (55) admits a full asymptotic expansion in √ h.

Moreover, there exists h 0 > 0 such that for all h ∈ (0, h 0 ),

u h = ± 1 1 + β(h) 2 ϕ 1 + β(h) 1 + β(h) 2 ϕ 2 , (56)
where β(h) is defined in (53) and (ϕ 1 , ϕ 2 ) is defined in (46) (notice that (56) holds in H 1 w (Ω)). Indeed, this is simply the relation (52) when ε(h) = 0. In addition, when ε(h) = 0 and α 2 (h) > α 1 (h) (the latter relation follows from [START_REF] Helffer | Puits multiples en limite semi-classique. II. Interaction moléculaire. Symétries. Perturbation[END_REF]), it holds u h = ±ϕ 1 , that is precisely the relation (56) since in this case β(h) is well defined and β(h) = 0 (see indeed (53)).

Since [H1] implies that lim h→0 ε(h) α1(h)-α2(h) = 0, one moreover obtains from (53) that in the limit h → 0:

β(h) = O |ε(h)| α 2 (h) -α 1 (h) . (57)
From ( 56), (57) together with u h > 0 on Ω, u 1 ≥ 0 on Ω, (42), and (46), one has, for every h small enough: u h , u 1 L 2 w = 1 + o(1) and then

u h = 1 + O β(h) 2 ϕ 1 + O |β(h)| ϕ 2 . (58)
Therefore, using (42), (46), and (57), there exists c > 0 such that for every h small enough:

u h = 1 + O β(h) 2 u 1 + O |β(h)| u 2 + O e -c h in L 2 w (Ω). ( 59
)
From (59), one deduces the following proposition which implies, using in addition ( 6) and (57), the asymptotic estimates ( 20) and ( 21) in Theorem 1.

i) When O ∩ {x 1 , x 2 } = {x 1 }, one has O u h e -2 h f = 1 + O β(h) 2 + O e -c h O u 1 e -2 h f = (hπ) d 4
det Hess f (x 1 )

1 4 e -1 h min Ω f 1 + O β(h) 2 + O(h) ,
where c > 0 is independent of h and β(h) satisfies (57).

ii

) When O ∩ {x 1 , x 2 } = {x 2 }, it holds O u h e -2 h f = h d 4 e -1 h min Ω f O |β(h)| + e -c h ,
where we recall β(h) satisfies (57) and c > 0 is independent of h.

iii

) When O ∩ {x 1 , x 2 } = ∅, it holds O u h e -2 h f = O e -1 h (min Ω f +c)
, where c > 0 is independent of h.

Proof. The relation (59) leads to

O u h e -2 h f = 1 + O β(h) 2 O u 1 e -2 h f + O (|β(h)|) O u 2 e -2 h f + O e -1 h (min Ω f +c) ,
where c > 0 is independent of h. In addition, one has, for i ∈ {1, 2}, ũi = χi χi L 2 w from (38) and it follows from the Laplace method that there exists c > 0 such that for any k ∈ {1, 2}, when h → 0,

O χ k i e -2 h f =        (h π) d 2 det Hess f (x i ) 1 2 e -2 h min Ω f 1 + O(h) if x i ∈ O O e -2 h (min Ω f +c) if x i / ∈ O. (60)
The statement of Proposition 5 follows easily.

We also deduce from (58) and Proposition 3 together with (57) the following estimates.

Proposition 6. Let us assume that the hypotheses [H-Well] and [H1] together with (16) are satisfied. Let u h be the principal eigenfunction of L D,(0) f,h satisfying [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 1[END_REF]. Let also ( u k ) k∈{1,2} and ψ ij (i,j)∈ 3 p=1 {p}×{1,...,np} be as in Proposition 2, and ψ ij (i,j)∈ 3 p=1 {p}×{1,...,np} be defined by (46). Then, there exists c > 0 such that in the limit h → 0:

i) For every j ∈ {1, . . . , n 1 }, ∇u h , ψ 1j L 2 w = -C 1,j,1 h -3 4 e -H h 1 + O β(h) 2 + O(h) ,
where C 1,j,1 is defined in (41) and β(h) satisfies (57).

ii) For every j ∈ {1, . . . , m 3 },

∇u h , ψ 3j L 2 w = O h -1 2 e -H h ,
iii) When i = 2 and j ∈ {1, . . . , n 2 } or, i = 3 and j ∈ {m 3 + 1, . . . , n 3 },

∇u h , ψ ij L 2 w = h -3 4 e -H h O |β(h)| + e -c h .
We are now in position to prove Theorem 1.

End of the proof Theorem 1. To conclude the proof Theorem 1, it remains to prove ( 22), ( 23) and [START_REF] Sorensen | Temperature-accelerated dynamics for simulation of infrequent events[END_REF]. Let assume that [H-Well] and [H1] hold with [START_REF] Helffer | Puits multiples en limite semi-classique. II. Interaction moléculaire. Symétries. Perturbation[END_REF] and let us consider F ∈ L ∞ (∂Ω, R). Let us recall that from [START_REF] Helffer | Multiple wells in the semi-classical limit III -Interaction through non-resonant wells[END_REF], one has

E ν h [F (X τΩ )] = - h 2λ 1 (h) ∂Ω F ∂ n u h e -2 h f Ω u h e -2 h f . Sharp asymptotic estimates when h → 0 of λ 1 (h) and Ω u h e -2
h f are respectively given in (54), (55) and in Proposition 5. Therefore, to prove ( 22), ( 23) and ( 24), it only remains to estimate when h → 0, for an open subset Σ of ∂Ω, the term

Σ F ∂ n u h e -2 h f .
Since the family (ψ ij ) (i,j)∈ 3 p=1 {p}×{1,...,np} introduced in (46) is an orthonormal basis of Ran π (1) h , it holds when h → 0, from the Parseval identity and from Propositions 4 and 6,

Σ F ∂ n u h e -2 h f = (i,j)∈ 3 p=1 {p}×{1,...,np} ∇u h , ψ ij L 2 w Σ F ψ ij • n e -2 h f (61) = (i,j)∈ 2 p=1 {p}×{1,...,np} ∇u h , ψ ij L 2 w Σ F ψ ij • n e -2 h f + O e -1 h (min ∂Ω f +H+c) , (62)
for some c > 0 independent of h. When Σ does not contain any of the z i,j 's for (i, j) ∈ 2 p=1 {p} × {1, . . . , n p }, one has, using again Propositions 4 and 6:

Σ F ∂ n u h e -2 h f = O e -1 h (min ∂Ω f +H+c) , (63)
where c > 0 is independent of h. Assume now that Σ does not contain any of the z 1,j 's for j ∈ {1, . . . , n 1 }. One then has in the limit h → 0, using Propositions 4 and 6 and defining

H := min ∂Ω f + H: Σ F ∂ n u h e -2 h f = n1 j=1 O h -3 4 e -H h O e -1 h (min ∂Ω f +c) + O e -1 h (H +c) + n2 j=1 O h -3 4 |β(h)| e -H h O h d-3 4 e -1 h min ∂Ω f = O e -1 h (H +c) + O h d-6 4 |β(h)|e -H h , (64)
where c > 0 is independent of h. Finally, let us assume that Σ∩{z 1,1 , . . . , z 1,n1 } = {z 1,j } and F is C ∞ around z 1,j . One then has, using (41), Propositions 4 and 6:

Σ F ∂ n u h e -2 h f = ∇u h , ψ 1j L 2 w Σ F ψ 1j • n e -2 h f + h d-6 4 e -H h O |β(h)| + e -c h = - 2 π d-2 4 ∂ n f (z 1,j ) det Hess f (x 1 ) 1 4 det Hess f | ∂Ω (z 1,j ) 1 2 h d-6 4 e -H h × F (z 1,j ) + O |β(h)| + h) . (65)
The estimates ( 22), ( 23) and ( 24), follows from ( 15) and ( 63)-(65), using in addition (57), ( 54), (55), and Proposition 5. This concludes the proof of Theorem 1.

Proofs of Theorems 2 and 3

Let us assume in this section that the hypotheses [H-Well] and [H2] are satisfied. We recall that [H2] means that there exists h 0 > 0 such that for all h ∈ (0, h 0 ),

ε(h) = 0 and lim h→0 α 1 (h) -α 2 (h) ε(h) = 0. (66)
We then deduce from (51) the following:

• when ∂C 1 ∩ ∂C 2 = ∅, using in addition [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach[END_REF] and the fact that ε(h) = O e -c h ) for some c > 0 (see [START_REF] Helffer | Semiclassical analysis for the Schrödinger operator and applications[END_REF]), it holds when h → 0:

λ 1 (h) = λ 2 (h) 1 + O e -c h ) , (67) and 2
√ h e 2 h H λ 1 (h) ∼ +∞ k=0 κ 1,k h k , (68)
• when ∂C 1 ∩ ∂C 2 = ∅, using in addition [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach[END_REF], the fact that ε(h) √ h (see [START_REF] Helffer | Semiclassical analysis for the Schrödinger operator and applications[END_REF]) and κ 1,0 = κ 2,0 (see (26), ( 27) and ( 13)) it holds for i ∈ {1, 2} when h → 0:

λ i (h) = κ 1,0 e -2 h H 2 √ h 1 + O( √ h) , (69)
where, the remainder term O √ h in (69) admits a full asymptotic expansion in √ h.

Remark 7. When there exists an isometry Φ : Ω → Ω satisfying (34), i.e. such that Φ(x 1 ) = x 2 , f • Φ = f , and Φ 2 = I, it necessarily holds n 1 = n 2 and Φ({z 1,1 , . . . , z 1,n1 }) = {z 2,1 , . . . , z 2,n2 }. For every h > 0 small enough, it follows moreover from the simplicity of the eigenvalues λ 1 (h) and λ 2 (h) (see Remark 1) and from the positivity of u h in Ω that u h • Φ = u h and u 2,h • Φ = -u 2,h , where u 2,h denotes any eigenvector of L D,(0) f,h associated with λ 2 (h). In addition, one can choose χ 1 and χ 2 such that χ 2 = χ 1 • Φ in (38). This leads, for h small enough, to π (0)

h u 1 + π (0) h u 2 ∈ Span(u h ), π (0) h u 1 -π (0) h u 2 ∈ Span(u 2,h ) and hence to π (0) h u 1 , π (0)
h u 1 L 2 w = π (0) h u 2 , π (0)
h u 2 L 2 w and L D,(0) f,h π (0) h u 1 , π (0)
h u 1 L 2 w = L D,(0) f,h π (0) h u 2 , π (0)
h u 2 L 2 w . It then follows from (42), (47), and (10) that for h small enough, α 1 (h) = α 2 (h) and hence, using λ 1 (h) = λ 2 (h), that ε(h) = 0. The relation (66) is thus in particular satisfied in this situation.

Moreover, there exists h > 0 such that for all h ∈ (0, h 0 ),

u h = ± 1 1 + β(h) 2 ϕ 1 + β(h) 1 + β(h) 2 ϕ 2 , (70)
where β(h) is defined in (53) and (ϕ 1 , ϕ 2 ) is defined in (46). This is indeed simply (52) since ε(h) = 0 according to (66). Using (66) and (53), one obtains moreover that when h → 0:

β(h) = - |ε (
Moreover, using (42), (46), and (72), the equality (73) implies that there exists c > 0 such that for every h small enough,

u h = 1 √ 2 1 + O (|µ(h)|) + O e -c h u 1 + 1 √ 2 1 + O (|µ(h)|) + O e -c h u 2 + O e -c h in L 2 w (Ω). ( 74
)
From (73), one deduces the following proposition which implies, using in addition ( 6) and (72), the asymptotic estimates (29) and (30) in Theorem 2. , where c > 0 is independent of h.

Proof. The proof of Proposition 7 is similar to that one of Proposition 5 using (74) instead of (59). satisfying [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 1[END_REF]. Let moreover ( u j ) j∈{1,2} and ( ψ ij ) (i,j)∈ 2 p=1 {p}×{1,...,np} be as in Proposition 2, and (ψ ij ) (i,j)∈ 2 p=1 {p}×{1,...,np} be defined by (46). Then, there exists c > 0 such that in the limit h → 0: i) For every k ∈ {1, 2} and j ∈ {1, . . . , n k },

∇u h , ψ kj L 2 w = - C k,j,k √ 2 h -3 4 e -H h 1 + O(|µ(h)|) + O(h) ,
where C k,j,k is defined in (41) and µ(h) satisfies (72).

ii) For every j ∈ {1, . . . , m 3 },

∇u h , ψ 3j L 2 w = O h -1 2 e -H h .
iii) For every j ∈ {m 3 + 1, . . . , n 3 }, ∇u h , ψ 3j L 2 w

= O e -1 h (H+c) .

End of the proofs Theorems 2 and 3. Let us assume that the hypotheses [H-Well] and [H2] are satisfied. It remains to prove the asymptotic estimates (32) and (33). We proceed as we did at the end of Section 2.2 to prove ( 22), [START_REF] Schwarz | Hodge decomposition -a method for solving boundary value problems[END_REF], and [START_REF] Sorensen | Temperature-accelerated dynamics for simulation of infrequent events[END_REF]. Let us then consider F ∈ L ∞ (∂Ω, R).

Let us first assume that Σ does not contain any of the z i,j 's for (i, j) ∈ 2 p=1 {p}× {1, . . . , n p }. Then, using (61) together with Propositions 4 and 8, one has in the limit h → 0:

Σ F ∂ n u h e -2 h f = O e -1 h (min ∂Ω f +H+c) , (75)
for some c > 0 independent of h.

Let us now assume that Σ ∩ z i,j , (i, j) ∈

where c > 0 is independent of h and µ(h) satisfies (72). The asymptotic estimates (32) and (33) are then straightforward consequences of ( 15) and ( 75), (76), using in addition (68), (69), and Proposition 7. This concludes the proof of Theorems 2 and 3.

[

  H-Well]: The function f belongs to C ∞ (Ω, R), |∇f | = 0 on ∂Ω, and f : Ω → R and f | ∂Ω are Morse functions. Moreover, the function f has only two local minima x 1 and x 2 in Ω which satisfy arg min

Figure 3 :

 3 Figure 3: Schematic representation of the connected components C1 and C2 of {f <min ∂Ω f } when the assumption [H-Well] is satisfied. In this representation, ∂C1 ∩ ∂C2 = {z3,1}, U ∂Ω 1 = {z1,1, z2,1, z3,2}, ∂C1 ∩ ∂Ω = {z1,1}, ∂C2 ∩ ∂Ω = {z2,1}, U Ω 1 = {z3,1} and arg min Ω f = {x1, x2}. Thus, n1 = 1, n2 = 1, m3 = 1 and n3 = 2.

  B 0 is given by

Proposition 5 .

 5 Let us assume that the hypotheses [H-Well] and [H1] together with (16) are satisfied. Let u h be the principal eigenfunction of L D,(0) f,h satisfying (7) and ( u 1 , u 2 ) be the functions introduced in (38). Then, for every open set O ⊂ Ω and for every h > 0 small enough:

2 w = 1 √ 2 + o( 1 )=-√ 2 1 +√ 2 1 +

 212111 ), (71) together with u h > 0 on Ω, u 1 , u 2 ≥ 0 on Ω, (42), and (46), one has, for every h small enough, u h , u 1 L and 0< u h , u 2 L 2 w |ε(h)| √ 2 ε(h) + o[START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF]. It follows that for every h small enough: ε(h) < 0,β(h) = 1 + µ(h) , where µ(h) = O |α 2 (h) -α 1 (h)| |ε(h)| → 0 when h → 0,(72)andu h = 1 O (|µ(h)|) ϕ 1 + 1 O (|µ(h)|) ϕ 2 .

Proposition 7 .u h e -2 h f = 1 √ 2 1 + 4 det 4 e - 1 h

 711441 Let us assume that the hypotheses [H-Well] and [H2] are satisfied. Let u h be the principal eigenfunction of Li) When, for some i ∈ {1, 2}, O ∩ {x 1 , x 2 } = {x i }, it holds O O(|µ(h)|) + O e -Hess f (x i ) 1 min Ω f 1 + O(|µ(h)|) + O(h) ,where c > 0 is independent of h and µ(h) satisfies (72).ii) When O ∩ {x 1 , x 2 } = ∅, one has O u h e -2 h f = O e -1 h (min Ω f +c)

Remark 8 .O1∪O2u h e -2 h f = √ 2 +Proposition 8 .

 828 Let us assume as in Remark 7 that there exists an isometry Φ : Ω → Ω satisfying (34) and denote by O 1 ⊂ Ω and O 2 ⊂ Ω two disjoint open sets such that x i ∈ O i for i ∈ {1, 2}. Using Proposition 7 and the fact that, for every h small enough,u 1 • Φ = u 2 , α 1 (h) = α 2 (h) and hence µ(h) = 0, it holds for i ∈ {1, 2}: Ω u h e -2 h f = 1 + O(e -c h ) O(e -c h ) Oi u i e -2 h f .This implies the first part of Theorem 3.From (73) and Proposition 3, one deduces the following estimates. Let us assume that the hypotheses [H-Well] and [H2] are satisfied. Let u h be the principal eigenfunction of L D,(0) f,h

2 p=1 6 4 e - 1 h 4 det×F

 2614 {p} × {1, . . . , n p } = {z p, } and that F is C ∞ around z p, . Then, using again (61) together with Propositions 4 and 8, one has when h → 0, definingH := min ∂Ω f + H, Σ F ∂ n u h e -2 h f = ∇u h , ψ p L 2 w Σ F ψ p • n e -2 h f + O h d-(H +c) = -√ 2 ∂ n f (z p, ) det Hess f (x p ) 1 Hess f | ∂Ω (z p, ) (z p, ) + O(|µ(h)|) + O(h) ,

They are actually bounded from below by some positive constant.

Actually, when z i,j ∈ ∂Ω and V denotes its corresponding neighborhood in Ω, (full) Dirichlet boundary conditions are considered on ∂V ∩ Ω while only tangential Dirichlet boundary conditions are considered on ∂V ∩ ∂Ω.

D,(0) f,h satisfying (7) and let ( u j ) j∈{1,2} be the functions introduced in (38). Then, for any open subset O of Ω and for h > 0 small enough:


Student Member, IEEE Bertrand Massot

Member, IEEE Claudine Gehin

Student Member, IEEE Ronald Nocua

Member, IEEE Andre Dittmar

Senior Member, IEEE Eric Mcadams

A wearable, low-power, health-monitoring instrumentation based on a Programmable System-on-Chip TM

Keywords: Ambulatory measurement, wireless sensors and systems, health monitoring, Programmable System-on-Chip (PSoC), autonomic nervous system (ANS)

Improvement in quality and efficiency of health and medicine, at home and in hospital, has become of paramount importance. The solution of this problem would require the continuous monitoring of several key patient parameters, including the assessment of autonomic nervous system (ANS) activity using non-invasive sensors, providing information for emotional, sensorial, cognitive and physiological analysis of the patient. Recent advances in embedded systems, microelectronics, sensors and wireless networking enable the design of wearable systems capable of such advanced health monitoring. The subject of this article is an ambulatory system comprising a small wrist device connected to several sensors for the detection of the autonomic nervous system activity. It affords monitoring of skin resistance, skin temperature and heart activity. It is also capable of recording the data on a removable media or sending it to computer via a wireless communication. The wrist device is based on a Programmable System-on-Chip (PSoC TM ) from Cypress: PSoCs are mixed-signal arrays, with dynamic, configurable digital and analogical blocks and an 8-bit Microcontroller unit (MCU) core on a single chip. In this paper we present first of all the hardware and software architecture of the device, and then results obtained from initial experiments.

I. INTRODUCTION

H EALTH monitoring already takes place in various settings: at home for prevention and in hospital for continuous assessment. It has become a pressing need for patients, to provide better quality of care, and also for society, to lead to more effective and lower cost health care provision. Unfortunately, ambulatory monitoring must overcome many constraints. These technical challenges include high integration for device size reduction, and low power consumption for extended patient autonomy. Each component has to be chosen according to these criteria, and PSoC is a preferential tool for reducing the number of components. Indeed, it can carry out all the steps from signal conditioning to transmission of formatted data, including analog to digital conversion and data processing, all within a single part component.

Manuscript created February 25, 2009; B. Massot, C. Gehin, A. Dittmar and E. McAdams are with the Biomedical Sensors Group, Lyon Institute of Nanotechnology, UMR 5270 CNRS-INSA Lyon, Villeurbanne, France. bertrand.masssot@insalyon.fr, claudine.gehin@insa-lyon.fr, andre.dittmar@insa-lyon.fr, eric.mcadams@insa-lyon.fr R. Nocua is with the AFIRM Group, TIMC-IMAG University Joseph Fourier, Grenoble, France. ronald.nocua@imag.fr Wireless communication is achieved by an XBee R ZB RF module. This module meets the specifications necessary for wearable applications: with low power consumption, resulting from ZigBee Pro Feature Set and IEEE 802.15.4 compliance, it facilitates the implementation of short range wireless communications. Embedded storage completes the ambulatory feature with the ability to record data on a micro-SD card, currently the smallest removable media with an acceptable storage capacity.

II. MATERIALS AND METHODS

A. System Architecture

System is mainly based on the PSoC. It is composed of an 8-bit Harvard architecture microprocessor (with a programmable frequency running in the present case at 12 MHz), a programmable analog array and a programmable digital array (Fig. 1). The analog blocks, based on configurable operational amplifier structures, perform sensor-interfacing and analog signal processing, especially for the ECG signal which requires a dedicated processing circuit. In addition, the analog array also includes a portion of the ADC, corresponding to the signal buffering. The firmware, working on the MCU core, retrieves digitized values through the API functions of the ADC. It carries out additional treatment (averaging for resistance and temperature, and band-pass filtering for the ECG), before transmitting data to communication interfaces in the form of formatted data frames. The digital blocks, based on configurable digital structures (timers, flip-flops, latches, etc.), are used to create timers (such as a millisecond clock) and communication interfaces: a SPI bus, to communicate with a micro Secure Digital flash card (micro SD), and UART to communicate with the XBee module. Due to the limited number of analog blocks, PSoC limitations were reached with the impossibility of creating three parallel and independent acquisition chains (amplification, filtering, and analog to digital conversion). The solution was to multiplex analog signals in the signal processing chain. This is enabled by the possibilities of dynamic settings of analog functions (amplifier gains, entries routing, voltage references, etc.). There is therefore only one acquisition chain for the measurement of three physiological parameters, each analog block's parameters being reset in real time following the measurement progress. Demux is done after A/D conversion, so each signal can receive its own digital processing.

B. Hardware Implementation

The ECG signal (Fig. 2), from sensors in a 'Lead II' limb lead configuration (Fig. 3), passes through an RFI filter to the input of a high common-mode-rejection ratio (CMRR) instrumentation amplifier in order to eliminate noise. To improve noise immunity in an ambulatory context, a "Driven-Right-Leg" circuit [START_REF] Winter | Driven-right-leg circuit design[END_REF] was implemented. From the output of the INSAMP, the signal passes through a high pass filter external to the PSoC (High Pass Filter). The reason for the use of discrete components for that filter is a result of the insufficient performance of PSoC filters when working at very low frequency (templates not respected, signal distortion from switch-capacitor blocks working at low frequency) [START_REF] Van Ess | Understanding switched capacitor filters[END_REF]. Finally, the signal is once more buffered and amplified by a programmable gain amplifier (PGA) before being digitized by a 12-bit analog-to-digital converter Electrodermal responses (EDR) are obtained by measuring skin resistance (Fig. 4). We measure between to electrodes the potential difference resulting from constant current injection. Thus the measured voltage is an image of the skin resistance. According to Edelberg [START_REF] Edelberg | Methods in psychophysiology[END_REF], an injected current less than 10 A.cm -2 is optimal. We built a constant current source according to Dave Van Ess' description [START_REF] Van Ess | Programmable bipolar analog current source[END_REF] from a differential amplifier (DIFFAMP) with a gain set to one, whose analog ground V ref is set to AGN D = V DD/2. Then the voltage drop across the setting resistor R set is constant thus setting the current flowing through it. Assuming that the operational amplifiers are ideal, i.e. the input impedance is infinite and the output is an ideal voltage source, the current flowing through R set is the same as that through the skin resistance. With V set = -0.917 V , V ref = 1.650 V and R set = 330 kΩ , we obtain I set = 2.13 µA. Electrodes used for measuring skin resistance are round Ag/AgCl discs of 0.8 cm diameter (i.e. an area of 0.5 = cm 2 ) resulting in a 4.24 µA.cm -2 current density, well within Edelberg's optimal range. A hydrogel pad was placed between the skin and the electrode in order to improve contact (avoiding variations in the area and quality of contact, and lowering contact impedance) and to avoid skin progressive moistening [START_REF] Mcadams | Encyclopedia of Medical Devices and Instrumentation[END_REF]. Electrodes were placed on the medial or distal phalanges of the index and middle fingers (Fig. 3), as these sites are characterized by pronounced electrodermal activity [START_REF] Boucsein | Electrodermal Activity[END_REF].

Skin temperature was measured using a BetaTHERM 10K3A1 thermistor (Fig. 5). Resistance changes were converted into voltages through a simplified Wheatstone bridge: The bias resistor was chosen in order to center voltage scale on AGN D reference (i.e. V DD/2). By selecting the same reference for the programmable gain amplifier (PGA), the second part of the bridge becomes unnecessary, thus reducing the number of discrete components.

C. Software Description

The firmware contains analog multiplexing control, digital filtering and digital communications. Analog multiplexing is controlled by means of the API of the analog input mux. Listing 1 is an example of the "on-the-fly" configuring process. Communication with the XBee module is made through an UART serial link with a rate of 57600 bauds, reading and writing being controlled by the UART API. An API also exists for SD-Card read/writing hence the communication with the micro-SD card is done with high level function calls. The clock for the SPI bus linked to the card is set at 4 M Hz and the read/write functions can reach a rate of 2250 Bytes.s -1 .

III. RESULTS

Fig. 6 shows a comparison made between the system and a GSR Amp of ADInstrument (measuring skin resistance with isolated alternating current). Six minutes tests were performed on two different subjects. An unexpected audio stimulus was applied after a five minute period of relaxation. Following signal standardization and trend removing (the electrodes used with the ADInstrument are only recommended for very short measurements), correlation between the signals was 0.75, showing the ambulatory system's ability to match an immovable instrumentation in terms of performance. Robustness of heart rate measurement was tested on an ECG measurement during several periods of patient effort (walking and sprinting). The ECG remained stable even when the subject ran and the excessive movements generated considerable signal noise. The peak detection algorithm recovered the full R-R intervals from the ECG (see traces on Fig. 7). Table I shows current consumption under different operating modes, with a power voltage of 3.3 V . As the system is used with a 2200 mAh battery placed in the bracelet, it can afford four day of continuous monitoring without recharging the battery, enabling long-term measurement to be carried out.

IV. DISCUSSION

Heart rate variability (HRV) [START_REF]Heart rate variability: standards of measurement, physiological interpretation and clinical use. task force of the european society of cardiology and the north american society of pacing and electrophysiology[END_REF] and skin resistance responses (SRR) [START_REF] Boucsein | Electrodermal Activity[END_REF] are the main indicators for the noninvasive emotional analysis. Although skin resistance is indicative of emotional, sensorial and cognitive stimuli, this response is also affected by thermoregulation in extreme conditions. This example demonstrates the need of also retrieving information concerning the subject's environment: skin temperature provides information on vasoconstriction and is also representative of the environmental temperature [START_REF] Collet | Autonomic nervous system response patterns specificity to basic emotions[END_REF]. The above three signals therefore need continuous monitoring and this has been achieved with the presented device. Furthermore, ZigBee mesh topologies enable the device to be integrated into a Wireless Wearable Body Area Network (WWBAN) as a wireless multi-sensor in a telemedicine system [START_REF] Jovanov | A wban system for ambulatory monitoring of physical activity and health status: Applications and challenges[END_REF].

V. CONCLUSION

This research involved the development of wearable and ambulatory monitoring systems [START_REF] Axisa | Wrist ambulatory monitoring system and smart glove for real time emotional, sensorial and physiological analysis[END_REF] [START_REF] Dittmar | Wearable medical devices using textile and flexible technologies for ambulatory monitoring[END_REF]. The ambulatory aspect of the device leads to a wide range of applications:

• In the study of vigilance when driving, where signals can be interpreted in real-time [START_REF] Ramon | An integrated platform to assess driver's physiological and functional states[END_REF] • During continuous monitoring under extreme conditions (firemen) • In the study of sports performance • For health monitoring at home, with a telemedicine system

The future of the project is to develop an intelligent system capable of providing alarms in various situations (stress, vigilance, critical states). We are working on offline algorithms for the extraction of new relevant indicators in order to quantify autonomic nervous system activity. These indicators should have low CPU consumption in order to be embedded on a PSoC architecture device for real-time detection.

Fig. 1 .

 1 Fig. 1. System Architecture

Fig. 2 .

 2 Fig. 2. Flowchart of Heart Rate measurement

Fig. 3 .

 3 Fig. 3. Electrode positioning

/

  / Load config for temperature measurement void Set_Tmp_Cfg(void) { // Set amplifier gain Gain_Amp_SetGain(Gain_Amp_G4_00); // Set input multiplexer AMUX8_InputSelect(AMUX8_PORT0_1); // Set amplifier reference (manually) Gain_Amp_GAIN_CR0 &= 0xfc; Gain_Amp_GAIN_CR0 |= 0x01; } Listing 1. Analog Mux Example Sample rate of the incremental ADC reaches 480 sps, so each signal is sampled at 160 sps and goes through digital filters. An example of these fast filters is shown on Listing 2, which is a 50 Hz Notch filter for ECG processing. int ECG_Filter(int x) { int y; static long x_1, x_2, y_1, y_2; y=(57344 * (x+x_2)-5243 * (x_1-y_1) -49152 * y_2)>>16; x_2=x_1; x_1=x; y_2=y_1; y_1=y; return y; } Listing 2. 50Hz 2 nd Order IIR Notch Filter

Fig. 5 .

 5 Fig. 5. Flowchart of skin temperature measurement

Fig. 6 .Fig. 7 .

 67 Fig. 6. Ambulatory and Ref. Instrumentation signals for skin resistance measurement

Fig. 8 .

 8 Fig. 8. Wrist-device preview


On the basis of actual studies of molecular biology, a dogma tends to be established: In mammals, odor molecular receptors (ORs) would be highly specific, each one recognizing only one odorant or a small set of structurally similar odor molecules, and a single olfactory receptor neurons (ORN) would express a single OR type. However studies driving the functional expression of ORs fail to demonstrate the low or high specificity of these ORs. Here for the first time in rats, ORNs' selectivity was addressed at the cellular level in physiological conditions using unitary extracellular recordings. Against all expectation, individual ORNs are poorly selective regarding qualitatively distinct odor compounds.

The chemical receptive field of ORNs mirrors directly the properties of their ORs equipment. This field is defined by the specificity of ORs regarding the structure of odor molecules and by the number and diversity of ORs expressed on the same neuron. Since ORNs were assumed to express only one receptor subtype (1), some authors have postulated that ORNs may have a narrow tuned specificity (2,3,4). Furthermore, the expression of ORs has been shown to be spatially segregated and such an organization was proposed as defining the chemotopy of the mucosa [START_REF] Nef | Proc. Natl. Sci. USA[END_REF].

Lastly, the rules governing the projection of ORNs to the olfactory bulb seem to maintain the spatial ORNs' segregation (6). Thus, if all these assumptions and results are confirmed, the question of qualitative discrimination of odor molecules would be solved as soon as the ORNs' level through a simple and seducing theory which propose that the olfactory system would function as a "labeled line system" [START_REF] Axel | Paper presented at the XIIIth ECRO meeting[END_REF].

As reviewed by functional data on the qualitative tuning of ORNs in vivo were mainly gathered in amphibians [START_REF]In mammals, very few is known on in situ ORNs' responses to odorants. To our knowledge, there are only two studies on separated heads of rat embryos and young rats (R.C. Gesteland, C.D.Sigward[END_REF] where they fail to support the narrow tuning assumption. Indeed, individual ORNs respond to structurally different odor molecules. This gap between molecular data obtained in mammals and cellular data obtained in amphibians may obviously be ascribed to the phylogenetic evolution, if one assumes that ORNs became more and more selective. However, such an explanation is not in agreement with the poor selectivity of amphibian (10) and mammalian (11) olfactory bulb mitral cells, the projection-neurons of ORNs, and it may only mask our lack of knowledge of response properties of individual ORNs in intact animals. Thus, the question of the range of the chemical receptive field of individual ORNs was addressed through the use of classical extracellular recording techniques in anaesthetized rats. Regarding the results, the necessity of a re-interpretation of the data of the molecular biology comes in as an open question that will guide the discussion of this paper.

For the first time, individual ORNs were recorded in in vivo freely breathing (n=19) or tracheotomized (n=16) rats ( 12). Ninety ORNs were recorded, generally in the endoturbinate II, during periods ranging from 20 min. to 2 hours. The electro-olfactogram (EOG) was simultaneously recorded as close as possible to the single unit recording site. EOG is a transepithelial potential resulting from the summed activity of numerous ORNs that gives through its kinetic and size a direct and global information on both the intensity of the ORNs' response and the number of responding neurons.

Sixteen pure odor compounds were utilized as stimuli. They were selected regarding their effectiveness and their molecular structure from those previously tested in the frog [START_REF] Duchamp-Viret | [END_REF]. They were chosen as members of the qualitative groups established through several studies by Duchamp and collaborators and belong mainly to the terpene, camphor, aromatic, and straight-chained keton groups (13). Ethyl vanillin that had never been tested in the frog in vivo was added to get information on the IP3 transduction pathway (14). Stimuli were odor pulses of 2-sec. duration delivered at 200ml/min. They were applied directly near the surface of the turbinate using a dynamic multistage olfactometer (15) which ensured a precise control of the concentration range and allowed delivering 12 discrete concentrations. Depending on their saturated vapor pressure (SV), compounds were delivered at concentrations ranging from 3 10 -8 /5 10 -7 (SV/562) to 2 10 -5 /3 10 -4 M/l (SV),

In rats, ORNs are spontaneously active. About 40% of them fired spontaneously at more than 100 spikes/min., which is a high rate as compared with rates reported in amphibians (10). Furthermore, they were highly responsive. 83% of neurons were excited by one odor at least. When considering all stimuli delivered (n=540), 53.5% induced excitatory responses, 5% inhibitory responses and only 41.5% did not evoke a response. Thus, on average, each ORN was excited by 4 odors out of the whole odor set ( 16). According to the nature of the stimulus, the excitatory power of the odorants varied from 40 to 60%. Only ethyl vanillin was clearly apart, with 15%. ORNs qualitative response spectra were poorly selective. Among the ORNs tested with the whole odor set, many of them were excited by several odors and some of them were even excited by the 16 odorants. The selectivity regarding the odor subset 1 ( 16) is illustrated in fig. 1. A little more than 25 % of ORNs responded to the six odorants. This is all the more significant as odorants are members of four distinct qualitative groups according to Duchamp and colleagues [START_REF] Axel | Paper presented at the XIIIth ECRO meeting[END_REF]10): Camphor for camphor group; limonene for terpene group, anisole and acetophenon for aromatic group and iso-amyl-acetate and methyl-amyl keton for straight-chained keton group. Lastly, whenever a cell responded to none odor of the subset, it never responded to any odor of the whole odor subset 1. Taken together, the present results and previous data gathered in the frog led us to propose the odors of the subset 1 as representative of the qualitative olfactory space and of molecular binding affinities of ORNs.

Twenty-one cells were tested using several concentrations of each odor and their response thresholds were estimated. About fifty percent of ORNs were observed to reach their response thresholds for concentrations higher than SV/10 (10 -6 /10 -5 M/l) while 32% showed supraliminary responses at the lowest available concentration SV/562 (10 -7 /10 -8 ). Some ORNs responded to different odors with thresholds dispersed over a wide concentration range (Fig. 2.) while others responded with thresholds that were tuned in a narrow concentration range (fig. 3). Increasing concentration was also utilized to gain an insight into the dynamics of ORNs' operation. For most of neurons, the burst response pattern changed progressively, but continuously with concentration: Bursts became more and more sustained and appeared earlier and earlier. Lastly, the responses evoked by the highest concentrations often consisted in an early high frequency and long duration burst or in a decremential initial burst followed by an incremential high frequency and sustained rebound, the delay between the two response phases increasing with concentration. Thus, such an evolution of the response pattern over the whole concentration range, from threshold to high intensities, provides evidence that ORNs did not work at saturation, but on the contrary within a dynamic phase of their excitability.

Simultaneous recordings of the EOG support this assertion since EOG amplitudes increased with concentration, mirroring the recruitment dynamics of ORNs which participate in odor coding.

This study brings for the first time functional evidences that the rat ORNs have broad tuned chemical receptive fields. Their selectivity and sensitivity are fully in agreement with those of mitral cells previously reported in the same animal species (11). By contrast, rats ORNs tend to display a broader qualitative profile and a lower sensibility than those of the frog (17).

As underlined by Zhao and coworkers (4): "Identifying the molecular receptive field of an olfactory receptor neuron is a critical first step in understanding how olfactory perception is achieved by higher brain centers". Here, the chemical receptive fields of ORNs are identified in a biological preparation where they worked as close as possible to physiological conditions.

How can the results of the molecular biology on olfactory receptor proteins be interpreted to take into account our functional results? First off all, it must be noted that the two molecular studies which have addressed the chemical tuning of odor receptors contain some divergent results. Indeed, in insect Sf9 cells transfected with the OR5 receptor, Raming and coworkers (18) have shown that several odor molecules increase IP3 responses and concluded that OR5 are rather poorly selective. By contrast, Zhao and colleagues (4) reported, for the first time in rat ORNs, that increasing the expression of a single gene led to greater responsiveness to octanal and few other compounds with a very close molecular structure. They concluded that they drove the expression of a gene coding for a selective olfactory receptor. It must be noted that the narrow tuned chemical field of their transfected ORNs fits with that of mouse ORNs as previously measured using calcium-imaging (3).

In our recording conditions, the qualitative response profiles of single ORNs directly mirror their ORs' equipment and thus the binding properties of these ORs. If the hypothesis that each ORN expresses one OR is true, the fact that most ORNs respond to several distinct odor molecules demonstrates that ORs are poorly selective. Another possibility is that each ORN would express several ORs of a given subfamily (19), so that its qualitative response spectra would be the sum of the individual receptive fields of its ORs. Lastly, given that some neurons display a differential sensitivity to different odors, we propose that, at the level of a single neuron, not only the categories of ORs and their specificity may differ, but also their number and their affinity for odor molecules.

6

At a single neuron level, in term of olfactory quality coding, our data are in complete agreement with those previously obtained in the frog. One may argue that in Vertebrates, at the cellular level, the odor/receptor binding process would result in an "across fiber pattern" specification of odor identity and intensity. In other words our results do not support the concept proposed by Axel in a recent congress [START_REF] Axel | Paper presented at the XIIIth ECRO meeting[END_REF] that there would be "a logic for olfactory perception in which ORNs expressing a given OR, and therefore responsive to a given odor, would project to precise loci in the olfactory bulb". However, the involvement of ORs in a topographical organization of the mucosa and then, in the primary axon targeting in the olfactory bulb is entirely coherent with our results if one ascribes to this topological organization a role in the perpetuation of glomerular innervation despite of the continual renewal of ORNs. Indeed, very recently, the large family of ORs was hypothesized as molecular-addressing code that may serve, not only in the olfactory system but during the ontogenesis, also throughout the entire brain and other organs (20). Duchamp, A. Holley, P. MacLeod, Chem. Senses. 3, 23-33 (1978); M.F. Revial, A. Duchamp, A. Holley, Chem. Senses. 3, 7-21 (1978); M.F. Revial, G. Sicard, A. Duchamp, A. Holley, Chem. Senses. 7, 175-190 (1982); M.F. Revial, G. Sicard, A. Duchamp, A. Holley, Chem. Senses. 8, 179-194 (1983).

11. Wellis, J.W. Scott, T.A. Harrison, J. Neurophysiol. 61, 116è-1177(1989);F. Motokizawa, Exp. Brain Res. 112, 24-34 (1996); M. Chalansonnet, M.A. Chaput, Chem. Senses 23, 1-9 (1998).. 12. Surgical methods: All experiments were performed following animal care guidelines. Adult Wistar rats (250-300 g) were anesthetized by an intraperitoneal injection of Equithesine (mixture of pentobarbital sodium and chloral hydrate) at the initial dose of 3ml/kg. Anesthetic was then supplemented as necessary to maintain a deep level of anesthesia. Rectal temperature was maintained at 37 ± 0.5°C by a homeothermic blanket (Harvard Apparatus, USA) and surgical wounds of the animals were regularly infiltrated with 2% Procaine. For recordings, anesthetized animals were secured in a sterotaxic apparatus. Recordings were performed in the Endoturbinate II.

Access to the olfactory mucosa was gained by removing the nasal bones and then gently slipping aside the dorsal recess underlying these bones. Recording procedures: Single-unit action potentials were recorded using metal-filled glass micropipettes (3 to 7 MOhms), and EOG with a glass micropipettes of 50-µm diameter filled with saline solution. The recorded signals were led through conventional amplifiers. Spike signals were filtered between 300 and 3,000 Hz. Data were stored on the DTR (Data Tape Recorder, by Biologic, France). During the experiment, the single unit nature of the recording was controlled on line by triggering the recorded cell near the background-noise. The activity was monitored on a memory oscilloscope. This allowed us to control the characteristics of the polyphasic spike of the cell studied in order to insure that the same cell was recorded over all the experimental procedure.

13. Studies of ORN qualitative discrimination properties in the frog [START_REF] Duchamp-Viret | [END_REF]10) have shown that the concept of the odor group has a fundamental meaning related to the structure of olfactory molecules.

14. Sklar et al., J. Bio. Chem. 33, 15538-15543 (1988) 15. Vigouroux, P. Viret, A. Duchamp, J. Neurosci. Methods. 24, 57-63 (1988).

16. Since recording a single ORN long enough to test all the 16 odorants was rather difficult, the whole odor-set was subdivided into 3 subsets. Subset 1 comprised camphor (CAM), limonene (LIM), iso-amyl acetate (ISO), acetophenon (ACE), methyl-amyl keton and anisole (MAK). Subset 2 comprised cineol (CIN), vanillin (VAN), p-cymene (CYM), cyclodecanon (CDN), cyclohexanon (HEX) and heptanol (HEP). Subset 3 comprised two pairs of enantiomers: l-and d-carvone and land d-citronellol. The three subsets were delivered in that order while odors of a given subset were delivered at random. According to this stimulation protocol, only neurons tested with subset 1, at least, were considered to analyze the qualitative discrimination properties of ORNs. 17. Duchamp-Viret, A. Duchamp, M. Vigouroux, J. Neurophysiol. 61, 1085-1094(1989); P. Duchamp-Viret, A. Duchamp, G. Sicard, Brain Research. 517, 256-262 (1990). 18. Raming et al., Nature 361, 353-356 (1993).

19. In in situ hybridization experiments, Ressler, Sullivan and Buck (Cell 73, 597-609 (1993)) have utilized antisense probes to hybridize ORs subfamilies. They observe a regional distribution of these subfamilies in the olfactory mucosa and hypothesize that each subfamily would code for ORs that would have identical or similar odor specificities. By contrast, our results suggest that a given OR subfamily may code for ORs expressing different odor binding properties. This hypothesis does not call the importance of the regionalization of ORs subfamilies into question since belonging to a subfamily would permit a precise targeting of epithelial zones to the olfactory bulb. 20. W.J. Dreyer, Proc. Natl. Acad. Sci. USA 95, 9072-9077, 1998. a small EOG, but did not modify significantly the ORN firing activity. Increasing the concentration to 5.9 10 -6 M/l (SV/20) induced a larger EOG and a ORN response characterized by a rhythmic discharge composed of 4 spikes. Thus the response threshold of this ORN is comprised between 1.9 and 5.9 10 -6 M/l (i.e. between SV/56.2 and SV/20). Increasing the concentration enhanced the ORN firing activity that became a sustained tonic response pattern, and then an initial high-frequency burst of activity followed by a silence and a rebound. EOG amplitude evolves in parallel : It was very small for 1. 9 10 -6 M/l and increased gradually, mirroring the global recruitment of the ORNs situated within the recording field of the electrode. Recordings also show that while the ORN burst discharge frequency increases with concentration, the latency of this discharge shortens with respect to the beginning of the odor-pulse and thus, appears earlier and earlier with respect to the EOG kinetics. Lastly, this figure demonstrates that the concentration range utilized in this study overlaps the dynamic range of rat ORNs' functioning.

Figure captions

Figure 1 :

 1 Figure 1: Selectivity of ORNs regarding to the odor subset 1 (16). Percentages of recorded ORNs are

Figure 2 :

 2 Figure 2: Spontaneous activity (upper trace) and representative responses of ORN50 to different

Figure 3 :

 3 Figure 3: Spontaneous activity (upper trace) and responses (lower traces) of ORN19 to different

Figure 4 :

 4 Figure 4: Spontaneous (upper pair of traces) and odor-evoked (lower pairs of traces) EOGs and single