Tony Lelièvre
email: tony.lelievre@enpc.fr

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

Boris Nectoux
email: boris.nectoux@enpc.fr

Exit event from a metastable state and Eyring-Kramers law for the overdamped Langevin dynamics

Keywords: Exit event, metastability, Eyring-Kramers and overdamped Langevin

. These results aim in particular at justifying the use of a Markov jump process parametrized by the Eyring-Kramers law to model the exit event from a metastable region.

The objective of this note is to give motivations (Section 1) and outlines of the proofs (Section 2) of results recently obtained in [START_REF] Di Gesù | Sharp asymptotics of the first exit point density[END_REF][START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points[END_REF][START_REF] Nectoux | Analyse spectrale et analyse semi-classique pour la métastabilité en dynamique moléculaire[END_REF]. These results justify the use of the Eyring-Kramers formulas together with a kinetic Monte Carlo model to model the exit event from a metastable state for the overdamped Langevin dynamics. Such results are particularly useful to justify algorithms and models which use such formulas to build reduced description of the overdamped Langevin dynamics.

1 Exit event from a metastable domain and Markov jump process

Overdamped Langevin dynamics and metastability

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 ∈ C ∞ (R d , R) is the potential function, h > 0 is the temperature and (B t ) t≥0 is a standard d-dimensional Brownian motion. The overdamped Langevin dynamics can be used for instance to describe the motion of the atoms of a molecule or the diffusion of impurities in a crystal (see for instance [52, Sections 2 and 3] or [START_REF] Chandrasekhar | Stochastic problems in physics and astronomy[END_REF]). The term -∇f (X t ) in [START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF] sends the process towards local minima of f , while thanks to the noise term √ h dB t , the process X t may jump from one basin of attraction of the dynamics ẋ = -∇f (x) to another one. If the temperature is small (i.e. h ≪ 1), the process (X t ) t≥0 remains during a very long period of time trapped around a neighborhood of a local minimum of f , called a metastable state, before going to another region. For that reason, the process ( 1) is said to be metastable. More precisely, a domain Ω ⊂ R d is said to be metastable for the probability measure µ supported in Ω if, when X 0 ∼ µ, the process (1) reaches a local equilibrium in Ω long before escaping from it. This will be made more precise below using the notion of quasi-stationary distribution (see Section 1.5). The move from one metastable region to another is typically related to a macroscopic change of configuration of the system. Metastability implies a separation of timescales which is one of the major issues when trying to have access to the macroscopic evolution of the system using simulations made at the microscopic level. Indeed, in practice, many transitions cannot be observed by integrating directly the trajectories of the process [START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF]. To overcome this difficulty, some algorithms use the fact that the exit event from a metastable region can be well approximated by a Markov jump process with transition rates computed with the Eyring-Kramers formula, see for example the Temperature Accelerated Dynamics method [START_REF] Sorensen | Temperature-accelerated dynamics for simulation of infrequent events[END_REF] that will be described below.

Markov jump process and Eyring-Kramers law

Kinetic Monte Carlo methods. Let Ω ⊂ R d be a domain of the configuration space and let us assume that the process (1) is initially distributed according to the probability measure µ (i.e. X 0 ∼ µ) which is supported in Ω and for which the exit event from Ω is metastable. Let us denote by (Ω i ) i=1,...,n the surrounding domains of Ω (see Figure 1), each of them corresponding to a macroscopic state of the system. Many reduced models and algorithms rely on the fact that the exit event from Ω, i.e. the next visited state by the process (1) among the Ω i 's as well as the time spent by the process [START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF] in Ω, is efficiently approximated by a Markov jump process using kinetic Monte Carlo methods [START_REF] Cameron | Computing the asymptotic spectrum for networks representing energy landscapes using the minimum spanning tree[END_REF][START_REF] Fan | Autonomous basin climbing method with sampling of multiple transition pathways: application to anisotropic diffusion of point defects in hcp Zr[END_REF][START_REF] Schütte | Conformational dynamics: modelling, theory, algorithm and application to biomolecules[END_REF][START_REF] Schütte | Metastability and Markov state models in molecular dynamics[END_REF][START_REF] Voter | Radiation Effects in Solids, chapter Introduction to the Kinetic Monte Carlo Method[END_REF][START_REF] Wales | Energy landscapes[END_REF]. Kinetic Monte Carlo methods simulate a Markov jump process in a discrete state space. To use a kinetic Monte Carlo algorithm in order to sample the exit event from Ω, one needs for i ∈ {1, . . . , n} the transition rate k i to go from the state Ω to the state Ω i . A kinetic Monte Carlo algorithm generates the next visited state Y among the Ω i 's and the time T spent in Ω for the process (1) as follows:

1. First sample T as an exponential random variable with parameter n i=1 k i , i.e.:

T ∼ E n i=1 k i .
(2)

2. Then, sample the next visited state Y independently from T , i.e

Y |= T (3)
using the following law : for all i ∈ {1, . . . , n}, Remark 1. Let us give an equivalent way to sample T and Y in a Monte Carlo method. Let (τ i ) i∈{1,...,n} be n independent random variables such that for all i ∈ {1, . . . , n}, τ i is exponentially distributed with parameter k i . Then, the couple (T, Y ) has the same law as (min j∈{1,...,n} τ j , argmin j∈{1,...,n} τ j ).

P Y = i = k i n ℓ=1 k ℓ . ( 4
Eyring-Kramers law. In practice, the transition rates (k i ) i∈{1,...,n} are computed using the Eyring-Kramers formula [START_REF] Hänggi | Reaction-rate theory: fifty years after Kramers[END_REF][START_REF] Voter | Radiation Effects in Solids, chapter Introduction to the Kinetic Monte Carlo Method[END_REF]:

k i = A i e -2 h (f (zi)-f (x0)) , (5)
where x 0 ∈ Ω is the unique global minimum of f in Ω and {z i } = argmin ∂Ω∩∂Ωi f , see Figure 1. We here assume for simplicity that the minimum is attained at one single point z i but the results below can be generalized to more general settings.

If Ω is the basin of attraction of x 0 for the dynamics ẋ = -∇f (x) so that z i is a saddle point of f (i.e. a critical point of index 1), then, for the overdamped Langevin dynamics (1), the prefactor A i writes:

A i = |λ(z i )| 2π det Hess f (x 0 ) |det Hess f (z i )| , ( 6
)
where λ(z i ) is the negative eigenvalue of the Hessian matrix of f at z i . Notice that the formula [START_REF] Bovier | Metastability in reversible diffusion processes. I. sharp asymptotics for capacities and exit times[END_REF] requires that x 0 and z i are non degenerate critical points of f . The formulas (5) and ( 6) have been first obtained in the small temperature regime by Kramers [START_REF] Kramers | Brownian motion in a field of force and the diffusion model of chemical reactions[END_REF] (see the review of the literature [START_REF] Hänggi | Reaction-rate theory: fifty years after Kramers[END_REF]).

Remark 2. In the Physics literature, the approximation of the macroscopic evolution of the system with a Markov jump process with transition rates computed with the Eyring-Kramers formula ( 5)-( 6) is sometimes called the Harmonic Transition State Theory [START_REF] Marcelin | Contribution à l'étude de la cinétique physico-chimique[END_REF][START_REF] Vineyard | Frequency factors and isotope effects in solid state rate processes[END_REF].

1.3

The temperature accelerated dynamics algorithm.

The temperature accelerated dynamics (TAD) algorithm proposed by M.R. Sørensen and A.F Voter [START_REF] Sorensen | Temperature-accelerated dynamics for simulation of infrequent events[END_REF] aims at efficiently approximating the exit event from a metastable domain for the dynamics (1) in order to have access to the macroscopic evolution of the system. We also refer to [START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF] for a mathematical analysis of this algorithm in a one-dimensional setting.

The basic idea of the TAD algorithm is the following: the exit time from the metastable domain Ω increases exponentially with the inverse of the temperature, see indeed ( 2)-( 5). The idea is then to simulate the process at higher temperature to accelerate the simulation of the exit event. Let us assume that the process (X t ) t≥0 , evolving at the temperature h low is at some time t 0 ≥ 0 in the domain Ω ⊂ R d which is metastable for the initial condition X t0 ∈ Ω.

Following [START_REF] Sorensen | Temperature-accelerated dynamics for simulation of infrequent events[END_REF], let us assume that the process instantaneously reaches the local equilibrium in Ω, i.e. that X t0 is distributed according to this local equilibrium.

The existence and the uniqueness of the local equilibrium in Ω as well as the convergence toward this local equilibrium is made more precise in Section 1.5 using the notion of quasi-stationary distribution. To ensure the convergence towards the local equilibrium in Ω, a decorrelation step may be used before running the TAD algorithm, see step (M1) in [1, Section 2.2]. As in the previous section, one denotes by (Ω i ) i=1,...,n the surrounding domains of Ω (see Figure 1), each of them corresponding to a macroscopic state of the system and, for i ∈ {1, . . . , n}, {z i } = argmin ∂Ω∩∂Ωi f . To sample the next visited state among the Ω i 's as well as the time T spent in Ω for the process (1), the TAD algorithm proceeds as follows. Let us introduce T sim = 0 (which is the simulation time) and T stop = +∞ (which is the stopping time), and iterate the following steps.

τ j (h low ) = τ j (h high ) e 2 1 h low -1 h high (f (zj)-f (x0)) , (7)
where we recall x 0 ∈ Ω is the unique global minimum of f in Ω. Then, update the minimum exit time τ min (h low ) among the τ j (h low )'s which have been observed so far. Finally, compute a new time T stop so that there is a very small probability (say α ≪ 1) to observe an exit event from Ω at the temperature h high which, using [START_REF] Bovier | Metastability in reversible diffusion processes. II. precise asymptotics for small eigenvalues[END_REF], would change the value of τ min (h low ). We refer to [START_REF] Sorensen | Temperature-accelerated dynamics for simulation of infrequent events[END_REF] or [START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF] for the computation of T stop . 2. If T sim ≤ T stop then go back to the first step starting from the local equilibrium in Ω at time T sim , else go to the next step.

3. Set T = τ min (h low ) and Y = ℓ where ℓ is such that τ ℓ (h low ) = τ min (h low ).
Finally, send X t0+T to Ω ℓ and evolve the process (1) with the new initial condition X t0+T .

Remark 3. In [START_REF] Sorensen | Temperature-accelerated dynamics for simulation of infrequent events[END_REF], when the process (Y t ) t≥Tsim leaves Ω, it is reflected back in Ω and it is then assumed that it reaches instantaneously the local equilibrium in Ω at temperature h high .

Remark 4. One can use a decorrelation step before running the TAD algorithm and the sampling of Y Tsim according to the local equilibrium in Ω at the beginning of the step 1 to ensure that the underlying Markov jump process is justified, see [START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF].

The extrapolation formula [START_REF] Bovier | Metastability in reversible diffusion processes. II. precise asymptotics for small eigenvalues[END_REF] which is at the heart of the TAD algorithm relies on the properties of the underlying Markov jump process used to accelerate the exit event from a metastable state and where transition times are exponentially distributed with parameters computed with the Eyring-Kramers formula, see Remark 1 and Equation [START_REF] Berglund | On the noise-induced passage through an unstable periodic orbit II: General case[END_REF]. In the algorithm TAD, it is indeed assumed that the exit event from Ω can be modeled with a kinetic Monte Carlo method where the transition rates are computed with the Eyring-Kramers law ( 5)- [START_REF] Bovier | Metastability in reversible diffusion processes. I. sharp asymptotics for capacities and exit times[END_REF]. Then, at high temperature, one checks that under this assumption, each τ i (h high )

(i ∈ {1, . . . , n}) is an exponential law of parameter A i e -2 h high (f (zi)-f (x0)) (see Remark 1
). The formula (7) allows to construct for all i ∈ {1, . . . , n}, an exit time τ i (h low ) which is an exponential law of parameter

A i e -2 h low (f (zi)-f (x0))
. By considering the couple (min i∈{1,...,n} τ i (h low ), argmin i∈{1,...,n} τ i (h low )), one has access to the exit event from Ω (see Remark 1).

Remark 5. There are other algorithms which use the properties of the underlying Markov jump process to accelerate the simulation of the exit event from a metastable state, see for instance [START_REF] Voter | A method for accelerating the molecular dynamics simulation of infrequent events[END_REF] and [START_REF] Voter | Parallel replica method for dynamics of infrequent events[END_REF].

Our objective is to justify rigorously that a Markov jump process with transition rates computed with the Eyring-Kramers formula (5) can be used to model the exit event from a metastable domain Ω for the overdamped Langevin process [START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF]. Before, let us recall mathematical contributions on the exit event from a domain and on the Eyring-Kramers formula (5).

Mathematical literature on the exit event from a domain and on the Eyring-Kramers formulas

In the mathematical literature, there are mainly two approaches to the study of the asymptotic behaviour of the exit event from a domain when h → 0: the global approaches and the local approaches.

Global approaches.

The global approaches study the asymptotic behaviours in the limit h → 0 of the eigenvalues of the infinitesimal generator

L (0) f,h = - h 2 ∆ + ∇f • ∇ (8)
of the diffusion (1) on R d . Let us give for example a result obtained in [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].

To this end, let us assume that the potential f : R d → R is a Morse function, has m local minima {x 1 , . . . , x m } and that for h small enough R d e -2 h f < +∞. Let us recall that φ : R d → R is a Morse function if all its critical points are non degenerate. For a Morse function φ : R d → R, we say that x is a saddle point of φ if x is a critical point of φ such that the Hessian matrix of φ at x has exactly one negative eigenvalue (i.e. x is a critical point of φ of index 1). Then, from [START_REF] Helffer | Puits multiples en mecanique semi-classique iv etude du complexe de witten[END_REF], the operator L (0) f,h has exactly m exponentially small eigenvalues {λ 1 , λ 2 , . . . , λ m } when h → 0 with λ 1 = 0 < λ 2 ≤ . . . ≤ λ m (i.e., when h → 0, for all i ∈ {1, . . . , m}, λ i = O(e -c h ) for some c > 0 independent of h). Moreover, sharp asymptotic estimates can be derived for the eigenvalues {λ 2 , . . . , λ m }. In [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], the following results are obtained. Let us assume that {x 1 } = argmin R d f . For k ∈ {2, . . . , m} and B k = {x ∈ {x 1 , . . . , x m } \ {x k }, f (x) ≤ f (x k )} (i.e. B k is the set of local minima of f which are lower in energy than x k ), one denotes by P(x k , B k ) the set of curves γ ∈ C 0 ([0, 1], R d ) such that γ(0) = x k and γ(1) ∈ B k . Let us finally assume that:

1. For all k ∈ {2, . . . , m}, there exists a unique saddle point z k (i.e. a critical point of f of index 1) such that f

(z k ) = inf γ∈P(x k ,B k ) sup t∈[0,1] f (γ(t)). 2. The values f (z k ) -f (x k ) k∈{2,...,m} are all distinct.
These assumptions imply that the map x k ∈ {x 2 , . . . , x m } → z k is injective. The set {x 2 , . . . , x m } is then labeled such that the sequence f (z k )-f (x k ) k∈{2,...,m} is strictly decreasing. The previous assumptions also imply the existence of a cascade of events, which occur with different timescales, to go from one local minimum x k of f to the global minimum x 1 of f in R d , see for instance Figure 2. Then, one has for k ∈ {2, . . . , m}, in the limit h → 0:

λ k = |λ(z k )| 2π det Hessf (x k ) | det Hessf (z k )| e -2 h (f (z k )-f (x k )) (1 + o(1)), (9)
where λ(z k ) is the negative eigenvalue of the Hessian matrix of f at z k . In the articles [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 a potential-theoretic approach, the sharp equivalent ( 9) is obtained and each of the eigenvalues λ k (for k ∈ {2, . . . , m}) is shown to be the inverse of the average time it takes for the process (1) to go from x k to B k . We also refer to [START_REF] Eckhoff | Precise asymptotics of small eigenvalues of reversible diffusions in the metastable regime[END_REF] for similar results. In [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach[END_REF], another proof of ( 9) is given using tools from semi-classical analysis. Let us also mention [START_REF] Michel | About small eigenvalues of Witten laplacian[END_REF] for a generalization of the results obtained in [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach[END_REF]. Notice that the results presented above do not provide any information concerning the average time it takes for the process (1) to go from the global minimum of f to a local minimum of f when h → 0. One also refers to [START_REF] Landim | A Dirichlet and a Thomson principle for nonselfadjoint elliptic operators, metastability in non-reversible diffusion processes[END_REF] for generalization of [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] for a class of non reversible processes when f has two local minima, and to [11-13, 37, 55] for related results. Remark 6. The global approaches have been used in [START_REF] Schütte | Conformational dynamics: modelling, theory, algorithm and application to biomolecules[END_REF][START_REF] Schütte | Metastability and Markov state models in molecular dynamics[END_REF] to construct a Markovian dynamics by projecting the infinitesimal generator

L (0)
f,h of the diffusion (1) with a Galerkin method onto the vector space associated with the m small eigenvalues {λ 1 , . . . , λ m }. This projection leads to a very good approximation of L (0) f,h in the limit h → 0. The question is then how to relate the transition events (or the trajectories) of the obtained Markov process to the exit events (or the trajectories) of the original one.

Local approaches. The local approaches consist in studying the asymptotic behaviour when h → 0 of the exit event (τ Ω , X τΩ ) from a domain Ω ⊂ R d , where τ Ω := inf{t ≥ 0, X t / ∈ Ω}.

One of the most well-known approaches is the large deviation theory developed by Freidlin and Wentzell in the 1970s. We refer to the book [START_REF] Freidlin | Random Perturbations of Dynamical Systems[END_REF] which summarizes their main contributions. This theory is based on the study of small pieces of the trajectories of the process defined with a suitable increasing sequence of stopping times. The rate function is fundamental in this theory: it quantifies the cost of deviating from a deterministic trajectory when h → 0. The rate functional was first introduced by Schilder [START_REF] Schilder | Some asymptotic formulas for Wiener integrals[END_REF] for a Brownian motion. Some typical results from [START_REF] Freidlin | Random Perturbations of Dynamical Systems[END_REF] to Ω) and that f has a unique non degenerate critical point x 0 in Ω such that f (x 0 ) = min Ω f . Then, for all x ∈ Ω:

lim h→0 h ln E x τ Ω = 2 inf ∂Ω f -f (x 0 ) .
The notation E x stands for the expectation given the fact that X 0 = x. Moreover, let x ∈ Ω such that f (x) < inf ∂Ω f . Then, for any γ > 0 and δ 0 > 0, there exist δ ∈ (0, δ 0 ] and h 0 > 0 such that for all h ∈ (0, h 0 ) and for all y ∈ ∂Ω:

e -2 h (f (y)-inf ∂Ω f ) e -γ h ≤ P x |X τΩ -y| < δ ≤ e -2 h (f (y)-inf ∂Ω f ) e γ h .
The notation P x stands for the probability given the fact that X 0 = x. Lastly, if the infimum of f on ∂Ω is attained at one single point y 0 ∈ ∂Ω, then for all δ > 0:

lim h→0 P x |X τΩ -y 0 | < δ = 1.
A result due to Day [START_REF] Day | On the exponential exit law in the small parameter exit problem[END_REF] (see also [START_REF] Mathieu | Zero white noise limit through Dirichlet forms, with application to diffusions in a random medium[END_REF][START_REF] Mathieu | Spectra, exit times and long time asymptotics in the zero-white-noise limit[END_REF]) concerning the law of τ Ω is the following. When h → 0, the exit time τ Ω converges in law to an exponentially distributed random variable and for all x ∈ Ω lim h→0

λ h E x τ Ω = 1,
where λ h is the principal eigenvalue of the infinitesimal generator of the diffusion (1) associated with Dirichlet boundary conditions on ∂Ω (see Proposition 2 below). The interest of this approach is that it can be applied to very general dynamics. However, when it is used to prove that the Eyring-Kramers formulas ( 5) can be used to study the exit distribution from Ω, it only provides the exponential rates (not the prefactor A i in ( 5)) and does not give error bounds when h → 0.

There are also approaches which are based on techniques developed for partial differential equations. In [START_REF] Matkowsky | The exit problem for randomly perturbed dynamical systems[END_REF][START_REF] Matkowsky | The exit problem: a new approach to diffusion across potential barriers[END_REF], using formal computations, when ∂ n f > 0 on ∂Ω and f has a unique non degenerate critical point x 0 in Ω such that f (x 0 ) = min Ω f , the following formula is derived: for any F ∈ C ∞ (∂Ω, R) and x ∈ Ω, one has when h → 0:

E x F X τΩ = ∂Ω F (z)∂ n f (z) e -2 h f (z) dz ∂Ω ∂ n f e -2 h f dσ + o(1). ( 10
)
The formal asymptotic estimate [START_REF] Chandrasekhar | Stochastic problems in physics and astronomy[END_REF] implies that the law of X τΩ concentrates on points where f attains its minimum on ∂Ω. Moreover, an asymptotic equivalent of E x τ Ω when h → 0 is also formulated in [START_REF] Naeh | A direct approach to the exit problem[END_REF] through formal computations.

These results are obtained injecting formal asymptotic expansions in powers of h in the partial differential equations satisfied by

x ∈ Ω → E x F X τΩ and x ∈ Ω → E x τ Ω .
We also refer to [START_REF] Matkowsky | The exit problem: a new approach to diffusion across potential barriers[END_REF], where using formal computations, asymptotic formulas are obtained concerning both the concentration of the law of X τΩ on argmin ∂Ω f and E x τ Ω when Ω is the union of basins of attraction of the dynamics d dt γ(t) = -∇f (γ(t)). When ∂ n f > 0 on ∂Ω and f has a unique non degenerate critical point x 0 in Ω such that f (x 0 ) = min Ω f , the formula ( 10) is proved rigorously by Kamin in [START_REF] Kamin | On elliptic singular perturbation problems with turning points[END_REF], and is extended to a non reversible diffusion process (Y t ) t≥0 solution to [START_REF] Day | On the asymptotic relation between equilibrium density and exit measure in the exit problem[END_REF][START_REF] Day | Recent progress on the small parameter exit problem[END_REF][START_REF] Kamin | Elliptic perturbation of a first order operator with a singular point of attracting type[END_REF][START_REF] Perthame | Perturbed dynamical systems with an attracting singularity and weak viscosity limits in hamilton-jacobi equations[END_REF] when Ω contains one attractor of the dynamics d dt γ(t) = b(γ(t)) and b(x) • n < 0 for all x ∈ ∂Ω. However, the results [START_REF] Day | On the asymptotic relation between equilibrium density and exit measure in the exit problem[END_REF][START_REF] Day | Recent progress on the small parameter exit problem[END_REF][START_REF] Kamin | Elliptic perturbation of a first order operator with a singular point of attracting type[END_REF][START_REF] Kamin | On elliptic singular perturbation problems with turning points[END_REF][START_REF] Perthame | Perturbed dynamical systems with an attracting singularity and weak viscosity limits in hamilton-jacobi equations[END_REF] do not provide any information on the probability to leave Ω through a point which is not a global minimum of f on ∂Ω.

dY t = b(Y t ) dt + √ h dB t in
Finally, let us mention [START_REF] Devinatz | Asymptotic behavior of the principal eigenfunction for a singularly perturbed dirichlet problem[END_REF][START_REF] Devinatz | The asymptotic behavior of the solution of a singularly perturbed dirichlet problem[END_REF][START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary[END_REF][START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF][START_REF] Peutrec | Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian[END_REF][START_REF] Mathieu | Zero white noise limit through Dirichlet forms, with application to diffusions in a random medium[END_REF][START_REF] Mathieu | Spectra, exit times and long time asymptotics in the zero-white-noise limit[END_REF] for a study of the asymptotic behaviour in the limit h → 0 of λ h and u h (see Proposition 2 below). The reader can also refer to [START_REF] Day | Mathematical Approaches to the Problem of Noise-Induced Exit[END_REF] for a review of the different techniques used to study the asymptotic behaviour of X τΩ when h → 0 and to [START_REF] Berglund | Kramers' law: Validity, derivations and generalisations[END_REF] for a review of the different techniques used to study the asymptotic behaviour of τ Ω when h → 0.

Remark 7. Some authors proved the convergence to a Markov jump process in some specific geometric settings and after a rescaling in time. We refer to [START_REF] Kipnis | The metastable behavior of infrequently observed, weakly random, one-dimensional diffusion processes[END_REF] for a one-dimensional diffusion in a double well and [START_REF] Galves | Metastability for a class of dynamical systems subject to small random perturbations[END_REF][START_REF] Mathieu | Spectra, exit times and long time asymptotics in the zero-white-noise limit[END_REF] for a study in higher dimension. In [START_REF] Sugiura | Metastable behaviors of diffusion processes with small parameter[END_REF], assuming that all the saddle points of f are at the same height, it is proved that a suitable rescaling of the time leads to a convergence of the diffusion process to a Markov jump process between the global minima of f . The results presented in this work (see [START_REF] Di Gesù | Sharp asymptotics of the first exit point density[END_REF][START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points[END_REF]) follow a local approach. The quasi-stationary distribution of the process (1) on Ω is the cornerstone of the analysis. They state that, under some geometric assumptions, the Eyring-Kramers formulas (with prefactors) can be used to model the exit event from a metastable state, and provide explicit error bounds.

Quasi-stationary distribution and transition rates

Local equilibrium. Let Ω be a C ∞ open bounded connected subset of R d and f ∈ C ∞ (Ω, R). Let us recall that τ Ω := inf{t ≥ 0, X t /
∈ Ω} denotes the first exit time from Ω. The quasi-stationary distribution of the process (1) on Ω is defined as follows.

Definition 1. A probability measure ν h on Ω is a quasi-stationary distribution of the process [START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF] on Ω if for all t > 0 and any measurable set A ⊂ Ω,

P ν h X t ∈ A t < τ Ω = ν h (A).
The notation P µ stands for the probability given the fact that the process ( 1) is initially distributed according to µ i.e. X 0 ∼ µ. The next proposition [START_REF] Champagnat | Quasi-stationary distribution for multidimensional birth and death processes conditioned to survival of all coordinates[END_REF][START_REF] Bris | A mathematical formalization of the parallel replica dynamics[END_REF] shows that the law of the process (1) at time t conditioned not to leave Ω on the interval (0, t) converges to the quasi-stationary distribution.

Proposition 1. Let Ω be a C ∞ open connected and bounded subset of R d and f ∈ C ∞ (Ω, R).
Then, there exist a unique probability measure ν h on Ω and c > 0 such that for any probability measure µ on Ω, there exist C(µ) > 0 and t(µ) > 0 such that for all t ≥ t(µ) and all measurable set A ⊂ Ω:

P µ X t ∈ A t < τ Ω -ν h (A) ≤ C(µ)e -ct . ( 11
)
Moreover, ν h is the unique quasi-stationary distribution of the process (1) on Ω.

Proposition 1 indicates that the quasi-stationary distribution ν h can be seen as a local equilibrium of the process [START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF] in Ω.

The quasi-stationary distribution ν h can be expressed with the principal eigenfunction of the infinitesimal generator

L (0)
f,h (see ( 8)) of the diffusion (1) associated with Dirichlet boundary conditions on ∂Ω. To this end, let us introduce the following Hilbert spaces L 2 w (Ω) = u : Ω → R, Ω u 2 e -2 h f < ∞ and for q ∈ {1, 2},

H q w (Ω) = u ∈ L 2 w (Ω), ∀α ∈ N d , |α| ≤ q, ∂ α u ∈ L 2 w (Ω) . ( 12
)
The subscript w in the notation L 2 w (Ω) and H q w (Ω) refers to the fact that the weight function x ∈ Ω → e -2 h f (x) appears in the inner product. Moreover, let us denote by H 1 0,w (Ω) = {u ∈ H 1 w (Ω), u = 0 on ∂Ω}. Let us recall the following result [START_REF] Bris | A mathematical formalization of the parallel replica dynamics[END_REF].

Proposition 2. Let Ω be a C ∞ open connected and bounded subset of R d and f ∈ C ∞ (Ω, R). Then, the operator L (0) f,h with domain H 1 0,w (Ω) ∩ H 2 w (Ω) on L 2 w (Ω), which is denoted by L D,(0) f,h , is self-adjoint, positive and has compact resolvent. Furthermore, the smallest eigenvalue λ h of L D,(0) f,h
is non degenerate and any eigenfunction associated with λ h has a sign on Ω.

In the following, one denotes by u h an eigenfunction associated with λ h . The smallest eigenvalue

λ h of L D,(0) f,h is called the principal eigenvalue of L D,(0) f,h
and u h a principal eigenfunction of L D,(0) f,h . Without loss of generality, one assumes that

u h > 0 on Ω and Ω u 2 h e -2 h f = 1. ( 13
)
Then, the quasi-stationary distribution ν h of the process [START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF] in Ω is given by (see [START_REF] Bris | A mathematical formalization of the parallel replica dynamics[END_REF]):

ν h (dx) = u h (x) e -2 h f (x) Ω u h e -2 h f dx. ( 14
)
Moreover, the following result shows that when X 0 ∼ ν h , the law of the exit event (τ Ω , X τΩ ) is explicitly known in terms of λ h and u h (see [START_REF] Bris | A mathematical formalization of the parallel replica dynamics[END_REF]).

Proposition 3. Let us assume that X 0 ∼ ν h , where ν h is the quasi-stationary distribution of the process [START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF] in Ω. Then, τ Ω and X τΩ are independent. Moreover, τ Ω is exponentially distributed with parameter λ h and for any open set Σ ⊂ ∂Ω, one has:

P ν h X τΩ ∈ Σ = - h 2λ h Σ ∂ n u h (z)e -2 h f (z) σ(dz) Ω u h e -2 h f , ( 15
)
where σ(dz) is the Lebesgue measure on ∂Ω.

Approximation of the exit event with a Markov jump process. Let us now provide justifications to the use of a Markov jump process with transition rates computed with the Eyring-Kramers formula (5) to model the exit event from a metastable domain Ω. In view of [START_REF] Davies | Dynamical stability of metastable states[END_REF], one can be more precise on the definition of the metastability of a domain Ω given in Section 1.1. For a probability measure µ supported in Ω, the domain Ω is said to be metastable if, when X 0 ∼ µ, the convergence to the quasi-stationary distribution ν h in (1) is much quicker than the exit from Ω. Since the process (1) is a Markov process, it is then relevant to study the exit event from Ω starting from the quasi-stationary distribution ν h , i.e. X 0 ∼ ν h . As a consequence of Proposition 3, the exit time is exponentially distributed and is independent of the next visited state. These two properties are the fundamental features of kinetic Monte Carlo methods, see indeed (2) and (3). It thus remains to prove that the transition rates can be computed with the Eyring-Kramers formula [START_REF] Berglund | On the noise-induced passage through an unstable periodic orbit II: General case[END_REF]. For that purpose, let us first give an expression of the transition rates. Recall that (Ω i ) i=1,...,n denotes the surrounding domains of Ω (see Figure 1). For i ∈ {1, . . . , n}, we define the transition rate to go from Ω to Ω i as follows:

k L i := 1 E ν h τ Ω P ν h X τΩ ∈ ∂Ω ∩ ∂Ω i , ( 16
)
where we recall, ν h is the quasi-stationary distribution of the process (1) in Ω.

The superscript L in [START_REF] Day | Recent progress on the small parameter exit problem[END_REF] indicates that the microscopic evolution of the system is governed by the overdamped Langevin process [START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF]. Notice that, using Proposition 3, it holds for all i ∈ {1, . . . , n}:

P ν h X τΩ ∈ ∂Ω ∩ ∂Ω i = k L i n ℓ=1 k L ℓ .
Thus, the expressions ( 16) are compatible with the use of a kinetic Monte Carlo algorithm, see ( 2) and (4). Indeed, starting from the quasi-stationary distribution ν h , the exit event from Ω can be exactly modeled using the rates ( 16): the exit time is exponentially distributed with parameter n ℓ=1 k L ℓ , independent of the exit point, and the exit point is in

∂Ω i ∩ ∂Ω with probability k L i / n ℓ=1 k L ℓ .
The remaining question is thus following: does the transition rate ( 16) satisfy the Eyring-Kramers law (5) in the limit h → 0?

Notice that, using Proposition 3, for i ∈ {1, . . . , n}, the transition rate defined by [START_REF] Day | Recent progress on the small parameter exit problem[END_REF] writes:

k L i = - h 2 ∂Ω∩∂Ωi ∂ n u h (z) e -2 h f (z) σ(dz) Ω u h e -2 h f , ( 17
)
where we recall, u h is the eigenfunction associated with the principal eigenvalue λ h of L D,(0) f,h . The remainder of this work is dedicated to the presentation of recent results in [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points[END_REF], [START_REF] Di Gesù | Sharp asymptotics of the first exit point density[END_REF] and [START_REF] Nectoux | Analyse spectrale et analyse semi-classique pour la métastabilité en dynamique moléculaire[END_REF] which aim at studying the asymptotic behaviour of the exit event (τ Ω , X τΩ ) from a metastable domain Ω in the limit h → 0. In particular, the results give a sharp asymptotic formula of the transition rates [START_REF] Day | Conditional exits for small noise diffusions with characteristic boundary[END_REF] when h → 0.

Remark 8. If one wants to recover the expression of the prefactor [START_REF] Bovier | Metastability in reversible diffusion processes. I. sharp asymptotics for capacities and exit times[END_REF], one has to multiply by 1 2 the expression ( 16). This can be explained as follows. Once the process (1) reaches ∂Ω ∩ ∂Ω i , it has, in the limit h → 0, a one-half probability to come back in Ω and a one-half probability to go in Ω i . If z i is a non degenerate saddle point of f , this result is not difficult to prove in dimension 1. Indeed, it is proved in [57, Section A.1.2.2], that when reaching ∂Ω ∩ ∂Ω i , the probability that the process (1) goes in Ω i is 1 2 + O(h) in the limit h → 0. To extend this result to higher dimensions, one can use a suitable set of coordinates around z i .

Main results on the exit event

In all this section, Ω ⊂ R d is C ∞ open, bounded and connected, and f ∈ C ∞ (Ω, R) 3 . The purpose of this section is to present recent results obtained in [START_REF] Di Gesù | Sharp asymptotics of the first exit point density[END_REF] and [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points[END_REF]. Both [START_REF] Di Gesù | Sharp asymptotics of the first exit point density[END_REF] and [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points[END_REF] are mainly concerned with studying the asymptotic behaviour when h → 0 of the exit law of a domain Ω of the process (1). In [START_REF] Di Gesù | Sharp asymptotics of the first exit point density[END_REF], when Ω only contains one local minimum of f and ∂ n f > 0 on ∂Ω, we obtain sharp asymptotic equivalents when h → 0 of the probability that the process (1) leaves Ω through a subset Σ of ∂Ω starting from the quasi-stationary distribution or from a deterministic initial condition in Ω. Then, these asymptotic equivalents are used to compute the asymptotic behaviour of the transition rates [START_REF] Day | Recent progress on the small parameter exit problem[END_REF]. In [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points[END_REF], we explicit a more general setting than the one considered in [START_REF] Di Gesù | Sharp asymptotics of the first exit point density[END_REF] where we identify the most probable places of exit of Ω as well as their relative probabilities starting from the quasi-stationary distribution or deterministic initial conditions in Ω. More precisely, we consider in [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points[END_REF] the case when Ω contains several local minima of f and |∇f | = 0 on ∂Ω.

Sharp asymptotic estimates on the exit event from a domain

In this section, we present the results of [START_REF] Di Gesù | Sharp asymptotics of the first exit point density[END_REF] which give sharp asymptotic estimates on the law of X τΩ and on the expectation of τ Ω when h → 0. These results give in particular the asymptotic estimates of the transition rates (k L j ) j=1,...,n defined in [START_REF] Day | Recent progress on the small parameter exit problem[END_REF].

f (z 1 ) = . . . = f (z n0 ) < f (z nn 0 +1 ) ≤ . . . ≤ f (z n ).
On Figure 3 outside Ω in a neighborhood of z i also implies that z i is a geometric saddle point of f as defined in Remark 9. In dimension one, such a construction was considered by Kramers in [START_REF] Kramers | Brownian motion in a field of force and the diffusion model of chemical reactions[END_REF] to derive formulas for transition rates, as explained in [START_REF] Matkowsky | A singular perturbation approach to kramers diffusion problem[END_REF]. Moreover, as in Remark 8, it can be proved in dimension 1 (exactly as in [57, Section A.1.2.2]), that when reaching ∂Ω ∩ ∂Ω i , the probability that the process (1) goes in Ω i is 1 2 + O(h) when h → 0. To extend this result to higher dimensions, one can use a suitable set of coordinates around z i .

Let us now define g :

Ω → R + by g(x) = ∇f (x) when x ∈ Ω and g(x) = ∇ T f (x) when x ∈ ∂Ω, ( 18
)
where

∇ T f is the tangential gradient of f in ∂Ω. Let us recall that for x ∈ ∂Ω, ∇ T f (x) is defined by ∇ T f (x) = ∇f (x) -(∇f (x) • n) n,
where n is the unit outward normal to ∂Ω at x. The assumptions one needs to state the results in this section depend on the Agmon distance in Ω between the points (z i ) i=1,...,n . The Agmon distance is defined as follows: for any x ∈ Ω and y ∈ Ω,

d a (x, y) := inf γ∈Lip(x,y) L(γ, (0, 1)), (19)
where Lip(x, y) is the set of Lipschitz curves γ : [0, 1] → Ω which are such that γ(0) = x and γ(1) = y, and where for γ ∈ Lip(x, y), L(γ, (0, 1)) =

1 0 g(γ(t))|γ ′ (t)|dt.
Remark 11. Let us give some common points and differences between the quasipotential V introduced in [26, Section 2] and the Agmon distance [START_REF] Day | Mathematical Approaches to the Problem of Noise-Induced Exit[END_REF]. Contrary to the quasipotential V , the Agmon distance ( 19) is symmetric. Moreover, let us consider x = y ∈ Ω such that there exists a curve γ :

[0, 1] → Ω with d dt γ(t) = -∇f (γ(t)), γ(0) = x and γ(1) = y.
Then, the Agmon distance ( 19) between x and y equals f

(x) -f (y) = V (y, x) > 0 but V (x, y) = 0 = d a (x, y).
Finally, let us define the following sets. For i ∈ {1, . . . , n}, B zi is the basin of attraction of z i for the dynamics d dt x(t) = -∇ T f x(t) in ∂Ω, i.e. B zi = {y ∈ ∂Ω, lim t→∞ x(t) = z i if x(0) = y} (see for instance Figure 3). Moreover, one defines for i ∈ {1, . . . , n}:

B c zi := ∂Ω \ B zi .
Main results. Let us now give the main results of this section.

Proposition 4. Let u h be the eigenfunction associated with the principal eigenvalue λ h of L D,(0) f,h which satisfies normalization (13). Let us assume that the hypotheses [H1], [H2], [H3]

are satisfied. Then, in the limit h → 0, one has:

λ h = det Hessf (x 0 ) √ πh n0 i=1 ∂ n f (z i ) det Hessf | ∂Ω (z i ) e -2 h (f (z1)-f (x0)) (1 + O(h)) (20) and Ω u h (x) e -2 h f (x) dx = π d 4 (det Hessf (x 0 )) 1/4 h d 4 e -1 h f (x0) (1 + O(h)). ( 21
)
Furthermore, one obtains the following theorem on the asymptotic behaviour of ∂ n u h , which is one of the main results of [START_REF] Di Gesù | Sharp asymptotics of the first exit point density[END_REF].

Theorem 1. Let us assume that [H1], [H2], and [H3]

are satisfied and that the following inequalities hold:

f (z 1 ) -f (x 0 ) > f (z n ) -f (z 1 ) (22)
and for all i ∈ {1, . . . , n},

d a (z i , B c zi ) > max[f (z n ) -f (z i ), f (z i ) -f (z 1 )]. ( 23
)
Let i ∈ {1, . . . , n} and Σ i ⊂ ∂Ω be an open set containing z i and such that Σ i ⊂ B zi . Let u h be the eigenfunction associated with the principal eigenvalue of L D,(0) f,h which satisfies [START_REF] Davies | Metastable states of symmetric Markov semigroups II[END_REF]. Then, in the limit h → 0:

Σi ∂ n u h e -2 h f = C i (h) e -2f (z i )-f (x 0 ) h (1 + O(h)) , ( 24
)
where

C i (h) = - (det Hessf (x 0 )) 1/4 ∂ n f (z i )2π d-2 4 det Hessf | ∂Ω (z i ) h d-6
4 .

These results have the following consequences.

Corollary 1.

Let us assume that all the assumptions of Theorem 1 are satisfied. Let i ∈ {1, . . . , n} and Σ i ⊂ ∂Ω be an open set containing z i and such that Σ i ⊂ B zi . Then, in the limit h → 0:

P ν h [X τΩ ∈ Σ i ] = ∂ n f (z i ) det Hessf | ∂Ω (z i ) n0 k=1 ∂ n f (z k ) det Hessf | ∂Ω (z k ) -1 × e -2 h (f (zi)-f (z1)) (1 + O(h)), ( 25
)
where ν h is the quasi-stationary distribution of the process [START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF] in Ω (see [START_REF] Day | On the exponential exit law in the small parameter exit problem[END_REF]). Moreover, if Σ i is the common boundary between the state Ω and a state Ω i , then, when h → 0

k L i = 1 √ πh ∂ n f (z i ) det Hessf (x 0 ) det Hessf | ∂Ω (z i ) e -2 h (f (zi)-f (x0)) (1 + O(h)), ( 26
)
where k L i is the transition rate (16) to go from Ω to Ω i . Notice that since z i is not a saddle point of f , the prefactor in ( 26) is not the prefactor 1 2 A i (see Remark 10 for the explanation of the multiplicative term 1 2 ), where A i is defined by ( 6), but it is actually the expected prefactor for a generalized saddle point of f (see Remarks 9 and 10).

The asymptotic estimate ( 25) is a consequence of Proposition 4, Theorem 1 together with [START_REF] Day | On the asymptotic relation between equilibrium density and exit measure in the exit problem[END_REF], and ( 26) is a consequence of Proposition 4, Theorem 1 and [START_REF] Day | Conditional exits for small noise diffusions with characteristic boundary[END_REF]. The main difficulty is to prove [START_REF] Eckhoff | Precise asymptotics of small eigenvalues of reversible diffusions in the metastable regime[END_REF] which requires a sharp equivalent of the quantity Σi ∂ n u h e -2 h f when z i is not a global minimum of f on ∂Ω, i.e. when i ∈ {n 0 + 1, . . . , n}. In [START_REF] Di Gesù | Sharp asymptotics of the first exit point density[END_REF], numerical simulations are provided to check that (25) holds and to discuss the necessity of the assumptions [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points[END_REF] to obtain [START_REF] Fan | Autonomous basin climbing method with sampling of multiple transition pathways: application to anisotropic diffusion of point defects in hcp Zr[END_REF]. Furthermore, in [START_REF] Di Gesù | Sharp asymptotics of the first exit point density[END_REF], the results ( 24) and ( 25) are generalized to sets Σ ⊂ ∂Ω which do not necessarily contain a point z ∈ {z 1 , . . . , z n }: this is the other main results of [START_REF] Di Gesù | Sharp asymptotics of the first exit point density[END_REF] which is not presented here. Moreover, with the help of "leveling" results on the function x → E x [F (X τΩ )], we generalized [START_REF] Fan | Autonomous basin climbing method with sampling of multiple transition pathways: application to anisotropic diffusion of point defects in hcp Zr[END_REF] to deterministic initial conditions in Ω (i.e. when X 0 = x ∈ Ω) which are the initial conditions considered in the theory of large deviations [START_REF] Freidlin | Random Perturbations of Dynamical Systems[END_REF]. The proofs of Proposition 4 and Theorem 1 are based on tools from semi-classical analysis and more precisely, they are based on techniques developed in [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary[END_REF][START_REF] Helffer | Multiple wells in the semi-classical limit I[END_REF][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[END_REF][START_REF] Helffer | Puits multiples en mecanique semi-classique iv etude du complexe de witten[END_REF][START_REF] Peutrec | Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian[END_REF].

Starting points of the proofs of Proposition 4 and Theorem 1. Let us recall that u h is the eigenfunction associated with the principal eigenvalue λ

h of L D,(0) f,h
which satisfies normalization [START_REF] Davies | Metastable states of symmetric Markov semigroups II[END_REF]. In view of [START_REF] Day | On the asymptotic relation between equilibrium density and exit measure in the exit problem[END_REF] and in order to obtain [START_REF] Fan | Autonomous basin climbing method with sampling of multiple transition pathways: application to anisotropic diffusion of point defects in hcp Zr[END_REF], one wants to study the asymptotic behaviour when h → 0 of ∇u h on ∂Ω. The starting point of the proofs of Proposition 4 and Theorem 1 is the fact that ∇u h is solution to an eigenvalue problem for the same eigenvalue λ h . Indeed, recall that u h is solution to

L (0) f,h u h = λ h u h in Ω and u h = 0 on ∂Ω. If one differentiates this relation, ∇u h is solution to          L (1) f,h ∇u h = λ h ∇u h in Ω, ∇ T u h = 0 on ∂Ω, - h 2 div + ∇f • ∇u h = 0 on ∂Ω, ( 27
)
where L

(1)

f,h = -h 2 ∆ + ∇f • ∇ +
h = n. ( 28
)
Therefore, from [START_REF] Galves | Metastability for a class of dynamical systems subject to small random perturbations[END_REF], it holds

∇u h ∈ Ran π (1) h , ( 29
)
and from [START_REF] Davies | Metastable states of symmetric Markov semigroups II[END_REF] and the fact that

L (0) f,h u h , u h L 2 w = h 2 ∇u h 2 L 2 w
, one has

λ h = h 2 ∇u h 2 L 2 w . ( 30
)
Thus, to study the asymptotic behaviour when h → 0 of λ h , u h and ∇u h , we construct a suitable orthonormal basis of Ran π

(1)

h . This basis is constructed using so-called quasi-modes.

Sketch of the proofs of Proposition 4 and Theorem 1.

Let us give the sketch of the proof of ( 25) which is the main result of [START_REF] Di Gesù | Sharp asymptotics of the first exit point density[END_REF]. Recall that from Proposition 2, one works in the Hilbert space L 2 w (Ω). The spaces L 2 w (Ω) and H 1 w (Ω) (see ( 12)) extend naturally on 1-forms as follows

Λ 1 L 2 w (Ω) := u = t (u 1 , . . . , u d ) : Ω → R d , ∀k ∈ {1, . . . , d}, Ω u 2 k e -2 h f < ∞ ,
and

Λ 1 H 1 w (Ω) := u = t (u 1 , . . . , u d ) : Ω → R d , ∀(i, k) ∈ {1, . . . , d} 2 , ∂ i u k ∈ L 2 w (Ω) .
In the following, one denotes by . L 2 w (resp. . H 1 w ) the norm of L 2 w (Ω) and of Λ 1 L 2 w (Ω) (resp. H 1 w (Ω) and Λ 1 H 1 w (Ω)). Finally, ., . L 2 w stands for both the scalar product associated with the norm of L 2 w (Ω) and with the norm of Λ 1 L 2 w (Ω). In view of ( 29) and ( 28), one has for all orthonormal basis (ψ j ) j∈{1,...,n} of Ran π

(1) h , in L 2
w (Ω):

∇u h = n j=1 ∇u h , ψ j L 2 w ψ j , ( 31
)
and from [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach[END_REF], it holds

λ h = h 2 n j=1 ∇u h , ψ j L 2 w 2 . ( 32
)
In particular, one has for all k ∈ {1, . . . , n},

Σ k ∂ n u h e -2 h f = n j=1 ∇u h , ψ j L 2 w Σ k ψ j • n e -2 h f , ( 33
)
where we recall that

Σ k is an open set of ∂Ω such that z k ∈ Σ k and Σ k ⊂ B z k .
Step 1: approximation of u h . Under [H1], [H2], and [H3], it is not difficult to find a good approximation of u h . Indeed, let us consider,

ũ := χ χ L 2 w , ( 34
)
where χ ∈ C ∞ c (Ω, R + ) and χ = 1 on {x ∈ Ω, d(x, ∂Ω) ≥ ε} where ε > 0. In particular, for ε small enough, χ = 1 in a neighboorhood of x 0 (which is assumed in the following). Let us explain why ũ is a good approximation of u h . Since L

D,(0) f,h is self adjoint on L 2 w (Ω), one has (1 -π (0) h )ũ 2 L 2 w ≤ C √ h L D,(0) f,h ũ, ũ L 2 w = Ch 2 √ h Ω |∇χ| 2 e -2 h f Ω χ 2 e -2 h f .
Since f (x 0 ) = min Ω f < min ∂Ω f and x 0 is the unique global minimum of f on Ω (see [H2]), one has using Laplace's method (x 0 is a non degenerate critical point of f and χ(x 0 ) = 1):

Ω χ 2 e -2 h f = (πh) d 2 detHessf (x 0 ) e -2 h f (x0) (1 + O(h)).
Therefore, for any δ > 0, choosing ε small enough, it holds when h → 0:

(1 -π (0) h )ũ 2 L 2 w = O(e -2 h (f (z1)-f (x0)-δ) ),
and thus:

π (0) h ũ = ũ + O(e -1 h (f (z1)-f (x0)-δ) ) in L 2 w (Ω).
From [START_REF] Gol'dshtein | Hodge decompositions with mixed boundary conditions and applications to partial differential equations on Lipschitz manifolds[END_REF] and since χ ≥ 0, one has for any δ > 0 (choosing ε small enough), when h → 0

u h = π (0) h ũ π (0) h ũ L 2 w = ũ + O(e -1 h (f (z1)-f (x0)-δ) ) in L 2 w (Ω). ( 35
)
Since ũ L 2 w = 1, this last relation justifies that ũ is a good approximation of u h in L 2 w (Ω). Notice that ( 35) implies [START_REF] Devinatz | The asymptotic behavior of the solution of a singularly perturbed dirichlet problem[END_REF].

Step 2: construction of a basis of Ran π

(1)

h to prove Theorem 1. In view of ( 33), the idea is to construct a family of 1-forms ( ψ j ) j∈{1,...,n} which forms, when projected on Ran π

(1)
h , a basis of Ran π (1) h and which allows to obtain sharp asymptotic estimates on ∂ n u h on all the Σ j 's when h → 0. In the literature, such a 1-form ψ j is called a quasi-mode (for L D, [START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF] f,h ). A quasi-mode for L D,(1) f,h is a smooth 1-form w such that for some norm, it holds when h → 0:

π (1) h w = w + o(1), ( 36
)
To prove Theorem 1, one of the major issues is the construction of a basis ( ψ j ) j∈{1,...,n} so that the remainder term o( 1) in ( 36), when w = ψ k , is of the order (see ( 23))

(1 -π (1) h ) ψ k H 1 w = O e -1 h max[f (zn)-f (z k ), f (z k )-f (z1)] . ( 37
)
This implies that π

(1) h ψ j j∈{1,...,n} is a basis of Ran π

(1) h and above all, after a Gram-Schmidt procedure on π

(1) h ψ j j∈{1,...,n} , when h → 0, that for all k ∈ {1, . . . , n} (see [START_REF] Helffer | Multiple wells in the semi-classical limit III-Interaction through non-resonant wells[END_REF]):

Σ k ∂ n u h e -2 h f = n j=1 ∇ũ, ψ j L 2 w Σ k ψ j • n e -2 h f + O e -2f (z k )-f (x 0 )+c h ( 38
)
and (see [START_REF] Helffer | Multiple wells in the semi-classical limit I[END_REF])

λ h = h 2 n j=1 | ∇ũ, ψ j L 2 w | 2 + O e -2 h (f (z1)-f (x0)+c) (39)
for some c > 0 independent of h. Here, we recall, ũ (see [START_REF] Helffer | Puits multiples en limite semi-classique. II. Interaction moléculaire[END_REF]) is a good approximation of u h (see [START_REF] Helffer | Puits multiples en mecanique semi-classique iv etude du complexe de witten[END_REF]). Let us now explain how we will construct the family ψ j j∈{1,...,n} in order to obtain [START_REF] Jakab | On the regularity of differential forms satisfying mixed boundary conditions in a class of Lipschitz domains[END_REF] and [START_REF] Kamin | Elliptic perturbation of a first order operator with a singular point of attracting type[END_REF]. Then, we explain how the terms Σj

ψ j • n e -2 h f j∈{1,...,n}
and ∇ũ, ψ j L 2 w j∈{1,...,n} appearing in [START_REF] Jakab | On the regularity of differential forms satisfying mixed boundary conditions in a class of Lipschitz domains[END_REF] and ( 39) are computed.

Step 2a: construction of the family ( ψ j ) j∈{1,...,n} . To construct each 1-form ψ j , the idea is to construct an operator L

(1)

f,h with mixed tangential Dirichlet and

Neumann boundary conditions on a domain Ωj ⊂ Ω which is such that {z 1 , . . . , z n } ∪ {x 0 } ∩ Ωj = {z j }. For j ∈ {1, . . . , n}, ψ j is said to be associated with the generalized saddle point z j . The goal of the boundary conditions is to ensure that when h → 0, each of these operators has only one exponentially small eigenvalue (i.e. this eigenvalue is O e -c h for some c > 0 independent of h), the other eigenvalues being larger than √ h. Then, we show that each of these small eigenvalues actually equals 0 using the Witten complex structure associated with these boundary conditions on ∂ Ωj . To construct such operators L (1) f,h with mixed boundary conditions on Ωj , the recent results of [START_REF] Jakab | On the regularity of differential forms satisfying mixed boundary conditions in a class of Lipschitz domains[END_REF] and [START_REF] Gol'dshtein | Hodge decompositions with mixed boundary conditions and applications to partial differential equations on Lipschitz manifolds[END_REF] are used. The 1-form ψ j associated with z j is then defined using an eigenform v (1) h,j associated with the eigenvalue 0 of the operator L [START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF] f,h associated with mixed boundary conditions on Ωj :

ψ j := χ j v (1) h,j χ j v (1) h,j L 2 w , ( 40
)
where χ j is a well chosen cut-off function with support in Ωj . Notice that for j ∈ {1, . . . , n}, the quasi-mode ψ j is not only constructed in a neighbourhood of z j : it has a support as large as needed in Ω. This is a difference with previous construction in the literature, such as [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary[END_REF]. We need such quasi-modes for the following reasons. Firstly, we compute the probability that the process [START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF] leaves Ω through open sets Σ j which are arbitrarily large in B zj . Secondly, we use the fact that the quasi-mode ψ j decreases very fast away from z j to get [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF]. This is needed to state the hypothesis [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 terms of Agmon distances, see next step.

Step 2b: Accuracy of the quasi-mode ψ j for j ∈ {1, . . . , n}. To obtain a sufficiently small remainder term in (36) (to get [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF] and then [START_REF] Jakab | On the regularity of differential forms satisfying mixed boundary conditions in a class of Lipschitz domains[END_REF]), one needs to quantify the decrease of the quasi-mode ψ j outside a neighboorhood of z j . This decrease is obtained with Agmon estimates on v

(1)
h,j which allow to localize ψ j in a neighboorhood of z j . For j ∈ {1, . . . , n}, we prove the following Agmon estimate on v

(1) h,j :

χ j v (1) h,j e 1 h da(.,zj) H 1 w = O(h -N ), (41)
for some N ∈ N and where d a is the Agmon distance defined in [START_REF] Day | Mathematical Approaches to the Problem of Noise-Induced Exit[END_REF]. To obtain (41), we study the properties of this distance. The boundary of Ω introduces technical difficulties. The Agmon estimate ( 41) is obtained adapting to our case techniques developed in [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary[END_REF][START_REF] Peutrec | Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian[END_REF]. For all j ∈ {1, . . . , n}, using the fact that (1π [START_REF] Kipnis | The metastable behavior of infrequently observed, weakly random, one-dimensional diffusion processes[END_REF], one shows that

(1) h ) ψ j 2 L 2 w ≤ C √ h L D,(1) f,h ψ j , ψ j L 2 w and (
(1 -π (1) h ) ψ j 2 L 2 w ≤ C h -q e -2 h infsupp∇χ j da(.,zj) ,
for some q > 0. Thus, in order to get [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF], the support of ∇χ j has to be arbitrarily close to x 0 and B c zj . This explains the assumptions ( 22) and ( 23), and the fact that the quasi-mode ψ j is not constructed in a neighboorhood of z j but in a domain Ωj arbitrarily large in Ω. This is one of the main differences compared with [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary[END_REF]. At the end of this step, one has a family ( ψ j ) j∈{1,...,n} which satisfies [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF]. This allows us to obtain, in the limit h → 0 (see [START_REF] Jakab | On the regularity of differential forms satisfying mixed boundary conditions in a class of Lipschitz domains[END_REF]), for some c > 0 independent of h and for all k ∈ {1, . . . , n}:

Σ k ∂ n u h e -2 h f = n j=1 ∇ũ, ψ j L 2 w Σ k ψ j • n e -2 h f + O e -2f (z k )-f (x 0 )+c h . Etape 3: computations of Σj ψ j •n e -2 h f j∈{1,...,n}
and ∇ũ, ψ j L 2 w j∈{1,...,n} .

In view of ( 38) and ( 39), for all j ∈ {1, . . . , n}, one needs to compute the terms

Σj ψ j • n e -2 h f and ∇ũ, ψ j L 2 w .
To do that, we use for all j ∈ {1, . . . , n} a WKB approximation of v

(1)
h,j , denoted by v

(1) zj ,wkb . In the literature we follow, v

(1) zj ,wkb is constructed in a neighboorhood of z j (see [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary[END_REF][START_REF] Peutrec | Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian[END_REF]). To prove Theorem 1, we extend the construction of v (1) zj ,wkb to neighbourhoods in Ω of arbitrarily large closed sets included in B zj (indeed, there is no restriction on the size of Σ j in B zj ). Then, the comparison between v are computed, one concludes the proof of (20) using [START_REF] Kamin | Elliptic perturbation of a first order operator with a singular point of attracting type[END_REF] and the proof of ( 24) using [START_REF] Jakab | On the regularity of differential forms satisfying mixed boundary conditions in a class of Lipschitz domains[END_REF].

Most probable exit points from a bounded domain

Setting and motivation. In this section, we present recent results from [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points[END_REF] on the concentration of the law of X τΩ on a subset of argmin ∂Ω f = {z ∈ ∂Ω, f (z) = min ∂Ω f } when h → 0 in a more general geometric setting than the one of Theorem 1. The main purpose of these results is to prove an asymptotic formula when h → 0 for the concentration of the law of X τΩ on a set of points of argmin ∂Ω f when Ω contains several local minima of f and when ∂ n f is not necessarily positive on ∂Ω.

Let Y ⊂ ∂Ω. We say that the law of

X τΩ concentrates on Y if for all neigh- borhood V Y of Y in ∂Ω, one has lim h→0 P [X τΩ ∈ V Y ] = 1,
and if for all x ∈ Y and all neighborhood V x of x in ∂Ω , it holds:

lim h→0 P [X τΩ ∈ V x ] > 0.
In [START_REF] Matkowsky | The exit problem for randomly perturbed dynamical systems[END_REF][START_REF] Matkowsky | The exit problem: a new approach to diffusion across potential barriers[END_REF][START_REF] Naeh | A direct approach to the exit problem[END_REF], when ∂ n f (x) = 0 for all x ∈ ∂Ω or when ∂ n f (x) > 0 for all x ∈ ∂Ω (and with additional assumptions on f ), it has been shown that the law of X τΩ concentrates on points where f attains its minimum on ∂Ω (see [START_REF] Chandrasekhar | Stochastic problems in physics and astronomy[END_REF]). Later on, it has been proved in [START_REF] Day | On the asymptotic relation between equilibrium density and exit measure in the exit problem[END_REF][START_REF] Day | Recent progress on the small parameter exit problem[END_REF][START_REF] Kamin | Elliptic perturbation of a first order operator with a singular point of attracting type[END_REF][START_REF] Kamin | On elliptic singular perturbation problems with turning points[END_REF][START_REF] Perthame | Perturbed dynamical systems with an attracting singularity and weak viscosity limits in hamilton-jacobi equations[END_REF] when ∂ n f > 0 on ∂Ω and f has a unique non degenerate critical point in Ω (which is necessarily its global minimum in Ω). Tools developed in semi-classical analysis allow us to generalize this geometric setting. For instance, we consider several critical points of f in Ω and we drop the assumptions ∂ n f > 0 on ∂Ω (however we do not consider the case when f has saddle points on ∂Ω). Assuming that f and f | ∂Ω are Morse functions, and |∇f | = 0 on ∂Ω, we raise the following questions:

-What are the geometric conditions ensuring that, when X 0 ∼ ν h , the law of X τΩ concentrates on points where f attains its minimum on ∂Ω (or a subset of these points)? -What are the conditions which ensure that these results extend to some deterministic initial conditions in Ω ?

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[END_REF] aim at answering these questions. Let us recall that when f and f | ∂Ω are Morse functions and when |∇f | = 0 on ∂Ω, the elements of the set

{z is a local minimim of f | ∂Ω } ∩ {z ∈ ∂Ω, ∂ n f (z) > 0} (42)
are the generalized saddle points of f on ∂Ω and play the role of saddle points of f on ∂Ω, see Remark 9. Before stating the main results 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], let us discuss the two questions above with one-dimensional examples.

Remark 12.

The assumption that the drift term b in (1) is of the form b = -∇f is essential here to the existence of a limiting exit distribution of Ω when h → 0. If it is not the case and when for instance the boundary of Ω is a periodic orbit of the dynamics d dt x(t) = b x(t) , the phenomenon of cycling discovered by Day in [START_REF] Day | Conditional exits for small noise diffusions with characteristic boundary[END_REF][START_REF] Day | Exit cycling for the van der Pol oscillator and quasipotential calculations[END_REF] prevents the existence of a limiting exit distribution when h → 0. We also refer to [START_REF] Berglund | Noise-induced phase slips, log-periodic oscillations, and the Gumbel distribution[END_REF][START_REF] Berglund | On the noise-induced passage through an unstable periodic orbit I: Two-level model[END_REF][START_REF] Berglund | On the noise-induced passage through an unstable periodic orbit II: General case[END_REF] for the study of this phenomenon of cycling.

One-dimensional examples.

To discuss the two questions raised in the previous section, one considers two one-dimensional examples.

Example 1. The goal is here to construct a one-dimensional example for which, starting from the global minimum of f in Ω or from the quasi-stationary distribution ν h , the law of X τΩ does not concentrate on points where f attains its minimum on ∂Ω. To this end, let us consider the function f represented in Figure 4 for which one has the following result. 4). Then, for all x ∈ (c, z 2 ], there exists ε > 0 such that when h → 0: Moreover, there exists ε > 0 such that when h → 0:

Proposition 5. Let z 1 < z 2 and f ∈ C ∞ ([z 1 , z 2 ], R) be a Morse function. Let us assume that f (z 1 ) < f (z 2 ), {x ∈ [z 1 , z 2 ], f ′ (x) = 0} = {c, x 1 } with z 1 < c < x 1 < z 2 and f (x 1 ) < f (z 1 ) < f (z 2 ) < f (c) (see Figure
P x [X τ (z 1 ,z 2 ) = z 1 ] = O(e -ε h ) and thus P x [X τ (z 1 ,z 2 ) = z 2 ] = 1 + O(e -ε h ). x1 • {f = min ∂Ω f } z1 • z2 • c •
P ν h [X τ (z 1 ,z 2 ) = z 1 ] = O(e -ε h ) and thus P ν h [X τ (z 1 ,z 2 ) = z 2 ] = 1 + O(e -ε h ),
where ν h is the quasi-stationary distribution of the process (1) in (z 1 , z 2 ).

The proof of Proposition 5 is based on the fact that in one dimension, explicit formulas can be written for x → P x [X τ (z 1 ,z 2 ) = z j ] (j ∈ {1, 2}), see [START_REF] Nectoux | Analyse spectrale et analyse semi-classique pour la métastabilité en dynamique moléculaire[END_REF]Section A.5.3.1] or [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points[END_REF]. According to Proposition 5, when h → 0 and when X 0 = x ∈ (c, z 2 ) or X 0 ∼ ν h , the process (1) leaves Ω = (z 1 , z 2 ) through z 2 . However, the generalized saddle point z 2 (see [START_REF] Kramers | Brownian motion in a field of force and the diffusion model of chemical reactions[END_REF]) is not the global minimum of f on ∂Ω. This fact can be explained as follows: the potential barrier f (c)f (x 1 ) is larger than the potential barrier f (z 2 )f (x 1 ). Thus, the law of X τΩ when X 0 = x ∈ (c, z 2 ) cannot concentrate on z 1 since it is less costly to leave Ω through z 2 rather than to cross the barrier f (c)f (x 1 ) to exit through z 1 . Moreover, it can be proved that the quasi-stationary distribution ν h concentrates in any neighborhood of x 1 in the limit h → 0, which explains why the law of X τΩ when X 0 ∼ ν h also concentrates on z 2 . Concerning the two questions raised in the previous section, this example indicates that in the small temperature regime, there exist cases for which the process (1), starting from the global minimum of f in Ω or from ν h , leaves Ω through a point which is not a global minimum of f | ∂Ω .

This example also suggests the following. If one wants the law of X τΩ to concentrate when h → 0 on points in ∂Ω where f attains its minimum, one should exclude cases when the largest timescales for the diffusion process in Ω are not related to energetic barriers involving points of ∂Ω where f | ∂Ω attains its minimum. In order to exclude such cases, we will assume in the following that the closure of each of the connected components of {f < min ∂Ω f } intersects ∂Ω.

Notice that if one modifies the function f in the vicinity of z 1 such that ∂ n f (z 1 ) > 0 and argmin Ω f = {x 1 }, z 1 is then a generalized order one saddle point and the previous conclusions remain unchanged.

Example 2. Let us construct a one-dimensional example for which the concentration of the law of X τΩ on argmin ∂Ω f is not the same starting from the global minima of f in Ω or from the quasi-stationary distribution ν h . For this purpose, let us consider

z 1 > 0, z 2 := -z 1 , z = 0 and f ∈ C ∞ ([z 1 , z 2 ], R) such that f is a Morse and even function, {x ∈ [z 1 , z 2 ], f ′ (x) = 0} = {x 1 , z, x 2 }, (43)
where

z 1 < x 1 < z < x 2 < z 2 , f (z 1 ) = f (z 2 ) > f (x 1 ) = f (x 2 ), f (z 1 ) < f (z). ( 44
)
A function f satisfying ( 43) and ( 44) is represented in Figure 5. One has the following result.

{f = min ∂Ω f } z1 • z2 • z • x1 • x2 • Fig. 5.
One-dimensional example where ( 43) and ( 44) are satisfied.

Proposition 6. Let z 1 > 0, z 2 := -z 1 , z = 0 and f ∈ C ∞ ([z 1 , z 2 ], R) which satisfies [START_REF] Kramers | Brownian motion in a field of force and the diffusion model of chemical reactions[END_REF] and [START_REF] Landim | A Dirichlet and a Thomson principle for nonselfadjoint elliptic operators, metastability in non-reversible diffusion processes[END_REF]. Then, one has for all h > 0,

P ν h [X τ (z 1 ,z 2 ) = z 1 ] = 1 2 and P ν h [X τ (z 1 ,z 2 ) = z 2 ] = 1 2 , ( 45
)
where ν h is the quasi-stationary distribution of the process (1) in (z 1 , z 2 ). Moreover, for all x ∈ (z 1 , z), there exists c > 0 such that when h → 0,

P x [X τ (z 1 ,z 2 ) = z 1 ] = 1 + O(e -c h ) and P x [X τ (z 1 ,z 2 ) = z 2 ] = O(e -c h ), (46)
and for all x ∈ (z, z 2 ), there exists c > 0 such that when h → 0

P x [X τ (z 1 ,z 2 ) = z 1 ] = O(e -c h ) and P x [X τ (z 1 ,z 2 ) = z 2 ] = 1 + O(e -c h ). ( 47
)
The asymptotic estimate ( 45) is a consequence of the fact that f is an even function (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]Section 1]). The asymptotic estimates ( 46) and ( 47) are proved exactly as Proposition 5, 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]Section 1]. Let us also mention that Proposition 6 is a consequence of the results [START_REF] Peutrec | Repartition of the quasi-stationary distribution and first exit point density for a double-well potential[END_REF]. Concerning the two questions raised in the previous section, Proposition 6 shows that, when f satisfies ( 43) and ( 44), the concentration of the law of X τΩ on {z 1 , z 2 } is not the same starting from x ∈ (z 1 , z 2 ) \ {z} or from ν h . This is due to the fact that in this case the quasistationary distribution ν h has an equal repartition in all disjoint neighboorhoods of x 1 and x 2 , i.e. for every (a (see [START_REF] Peutrec | Repartition of the quasi-stationary distribution and first exit point density for a double-well potential[END_REF]). When X 0 = x ∈ (z 1 , z 2 ) \ {z}, the asymptotic estimates ( 46) and ( 47) can be explained by the existence of a barrier f (z)f (x 1 ) which is larger than f (z 1 )f (x 1 ). In order to exclude such cases, we will assume in the following that there exists a connected component

C of {f < min ∂Ω f }, such that argmin Ω f ⊂ C.
Main results on the exit point distribution. In this section, a simplified version of 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[END_REF] is presented. The aim is to exhibit a simple geometric setting for which, on the one hand, the law of X τΩ concentrates on the same points of ∂Ω when X 0 ∼ ν h or X 0 = x ∈ Ω for some x ∈ {f < min ∂Ω f } and, on the other hand, this concentration occurs on generalized saddle points of f which belong to argmin ∂Ω f . To this end, let us define the two following assumptions:

-[H-Morse] The function f : Ω → R is C ∞ . The functions f : Ω → R
} ⊂ {z ∈ ∂Ω, ∂ n f (z) > 0} ∩ argmin ∂Ω f. ( 49
)
Remark 14. Under [H-Min], the normal derivative of f can change sign and the function f can have saddle points in Ω higher than min ∂Ω f , see for instance Figure 6.

As shown in the following theorem, the assumption [H-Min] ensures that the quasi-stationary distribution ν h concentrates in neighborhoods of the global minima of f in C and, starting from x ∈ C or from ν h , that the concentration of the law of X τΩ when h → 0 occurs on the set of generalized saddle points {z 1 , . . . , z k0 } (see [START_REF] Marcelin | Contribution à l'étude de la cinétique physico-chimique[END_REF]). Notice that the assumption [H-Min] is not satisfied in the two examples given in the previous section (see Figures 4 and5). (see [START_REF] Day | On the exponential exit law in the small parameter exit problem[END_REF]). Let V be an open subset of Ω. Then, if V ∩ argmin C f = ∅, one has in the limit h → 0:

Theorem 2. Let us assume that the hypotheses [H-Morse] and [H-Min

ν h V = x∈V∩argmin C f det Hessf (x) -1 2 x∈argmin C f det Hessf (x) -1 2 1 + O(h) .
When V ∩ argmin C f = ∅, there exists c > 0 such that when h → 0:

ν h V = O e -c h .
In addition, let F ∈ C ∞ (∂Ω, R). Then, when h → 0:

E ν [F (X τΩ )] = k0 i=1 F (z i ) a i + O(h 1 4 ), ( 50
)
where for i ∈ {1, . . . , k 0 },

a i = ∂ n f (z i ) det Hessf ∂Ω (z i )   k0 j=1 ∂ n f (z j ) det Hessf ∂Ω (z j )   -1 . ( 51
)
Finally, (50) holds when X 0 = x ∈ C.

Remark 15. In [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points[END_REF], one also gives sharp asymptotic estimates of λ h and ∂ n u h in a more general setting than the one of Theorem 2 (for instance, we study the case when f has local minima higher than min ∂Ω f ). However, in [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points[END_REF], we do not study the precise asymptotic behaviour of X τΩ when h → 0 near generalized saddle points z of f on ∂Ω which are such that f (z) > min ∂Ω f as we did in [START_REF] Di Gesù | Sharp asymptotics of the first exit point density[END_REF] (see Corollary 1). Finally, in [START_REF] Di Gesù | The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points[END_REF], the optimality of the remainder term O(h 1 4 ) in ( 50) is discussed and improved in some situations.

Ideas and sketch of the proof of Theorem 2. In this section, one gives the sketch of the proof of (50) which is the main result of Theorem 2. Recall that from [START_REF] Day | On the asymptotic relation between equilibrium density and exit measure in the exit problem[END_REF]

, for F ∈ C ∞ (∂Ω, R) E ν h F (X τΩ ) = - h 2λ h Σ F ∂ n u h e -2 h f Ω u h e -2 h f
, where u h is the eigenfunction associated with the principal eigenvalue λ h of L D,(0) f,h . Therefore, to prove [START_REF] Mathieu | Spectra, exit times and long time asymptotics in the zero-white-noise limit[END_REF], one studies the asymptotic behaviour when h → 0 of the following quantities

λ h , ∂ n u h and Ω u h e -2 h f . ( 52
)
Under the assumptions [H-Morse] and [H-Min], one defines

m 0 := Card {z ∈ Ω, z is a local minimum of f } and m 1 : = Card {z is a local minimum of f | ∂Ω } ∩ {z ∈ ∂Ω, ∂ n f (z) > 0} + Card {z is saddle point of f } . ( 53
)
The integer m 1 is the number of generalized saddle points of f in Ω (see [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary[END_REF]Section 5.2]). To study the asymptotic behaviour when h → 0 of the quantities involved in [START_REF] Matkowsky | The exit problem: a new approach to diffusion across potential barriers[END_REF], the starting point is to again observe that ∇u h is solution to an eigenvalue problem for the same eigenvalue λ h (as already explained at the end of Section 2.1). Indeed, ∇u h is solution to (see [START_REF] Galves | Metastability for a class of dynamical systems subject to small random perturbations[END_REF])

         L (1)
f,h ∇u h = λ h ∇u h in Ω, ∇ T u h = 0 on ∂Ω,

- h 2 div + ∇f • ∇u h = 0 on ∂Ω, (54)
where we recall that L

(1)

f,h = -h 2 ∆ + ∇f • ∇ + Hess f is an operator acting on 1-forms. Let us also recall that the operator L The aim of the map j is to associate each local minimum x of f with the connected component of {f < f (j(x))} which contains x.

The second step consists in constructing bases of Ran π To construct the family of 1-forms ( ψ j ) j∈{1,...,m1} , we proceed as follows. For each saddle point z of f in Ω, following the procedure of [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach[END_REF], one constructs a 1-form supported in a neighboorhood of z in Ω. For a local minimum z of f | ∂Ω such that ∂ n f (z) > 0, one constructs a 1-form supported in a neighboorhood of z in Ω as made in [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary[END_REF]. To construct the family of functions ( u k ) k∈{1,...,m0} , one constructs for each local minimum x of f a smooth function whose support is almost j(x) (this construction is close to the one made in [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][START_REF] Hérau | Tunnel effect and symmetries for Kramers-Fokker-Planck type operators[END_REF][START_REF] Peutrec | Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian[END_REF][START_REF] Michel | About small eigenvalues of Witten laplacian[END_REF]). The next step consists in finding a sharp asymptotic equivalent for λ h when h → 0. The quantity 2 h λ h equals the square of the smallest singular values of the finite dimensional operator

∇ : Ran π (0) h → Ran π (1)
h .

To study the asymptotic behaviour when h → 0 of this smallest singular value, one uses the bases of Ran π (0) h and Ran π [START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF] h which have been constructed previously. The analysis of this finite dimensional problem is inspired by [START_REF] Hérau | Tunnel effect and symmetries for Kramers-Fokker-Planck type operators[END_REF] and also yields the asymptotic equivalent of Ω u h e -2 h f when h → 0.

Then, we study the asymptotic behaviour of the normal derivative of u h on ∂Ω when h → 0 to deduce that the law of X τΩ concentrates when h → 0 on C ∩∂Ω = {z 1 , . . . , z k0 } when X 0 ∼ ν h . Lastly, one proves "leveling" results on the function

x → E x [F (X τΩ )]
to obtain that when X 0 = x ∈ C, the law of X τΩ also concentrates when h → 0 on {z 1 , . . . , z k0 }.

To conclude, the main results 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] are the following:

1. One uses techniques from semi-classical analysis to study the asymptotic behaviours of λ h and ∂ n u h when h → 0, and then, the concentration of the law of X τΩ on a subset of argmin ∂Ω f when X 0 ∼ ν h . 2. One identifies the points of argmin ∂Ω f where the law of X τΩ concentrates when X 0 ∼ ν h : this set of points is {z 1 , . . . , z k0 }. Moreover, explicit formulas for their relative probabilities are given (see indeed [START_REF] Matkowsky | The exit problem for randomly perturbed dynamical systems[END_REF]) as well as precise remainder terms. 3. One extends the previous results on the law of X τΩ to a deterministic initial condition in Ω: X 0 = x where x ∈ C. 4. These results hold under weak assumptions on the function f and onedimensional examples are given to explain why the geometric assumptions are needed to get them.

Conclusion.

We presented recent results which justify the use of a kinetic Monte Carlo model parametrized by Eyring-Kramers formulas to model the exit event from a metastable state Ω for the overdamped Langevin dynamics (1). Our analysis is for the moment limited to situations where |∇f | = 0 on ∂Ω, which does not allow to consider order one saddle points on ∂Ω. The extensions of [START_REF] Di Gesù | Sharp asymptotics of the first exit point density[END_REF] and [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 are currently under study are the following: the case when f has saddle points on ∂Ω and the case when the diffusion process X t = (q t , p t ) is solution to the Langevin stochastic differential equation dq t = p t dt, dp t = -∇f (q t )dtγ p t dt + hγ dB t ,

Fig. 1 .

 1 Fig. 1. Representation of the domain Ω, the surrounding domains (Ωi)i=1,...,4 of Ω, the global minimum x0 of f in Ω and {zi} = argmin ∂Ω∩Ω i f (i ∈ {1, 2, 3, 4}).

Fig. 2 .

 2 Fig. 2. Examples of two labelings of the local minima {x1, x2, x3} of f in dimension one.

  Geometric setting. Let us give the geometric setting which is considered in this section: -[H1] The function f : Ω → R and the restriction of f to Ω, denoted by f | ∂Ω , are Morse functions. Moreover, |∇f |(x) = 0 for all x ∈ ∂Ω. -[H2] The function f has a unique global minimum x 0 in Ω and min ∂Ω f > min Ω f = min Ω f = f (x 0 ). The point x 0 is the unique critical point of f in Ω. The function f | ∂Ω has exactly n ≥ 1 local minima which are denoted by (z i ) i=1,...,n . They are ordered such that f (z 1 ) ≤ . . . ≤ f (z n ). -[H3] ∂ n f (x) > 0 for all x ∈ ∂Ω. Under the assumption [H2], one denotes by n 0 ∈ {1, . . . , n} the number of global minima of f | ∂Ω , i.e.:

Remark 9 . 4 Fig. 3 .Remark 10 .

 94310 Fig. 3. Schematic representation in dimension 2 of a function f satisfying the assumptions [H1], [H2], and [H3], and of its restriction f | ∂Ω to ∂Ω. On the figure, n = 4 and n0 = 2.

  is also extended to neighbourhoods in Ω of arbitrarily large closed sets included in B zj . Once the terms Σj ψ j • n e -2 h f j∈{1,...,n} and ∇ũ, ψ j L 2 w j∈{1,...,n}

Fig. 4 .

 4 Fig. 4. Example of a function f such that, starting from the global minimum x1 of f in Ω or from the quasi-stationary distribution ν h , the law of Xτ Ω concentrates on z2 whereas f (z2) > min ∂Ω f = f (z1) .

Fig. 6 .

 6 Fig. 6. A one-dimensional example where [H-Morse] and [H-Min] are satisfied, the normal derivative of f changes sign and the function f has a saddle point in Ω higher than min ∂Ω f . In this example, {f < min ∂Ω f } is connected and thus C = {f < min ∂Ω f }. Moreover, C ∩ ∂Ω = {z1}.

has exactly m 1 eigenvalues smaller than √ h 2 (h

 2 with tangential boundary conditions[START_REF] Michel | About small eigenvalues of Witten laplacian[END_REF] is denoted by L D,[START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF] f,h . From[START_REF] Michel | About small eigenvalues of Witten laplacian[END_REF], ∇u h is an eigenform of L D,[START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF] f,h associated with λ h . The second ingredient is the following result: under the assumptions [H-Morse] and [H-Min] and when h → 0, the operator L see[START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary[END_REF] Chapter 3]). Actually, all theses small eigenvalues are exponentially small when h → 0, i.e. they are all O e -c h for some c > 0 independent of h. In particular λ h is an exponentially small eigenvalue of L ) the orthogonal projector in L 2 w (Ω) onto the m 0 (resp. m 1 ) smallest eigenvalues of L ). Then, according to the foregoing, one has when h → 0:dim Ran π (0) h = m 0 , dim Ran π Let us now explain how we prove Theorem 2. To this end, let us introduce the set of local minima of f in Ω,U Ω 0 := {x ∈ Ω, x is a local minimum of f },and the set of generalized saddle points of f in Ω,U Ω 1 = {z is a local minimum of f | ∂Ω } ∩ {z ∈ ∂Ω, ∂ n f (z) > 0} {z is a saddle point of f }.Let us recall that m 0 = Card U Ω 0 and, from[START_REF] Matkowsky | A singular perturbation approach to kramers diffusion problem[END_REF], that m 1 = Card U Ω 1 . The first step to prove Theorem 2 consists in constructing two maps j and j. The goal of the map j is to associate each local minimum x of f with a set of generalized saddle points j(x) ⊂ U Ω 1 such that ∀z, y ∈ j(x), f (z) = f (y), and such that, in the limit h → 0, there exists at least one eigenvalue of L D,(0) f,h whose exponential rate of decay is 2 f (j(x))f (x) i.e.∃λ ∈ σ L D,(0) f,h such that lim h→0 h ln λ = -2 f (j(x))f (x) .

  this end, one constructs two families of quasi-modes, denoted by ( u k ) k∈{1,...,m0} and ( ψ j ) j∈{1,...,m1} , which are then respectively projected onto Ran π

  (see Theorem 2.1, Theorem 4.1, and Theorem 5.1 there) are the following. Let Ω be a C ∞ open and connected bounded subset of R d . Let us assume that ∂ n f > 0 on ∂Ω (where ∂ n is the outward normal derivative

  Hess f is an operator acting on 1-forms (namely on vector fields). In the following the operator L

		(1) f,h with tangential boundary
	conditions (27) is denoted by L of L D,(1) f,h associated with λ h . For p ∈ {0, 1}, let us denote, by π D,(1) f,h . From (27), ∇u h is therefore an eigenform (p) h the orthogonal projector of L D,(p) f,h associated with the eigenvalues of L D,(p) f,h smaller than √ h 2 . Another crucial ingredient for the proofs of Proposition 4 and Theorem 1 is the
	fact that, from [31, Chapter 3],
	Ran π	(0)

h = Span u h and dim Ran π

[START_REF] Aristoff | Mathematical analysis of temperature accelerated dynamics[END_REF]

  1 , b 1 ) ⊂ (z 1 , z) and (a 2 , b 2 ) ⊂ (z, z 2 ) such that a 1 < x 1 < b 1 and a 2 < x 2 < b 2 ,it holds for any j ∈ {1, 2}, lim h→0

	bj aj ν h = 1 2

Actually, all the results presented in this section are proved in[START_REF] Di Gesù | Sharp asymptotics of the first exit point density[END_REF] and[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 the more general setting: Ω = Ω ∪ ∂Ω is a C ∞ oriented compact and connected Riemannian manifold of dimension d with boundary ∂Ω.

Acknowledgements. This work is supported by the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement number 614492.

where (q t , p t ) ∈ Ω × R d , Ω being a bounded open subset of R d .