Sébastien Dutercq

Metastability in reversible diffusion processes invariant under a symmetry group

Keywords: 2010 Mathematical Subject Classification. 60J60 (primary), 20C35 (secondary) Metastability, Kramers' law, stochastic exit problem, spectral gap, diffusion processes, spectral theory, potential theory, capacity, symmetry group, representation theory, firstpassage time

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Metastability in reversible diffusion processes invariant under a symmetry group

Introduction

In this present work, we extend the analysis of the problem of metastability for Markovian jump processes with symmetries, initiated in [START_REF] Berglund | The Eyring-Kramers law for Markovian jump processes with symmetries[END_REF], to diffusion processes. We consider the stochastic differential equation

dX ε (t) = -∇V (X ε (t))dt + √ 2εdW (t), (1.1)
where V : R d → R is a confining potential, and W t is a d-dimensional standard Brownian motion. Recall that if V has just two quadratic local minima x * and y * , separated by a quadratic saddle z * , the Eyring-Kramers law, which characterises the mean transition times between local minima of a diffusion in a potential landscape, states that the expected first-passage time τ from x * to a small ball around y * is given by

E x * [τ ] = 2π |λ -(z )| |det(∇ 2 V (z ))| det(∇ 2 V (x )) e [V (z )-V (x )]/ε [1 + O( √ ε|log ε| 3/2 )] , (1.2)
where ∇ 2 V (x) denote the Hessian matrices of V at x, and λ -(z ) is the unique negative eigenvalue of ∇ 2 V (z ). We will consider a potential V with N > 2 local minima, and in this case the characterisation of metastable timescales becomes more involved. It has been known for a long time that the diffusion's generator admits N exponentially small eigenvalues, and that they are connected to metastable timescales [START_REF] Mathieu | Spectra, exit times and long time asymptotics in the zero-white-noise limit[END_REF][START_REF] Miclo | Comportement de spectres d'opérateurs de Schrödinger à basse température[END_REF][START_REF] Kolokol | Asymptotic spectral analysis of a small diffusion operator and the life times of the corresponding diffusion process[END_REF][START_REF] Kolokoltsov | Semiclassical analysis for diffusions and stochastic processes[END_REF]. The generator of the diffusion process (1.1) is the linear operator L ε of the form:

L ε = -ε e V (•)/ε ∇ e -V (•)/ε ∇ = -ε∆ + ∇V (•), ∇ . (1.3) 1 (1)
(2)

Figure 1. Examples of potentials : in [START_REF] Berglund | The Eyring-Kramers law for Markovian jump processes with symmetries[END_REF] we get all eigenvalues, in (2) our theory doesn't work but [START_REF] Landim | Metastability of reversible random walks in potential fields[END_REF] work.

Remark that the sign of the generator was reversed intentionally because this will make all the eigenvalues positive.

Results in [START_REF] Bovier | Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times[END_REF] show that one can order the local minima x 1 , . . . , x N of V in such a way that the mean transition time from each x k to the set {x 1 , . . . , x k-1 } of its predecessors is close to the inverse of the kth small eigenvalue. But these results have a limitation, they require that all relevant saddle heights have to be different (see subsection 2.1 for a precise formulation).

In our case, this condition fails because we will suppose that the potential V is invariant under a symmetry group G. Since we can't apply some parts of the results in [START_REF] Bovier | Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times[END_REF], we will use the representation theory of finite groups to avoid the condition that all relevant saddle heights have to be different.

Results obtained in this work allow us to say that the stochastic system behaves for small noise intensity ε like a Markovian jump process, which in particular proves the results announced in Corollary 4.5 and Theorem 5.1 in [START_REF] Berglund | Interface dynamics of a metastable mass-conserving spatially extended diffusion[END_REF].

Our approach is similar to [START_REF] Landim | Metastability of reversible random walks in potential fields[END_REF] in the sense that we make packages of local minima by group orbits which allows us to compute all N small eigenvalues of the generator (for example the potential (1) in Figure 1) instead of only part of them as in [START_REF] Landim | Metastability of reversible random walks in potential fields[END_REF]. But on the other hand, our hypothesis does not allow us to deal with, for example, the potential (2) in Figure 1 while the approach in [START_REF] Landim | Metastability of reversible random walks in potential fields[END_REF] can deal with this situation.

The remainder of the article is organised as follows. In subsection 2.1, we define the main setting and recall some elements of representation theory of finite groups. Subsections 2.2, 2.3 and 2.4 contain the results on eigenvalues and transition times for the processes. These results are illustrated in Subsection 2.5 on an example. In Section 3 we explain the main strategy that we will follow and recall elements of potential theory. In Section 4 we define the capacity matrix and make the connection between this matrix and the eigenvalues of the generator. The last section contains the proofs of the main results.

Notations: If i j are integers, Ji, jK denotes the set {i, i + 1, . . . , j}. The cardinality of a finite set A is denoted by |A|. The spectrum of a matrix is denoted by Sp.

Results

Setting

In this subsection we will start by giving the necessary definitions and assumptions, from potential theory and representation theory of finite groups, which will allow us to state the main results. Let us start by some smoothness on the potential V . Assumption 2.1. We suppose that V ∈ C 3 (R d ) and

(1) [START_REF] Berglund | Interface dynamics of a metastable mass-conserving spatially extended diffusion[END_REF] (3) In all this paper, • denotes the classic scalar product on R d and • 2 denotes the norm on L 2 (R d , e -V (x)/ε dx).

Remark 2.2. Assumption 2.1 ensures that the resolvent of the generator L ε is compact for ε sufficiently small. Indeed using the relation

B R ∇u(x) 2 e -V (x)/ε dx = B R ∇g(x) 2 + 1 4ε 2 g 2 ∇V (x) 2 + 1 2ε ∇g 2 (x), ∇V (x) dx, (2.1)
where B R is the ball of radius R and centred on 0, u ∈ H 1 (R d , e -V (x)/ε dx) and g = u e -V /2ε we get

R d ∇u(x) 2 e -V (x)/ε dx ≥ C R d u(x) 2 e -V (x)/ε dx, (2.2)
for ε sufficiently small, which leads to the coercivity of L ε and yields compactness using the Lax-Milgram theorem. Moreover, it implies that V has exponentially tight level in sense that for all a∈ R y : V (y)≥a e -V (y)/ε dy ≤ C e -a/ε , (

where

C = C(a) < ∞ is uniform in ε ≤ 1.
The set of saddle points is intuitively the subset of the level set G(A, B) = {z ∈ R d | V (z) = V (A, B)} that cannot be avoided by any paths ω that try to stay as low as possible. We define this set as follows: Definition 2.3. Let A, B ⊂ R d be two disjoint sets.

1. We define V (A, B), the height of the saddle between A and B, by

V (A, B) = inf ω:[0,1]→R d ω(0)∈A,ω(1)∈B sup t∈[0,1]
V (ω(t)), (2.4) where ω is a continuous path. By abuse of notation we write V (x, B) instead of V ({x}, B).

We define the communication height from

x ∈ R d to B by H(x, B) = V (x, B) -V (x) (2.5) h 23 h 32 h 31 h 13 2 3 1 2 3 1 Figure 3.
Example of a graph associated to a potential.

3. We define P(A, B), the set of minimal paths from A to B, by

P(A, B) = {ω ∈ C([0, 1], R d ) | ω(0) ∈ A , ω(1) ∈ B , sup t∈[0,1] V (ω(t)) = V (A, B)}.
(2.6)

4. A gate G * (A, B) is a minimal subset of G(A, B)
with the property that all minimal paths intersect G(A, B). Note that G * (A, B) is in general not unique (see Figure 2). Then the set S(A, B) of saddle points is the union of all gates G * (A, B).

To avoid some complications, we will make the assumption that all saddle points are non-degenerate in the following sense : Assumption 2.4.

1. The set M, of local minima of V , is finite. For any two local minima x, y of V , any set G * (x, y) consists of a finite set of isolated points z * i (x, y).

2. The Hessian matrix of V at all local minima x i ∈ M and all saddle points z * i is non-degenerate (i.e. has only non-zero eigenvalues).

Note that the definition of saddle point using paths is equivalent to the definition of saddle points from an analytical point of view, based on the Morse lemma (see [START_REF] Matsumoto | An introduction to Morse theory[END_REF] for Morse lemma and [START_REF] Berglund | The Eyring-Kramers law for potentials with nonquadratic saddles[END_REF] for a proof of the equivalence).

Remark that under Assumption 2.4, all eigenvalues of a local minimum are strictly positive and all eigenvalues of a saddle point are strictly positive except one which is strictly negative.

As in [START_REF] Berglund | The Eyring-Kramers law for Markovian jump processes with symmetries[END_REF], it will be convenient to consider the undirected graph associated to the potential V . Definition 2.5. Let G = (M, E) be the undirected graph associated to the potential V where the edges are defined as follows : for all x ∈ M let D x , the basin of attraction of x, be the set of points y ∈ R d such that the solution of the differential equation d dt y(t) = -∇V (y(t)), with initial condition y(0) = y, converges to x. Then

E = {(x 1 , x 2 ) ∈ M 2 | ∃ω ∈ P({x 1 }, {x 2 }) , ∀y ∈ M \ {x 1 , x 2 } , Im ω ∩ D y = ∅} (2.7)
In words, this means that there is a minimal path from x 1 to x 2 which does not pass through any other basin of attraction than D x 1 and D x 2 . We will say that x 1 and x 2 are neighbours, denoted x 1 M ∼ x 2 , if (x 1 , x 2 ) ∈ E. Moreover, to each edge (x 1 , x 2 ) ∈ E we will associate the communication height H(x 1 , x 2 ) (see Figure 3 for an example).

Recall the algorithm of removing vertices in [START_REF] Berglund | The Eyring-Kramers law for Markovian jump processes with symmetries[END_REF] subsection 2.3. If we remove a set of vertices {y 1 , . . . , y k } and recompute the edges, then we get a new graph G = (M \ {y 1 , . . . , y k }, E ). We will say that x 1 and x 2 are neighbours with respect to M\{y 1 , . . . , y k },

denoted x 1 M\{y 1 ,...,y k } ∼ x 2 , if (x 1 , x 2 ) ∈ E .
For example, in Figure 3 we have 1

{1,2}
∼ 2. Remark that we can see this definition as follows : for all a ∈ M we have

x M\{a} ∼ y ⇔ x M ∼ a and a M ∼ y.
(2.8)

Unlike in [START_REF] Bovier | Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues[END_REF][START_REF] Bovier | Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times[END_REF] we do not make the assumptions on the fact that communication heights are all different since we want to make a generalisation to the symmetric case. Finally we give some elements of representation theory of finite groups. Let G be a finite group of isometries g : R d → R d such that the potential V is invariant under the group G in the following sense ∀g ∈ G , V • g = V.

(2.9)

We denote by π(g) ∈ R |M|×|M| the permutation matrix

π(g) ab = 1 if g(a) = b , 0 otherwise . ( 2.10)
Let us recall a few definitions from basic group theory.

Definition 2.6.

1. For a ∈ M, O a = {g(a) : g ∈ G} ⊂ M is called the orbit of a. 2. For a ∈ M, G a = {g ∈ G : g(a) = a} ⊂ G is called the stabiliser of a. 3. For g ∈ G, M g = {a ∈ M : g(a) = a} ⊂ M is called the fixed-point set of g.
The following facts are well known:

• The orbits form a partition of M, denoted M/G.

• For any a ∈ M, the stabiliser G a is a subgroup of G.

• For any a ∈ M, the map ϕ : We will denote the orbits of G by A 1 , . . . , A n G . The value of the communication height H(a, A j ) is the same for all a ∈ A i , and we will denote it H(A i , A j ). Similarly, we write V A i for the common value of all V (a), a ∈ A i . We shall make the following non-degeneracy assumption:

Assumption 2.7 (Metastable order of orbits). Let M k = A 1 ∪ • • • ∪ A k .
One can order the orbits in such a way that

H(A k , M k-1 ) min i<k H(A i , M k \ A i ) -θ , k = 2, . . . , n G (2.11)
for some θ > 0. We indicate this by writing

A 1 ≺ A 2 ≺ • • • ≺ A n G .
In Figure 3 we have 1 ≺ 2 ≺ 3 . See [7, 6, section 4.3] for an algorithm determining the metastable hierarchy.

In [START_REF] Berglund | The Eyring-Kramers law for Markovian jump processes with symmetries[END_REF] we made an assumption of absence of accidental degeneracy to simplify the expression of eigenvalues. Now we do the same for diffusion processes. Assumption 2.8 (Absence of accidental degeneracy). Whenever there are elements

x i 1 M ∼ x i 2 , x j 1 M ∼ x j 2 ∈ M such that V (x i 1 , x j 1 ) -V (x i 1 ) = V (x i 2 , x j 2 ) -V (x i 2 ), there exist g ∈ G such that g(x i 1 ) = x i 2 and g(x j 1 ) = x j 2 .
Remark 2.9. Assumption 2.8 implies that all communication height between orbits are different and also that we can only have at most two saddles in series. Assumptions 2.1, 2.4, 2.7 and 2.8 will be assumed to hold throughout this paper. The map π defined by (2.10) is a morphism from G to GL(|M|, C), and thus defines a representation of G (of dimension dim π = n). In what follows, we will draw on some facts from representation theory of finite groups (see for instance [START_REF] Serre | Linear representations of finite groups[END_REF]):

• A representation of G is called irreducible if there is no proper subspace of C n which is invariant under all π(g). • Two representations π and π of dimension d of G are called equivalent if there exists a matrix S ∈ GL(d, C) such that Sπ(g)S -1 = π (g) for all g ∈ G.

• Any finite group G has only finitely many inequivalent irreducible representations π (0) , . . . , π (r-1) . Here π (0) denotes the trivial representation, π (0) (g) = 1 ∀g ∈ G. • Any representation π of G can be decomposed into irreducible representations:

π = r-1 p=0 α (p) π (p) , α (p) 0 , r-1 p=0 α (p) dim(π (p) ) = dim(π) = n . (2.12)
This means that we can find a matrix S ∈ GL(n, C) such that all matrices Sπ(g)S -1 are block diagonal, with α (p) blocks given by π (p) (g). This decomposition is unique up to equivalence and the order of factors. • For any irreducible representation π (p) contained in π, let χ (p) (g) = Tr π (p) (g) denote its characters. Then

P (p) = dim(π (p) ) |G| g∈G χ (p) (g)π(g) (2.13)
is the projector on the invariant subspace of C n associated with π (p) . In particular,

α (p) dim(π (p) ) = Tr P (p) = dim(π (p) ) |G| g∈G χ (p) (g)χ(g) , (2.14)
where χ(g) = Tr π(g). Note that for the representation defined by (2.10), we have

χ(g) = |M g |.
Example 2.10 (Irreducible representations of the permutation group S 4 , [START_REF] Fulton | Representation Theory: A First Course[END_REF]). The permutation group S 4 can be seen as the group of orientation-preserving symmetries of a cube and octahedron. It is generated by the transpositions (1, 2), (2, 3) and [START_REF] Berglund | The Eyring-Kramers law for potentials with nonquadratic saddles[END_REF][START_REF] Bovier | Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues[END_REF]. S 4 has :

• 2 irreducible representations of dimension 1, the trivial one (π (0) (g) = 1 for all g ∈ G) and the signature (π (1) (g) = σ(g) = sign(g) for all g ∈ G),

• 1 irreducible representation π (2) = θ of dimension 2, equivalent to

θ((1, 2)) = θ((3, 4)) = 1 0 0 -1 , θ((2, 3)) = cos( 2π 3 ) sin( 2π 3 ) sin( 2π 3 ) -cos(( 2π 3 ))
.

(2.15)

• 2 irreducible representations of dimension 3, π (3) = ϕ 1 equivalent to

ϕ 1 ((1, 2)) =   1 0 0 0 0 -1 0 -1 0   , ϕ 1 ((2, 3)) =   0 1 0 1 0 0 0 0 1   , ϕ 1 ((3, 4)) =   1 0 0 0 0 1 0 1 0   , (2.16)
and π (4) = ϕ 2 equivalent to ϕ 2 (g) = σ(g)ϕ 1 (g) for all g ∈ G.

The associated characters are given by id (ab) (abc) (ab)(cd) (abcd)

χ 0 1 1 1 1 1 χ 1 1 -1 1 1 -1 χ 2 2 0 -1 2 0 χ 3 3 1 0 -1 -1 χ 4 3 -1 0 -1 1
for any a, b, c, d such that {a, b, c, d} = {1, 2, 3, 4}. There are no irreducible representations of dimension larger than 3.

We use representation theory of finite groups because, each irreducible representation of G will give us a subset of the eigenvalues and eigenfunctions more easily. Indeed each irreducible representations of G will provide us a subspace invariant under L ε which will lead to find eigenvalues and eigenfunctions. At the end, we will see that this procedure indeed gives as many eigenvalues as there are local minima, having that all exponentially small eigenvalues have been found. In the next three subsections we will denote by

B i = B ε (x i ) the ball of radius ε centred on local minima x i ∈ M and S k = k j=1 i:x i ∈A j B i for k ∈ J1, n G K.

The trivial representation

Let us start by the trivial representation π (0) . Theorem 2.11. For ε small enough, the spectrum of L ε contains λ π (0) 1 = 0 and

λ π (0) k = n k |λ -(s)| 2π det(∇ 2 V (a)) |det(∇ 2 V (s))| e -H(A k ,M k-1 )/ε (1 + O( √ ε ln ε 3/2 )) (2.17)
where k ∈ J2, n G K, ∇ 2 V (x) denotes the Hessian matrix of V at x, a is any element of A k , n k is the number of optimal saddles between a and M k-1 , s any of these saddles and λ -(s) is the unique negative eigenvalue of ∇ 2 V (s). Moreover the normalized eigenfunction associated to λ π (0)

1 is φ π (0) 1 = 1 √ |A k | h i 2
and the normalized eigenfunction associated to λ π (0) k for k ∈ J2, n G K is given by

φ π (0) 1 (y) = 1 |A k | h i 2 j∈A k h j (y)(1 + O(e -δ/ε )) + j∈M k-1 h j (y) h j 2
O(e -δ/ε ), (2.18) where δ > 0, h i (y) = P y τ B i < τ S k \B i and τ A is the first hitting time of a set

A ⊂ R d . Furthermore, for k ∈ J2, n G K E x τ S k-1 = 1 λ π (0) k 1 + O(e -δ/ε ) (2.19) holds for all x ∈ S n G \ S k-1 .
Remark 2.12. the integer n k is the main difference with the results of [START_REF] Bovier | Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues[END_REF]. Another way to define n k is that

n k = |G * (A k , M k-1
)| or as in [START_REF] Berglund | The Eyring-Kramers law for Markovian jump processes with symmetries[END_REF] if we take a ∈ A k , a minimal path γ from a to M k-1 in G and let the unique i, j ∈ M such that i and j are in the minimal path γ and the highest saddle point of γ is between i and j, then

n k = |G a | |G i ∩ G j | .
(2.20)

Other irreducible representations of dimension 1

Theorem 2.11 only accounts for a small subset of n G eigenvalues of the generator, associated with distributions that are uniform on each orbit A i . All other eigenvalues of L will be associated to the rate at which non-uniform initial distributions approach the uniform one.

We first determine eigenvalues associated with nontrivial representations of dimension 1, which are easier to obtain. The following lemma shows that given such a representation, only part of the orbits may be present in the image of the associated projector.

Lemma 2.13 ([1], Lemma 3.3). Let π (p) be an irreducible representation of dimension 1 of G, let A i be an orbit of G and fix any a ∈ A i . Denote by π i (g) the permutation induced by g ∈ G on A i and let P (p) i be the associated projector, cf. (2.13). Then one of two following cases holds: Let us call active (with respect to the representation π (p) ) the orbits A i such that Tr P (p) i = 1, and inactive the other orbits. We denote n π (p) the number of active orbits with respect to the representation π (p) . From the representation π (p) we can deduce a number of eigenvalues and eigenfunctions equal to the number of active orbits. We need to define one additional object before getting the result Definition 2.14. For any a ∈ M let

• either π (p) (h) = 1 for all h ∈ G a ,
C a = {g(a) | π (p) (g) = 1}.
(2.21)

For representations other than the trivial, we will always have two possible cases. The first one (l = 1 in the Theorem) with correspond to optimal paths between C a and M k \ C a are paths between x ∈ C a and y ∈ M k-1 , that is to say there is no optimal path inner the orbit A k . And the first one (l = 2 in the Theorem) with correspond to optimal paths between C a and M k \ C a are paths between x ∈ C a and y ∈ A k \ C a , that is to say there is optimal path inner the orbit A k . Now we can give the result, which reads as follows Theorem 2.15. Let π (p) be an irreducible representation of dimension 1 of G. For ε small enough, the spectrum of L ε contains n π (p) eigenvalues of geometric multiplicity 1 given by

λ π (p) k = n l k |λ -(s l )| 2π D l det(∇ 2 V (a)) |det(∇ 2 V (s l ))| e -H l /ε (1 + O( √ ε ln ε 3/2 )) (2.22)
where

l = 1 if H(C a , M k \ C a ) = H(A k , M k-1 ), 2 if H(C a , M k \ C a ) < H(A k , M k-1 ), H l = H(A k , M k-1 ) if l = 1, H(C a , M k \ C a ) if l = 2, (2.23) k ∈ J1, n G K such that A k is active, ∇ 2 V (x) denotes the Hessian matrix of V at x, a is an element of A k , n 1 k (respectively n 2 k
) is the number of optimal saddles between a and M k-1

(respectively M k \ C a ), s 1 (respectively s 2 ) any of these saddles, λ -(s) is the unique negative eigenvalue of ∇ 2 V (s), D 1 = 1 and D 2 =                1 N k g∈G/Ga g(a) M k ∼ a (1 -π(g)) if there is an unique choice of gate G * (C a , S k \ C a ), 1 4N k g∈G/Ga g(a) M k ∼ a (1 -π(g)) otherwise.
(2.24)

Here N k = |{g ∈ G/G a | π(g) = 1 and a M k ∼ g(a)}|, (2.25)
moreover the associated normalized eigenfunction to

λ π (p) k is φ π (p) k (y) = g∈G/Ga π(g) |A k | h g(a) (y) h a 2 (1 + O(e -δ/ε )) + j∈M k-1 h j (y) h j 2 O(e -δ/ε ), (2.26)
where δ > 0 and h i (y

) = P y τ B i < τ S k \B i .
Remark 2.16. n 1 k and n 2 k can be also defined as in Remark 2.12. Because of Assumption 2.7 it is easy to see that we always have

H(C a , M k \ C a ) ≤ H(A k , M k-1 ).

Irreducible representations of dimension larger than 1

We dealt with all irreducible representations of dimension 1 now it remains irreducible representations of dimension larger than 1. As for irreducible representations of dimension 1, let us start with a lemma shows that given such a representation, only part of the orbits may be present in the image of the associated projector.

Lemma 2.17 ([1], Lemma 3.6). Let π (p) be an irreducible representation of G of dimension d 2, and let A i be an orbit of G. Denote by π i the permutation induced by G on A i , and let P (p) i be the associated projector, cf. (2.13). Then for arbitrary a ∈ A i ,

Tr(P (p) i ) = dα (p) i , α (p) i = 1 |G a | h∈Ga χ (p) (h) ∈ {0, 1, . . . , d} .
(2.27)

Here χ (p) (h) = Tr π (p) (h) denotes the characters of the irreducible representation.

Let us again call active (with respect to the irreducible representation π (p) ) those orbits for which Tr(P (p) i ) > 0. We denote again n π (p) the number of active orbits with respect to the representation π (p) . Definition 2.18. For any a ∈ M let

c h (a) = g∈Ga χ (p) (gh),
(2.28)

C a = {h(a) | c h (a) = c id (a)}. (2.29) Choose dα i elements a i l∈J1,dα i K of A i such that if we define J(a i l ) = {h(a i l ) | h ∈ G/G a i l and c h (a i l ) = 0}, (2.30) ĉh (a i l ) = c h (a i l ) ( k∈G/G a i l (c k (a i l )) 2 ) 1/2 , (2.31)
then the l vectors ĉl,i are linearly independent.

Remark 2.19. This new definition of C a in (2.29) is a generalisation of (2.21). Indeed for an irreducible representation of dimension 1 of G we have

c h (a) = c id (a) ⇔ g∈Ga χ (p) (gh) = g∈Ga χ (p) (g) ⇔ g∈Ga π(g)π(h) = g∈Ga π(g) ⇔ π(h) = 1
With this definition we can give the remaining eigenvalues of L ε Theorem 2.20. Let π (p) be an irreducible representation of dimension larger than 1 of G.

For ε small enough, the spectrum of L ε contains n π (p) eigenvalues of geometric multiplicity dα k given by

λ π (p) k = n l k |λ -(s l )| 2π D l det(∇ 2 V (a)) |det(∇ 2 V (s l ))| e -H l /ε (1 + O( √ ε ln ε 3/2 )) (2.32) where l = 1 if H(C a , M k \ C a ) = H(A k , M k-1 ), 2 if H(C a , M k \ C a ) < H(A k , M k-1 ), H l = H(A k , M k-1 ) if l = 1, H(C a , M k \ C a ) if l = 2, (2.33) k ∈ J1, n G K such that A k is active, ∇ 2 V (x) denotes the Hessian matrix of V at x, a is an element of A k , n 1 k (respectively n 2 k
) is the number of optimal saddles between a and M k-1

(respectively M k \ C a ), s 1 (respectively s 2 ) any of these saddle, λ -(s) is the unique negative eigenvalue of ∇ 2 V (s), D 1 = 1 and D 2 =                1 N k q∈G/Ga q(a) M k ∼ a 1 - cq(a) c id (a) if G * (C a , S k \ C a ) is unique, 1 4N k q∈G/Ga q(a) M k ∼ a 1 - cq(a) c id (a)
otherwise.

(2.34)

Here

N k = |{q ∈ G/G a | c q (a) = c id (a) and a M k ∼ q(a)}|, (2.35) moreover the associated normalized eigenfunctions to λ π (p) k are φ π (p) k,l (y) = g∈G/G a k l ĉg (a k l ) h g(a k l ) (y) h a k l 2 (1 + O(e -δ/ε )) + j∈M k \J(a k l ) h j (y) h j 2 O(e -δ/ε ), (2.36)
where δ > 0 and h i (y

) = P y τ B i < τ S k \B i .
Remark 2.21. The new definition of N k in (2.35) is also a generalisation of (2.25).

Example

To illustrate these results, as in [START_REF] Berglund | The Eyring-Kramers law for Markovian jump processes with symmetries[END_REF], we will consider the following potential

V (x) = d i=1 1 4 x 4 i - 1 2 x 2 i , (2.37)
restricted to the space {x ∈ R d :

x i = 0}. We have seen in [START_REF] Berglund | Interface dynamics of a metastable mass-conserving spatially extended diffusion[END_REF] that the set of local minima consists of elements with two coordinates equal to 1 and two coordinates equal to -1, and the set of saddle points consists of elements with two coordinates equal to 0, one coordinate equal to 1 and one coordinate equal to -1. The graph associated to this potential is

(1,-1,-1,1)= 2 (1,1,-1,-1)= 3 (-1,1,1,-1)= 4 (-1,-1,1,1)= 1 (1,-1,1,-1)= 5 (-1,1,-1,1)= 6
We can take G = S 4 × Z/2Z = (ab), (bc), (cd) × f which can be seen as the group of symmetries of an octahedron where • a represents the line passing through the two barycentres of {1, 4, 5} and of {2, 3, 6},

• b represents the line passing through the two barycentres of {3, 4, 5} and of {1, 2, 6},

• c represents the line passing through the two barycentres of {2, 3, 5} and of {1, 4, 6},

• d represents the line passing through the two barycentres of {1, 2, 5} and of {3, 4, 6},

• f is a point reflection in the barycentre of the octahedron.

We use the notation (ab) for permutation of the lines a and b, and similarly for (bc) and (cd). We have

∀x ∈ M , V (x) = -1, ∀s ∈ S , V (s) = - 1 2 .
(2.38)

We have only one orbit A = M = {1, 2, 3, 4, 5, 6}. The elements of G can be seen as permutations of local minima and are given by this table :

(ab) (26)(45)(13) (ab)f (25)(46) (bc) (35)(16)(24) (bc)f (15)(36) (cd) (25)(46)(13) (cd)f (26)(45)
Then the stabiliser of 1 is

G 1 = {id, (ab)f, (cd)f, (ab)(cd), (ac)(bd)f, (ac)(bd)f, (acbd), (adbc)}. (2.39) Using example 2.10, the irreducible representations of G are ∀i ∈ J0, 4K , ∀g ∈ S 4 , π i± (gf j ) = (±1) j π (i) (g) (2.40)
By applying Theorem 2.11 we get the eigenvalue λ 0 = 0. The orbit A is inactive for all other irreducible representations of dimension 1. The orbit A is active only for the following two representations :

• for π 2+ , Theorem 2.20 gives l = 2, H 2 = 1 2 , n 2 2 = 4, N 2 = 4, D 2 = 3 2 and the eigenvalue λ 2 = 6 √ 2 π e -1 2ε (1+O( √ ε ln ε 3/2
)) of geometric multiplicity 2 with associated eigenfunctions :

φ π 2+ 1 (y) =   √ 3 3 i∈{1,3} h i (y) h i 2 - √ 3 6 j∈{2,4,5,6} h j (y) h j 2   (1 + O(e -δ/ε )) φ π 2+ 2 (y) =   √ 3 3 i∈{2,4} h i (y) h i 2 - √ 3 6 j∈{1,3,5,6} h j (y) h j 2   (1 + O(e -δ/ε )) (2.41) • for π 4-, Theorem 2.20 gives l = 2, H 2 = 1 2 , n 2 1 = 4, N 1 = 4, D 2 = 1 and the eigenvalue λ 1 = 4 √ 2 π e -1 2ε (1+O( √ ε ln ε 3/2
)) of geometric multiplicity 3 with associated eigenfunctions :

φ π 3+ 1 (y) = √ 2 2 h 1 (y) h 1 2 - h 3 (y) h 3 2 (1 + O(e -δ/ε )) + i∈{2,4,5,6} h i (y) h i 2 O(e -δ/ε ) φ π 3+ 2 (y) = √ 2 2 h 2 (y) h 2 2 - h 4 (y) h 4 2 (1 + O(e -δ/ε )) + i∈{1,3,5,6} h i (y) h i 2 O(e -δ/ε ) φ π 3+ 5 (y) = √ 2 2 h 5 (y) h 5 2 - h 6 (y) h 6 2 (1 + O(e -δ/ε )) + i∈{1,2,3,4} h i (y) h i 2 O(e -δ/ε ) (2.42)
3 Strategy

Boundary value problem

Let us explain the main strategy that we will follow to get the main results. Consider balls

B i = B ε (x i ) of radius ε centred on local minima for i ∈ J1, |M|K. Let λ k be the principal eigenvalue of the Dirichlet problem (L ε -λ)f (x) = 0, x ∈ R d \ ∂S k , f (x) = 0, x ∈ ∂S k , (3.1)
where

S k = k j=1 i:x i ∈A j B i and k ∈ J1, n G K.
Note that the definition of S k is slightly different from that in [START_REF] Bovier | Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues[END_REF], since because of the symmetry we need to work with unions by orbits instead of elements. Consider, for λ < λ k , the solution of the Dirichlet problem

(L ε -λ)f λ (x) = 0, x ∈ R d \ ∂S k , f λ (x) = φ(x), x ∈ ∂S k . (3.2)
The idea is to construct an eigenfunction of the operator on all R d as a solution of the problem (3.2) for a well chosen function φ. In fact we can see that, if λ an eigenvalue of L ε and if we choose φ as the eigenfunction associated to the eigenvalue λ, then f λ = φ on R d . To see this, remark that if we have the equality on ∂S k , then for all x ∈ R d \ S k we have

(L ε -λ)(f λ -φ)(x) = 0. (3.3)
But if λ is not in the spectrum of (3.1), then (L ε -λ) is invertible so we obtain f λ = φ on R d \ S k . The same argument works on the interior of S k so we have the equality on all R d . Thus λ < λ k is an eigenvalue of L ε if and only if we can find a function on ∂S k such that the solution of the Dirichlet problem (3.2) is an eigenfunction of L ε associated to the eigenvalue λ. So any eigenfunction associated to an eigenvalue λ < λ k can be represented as a solution of the Dirichlet problem (3.2).

The problem of finding eigenvalues boils down to finding for what value λ for a well chosen function φ on the boundary

∂B i we have (L ε -λ)f λ = 0 on all R d . One can interpret (L ε -λ)f λ as a measure concentrated on ∂S k . That means, for any test function g ∈ C ∞ c (R d ) R d e -V (y)/ε g(y)(L ε -λ)f λ (y)dy = R d e -V (y)/ε f λ (y)(L ε -λ)g(y)dy = R d \S k e -V (y)/ε f λ (y)(L ε -λ)g(y)dy + S k e -V (y)/ε f λ (y)(L ε -λ)g(y)dy =ε ∂S k e -V (y)/ε (g(y)∂ n(y) f λ (y) -f λ (y)∂ n(y) g(y))dσ S k (y) + ε ∂S k e -V (y)/ε (g(y)∂ -n(y) f λ (y) -f λ (y)∂ -n(y) g(y))dσ S k (y) =ε ∂S k e -V (y)/ε (g(y)∂ n(y) f λ (y) + g(y)∂ -n(y) f λ (y))dσ S k (y), (3.4)
where ∂ n(y) denotes the derivative in the direction of the exterior normal vector to the surface ∂S k .Requesting that the measure (L ε -λ)f λ vanishes is equivalent to requesting that the measure ε e -V (y)/ε (∂ n(y) f λ (y) + ∂ -n(y) f λ (y))dσ S k (y), (3.5) vanishes. Remark that the problem would be easier if the function φ were constant on the boundary ∂B i . Although this is not true, we will see that either the function φ does not change sign around a minimum, and therefore we will be able to use Harnack and Hölder inequalities. Or the function φ changes sign and in this case it is exponentially small. Now it will be easier than in [START_REF] Bovier | Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues[END_REF]. Indeed, we will easily find the eigenfunctions because the values around the minima can be deduced from the capacity matrix (defined in section 4) which is invariant under the symmetry group G. So using the irreducible representation of G (see [START_REF] Berglund | The Eyring-Kramers law for Markovian jump processes with symmetries[END_REF]), we already have a good candidate for the eigenfunctions, so all we have to do is to prove that indeed they are the eigenfunctions and at the same time get the eigenvalues.

Potential theory

Let us recall some important facts from potential theory which come from [START_REF] Bovier | Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times[END_REF].

Proposition 3.1. By construction the operator L ε given in (1.3) is symmetric on L 2 (R d , e -V (x)/ε dx), i.e. L ε f, g := ε R d ∇f (x)∇g(x) e -V (x)/ε dx = f, L ε g , (3.6)
for any function f, g ∈ H 1 (R d , e -V (x)/ε dx) in weak sense.

We want to find eigenvalues and eigenfunctions of the operator L ε , so we introduce a useful tool, the Green's function.

(L ε -λ)f (x) = g(x), x ∈ Ω, f (x) = 0, x ∈ Ω c , (3.7)
where Ω ⊂ R d .We call Green function G λ Ω (x, y) associated to the inverse of the operator L ε -λ, the function which satisfies :

(L ε -λ)G λ Ω (x, y) = δ(x -y). (3.8)
Proposition 3.3. The solution f , of the Dirichlet problem (3.7), can be written

f (x) = Ω G λ Ω (x, y)g(y)dy. (3.9)
Moreover the Green function is symmetric with respect to the measure e -V (x)/ε , i.e.

G λ Ω (x, y) = e -V (y)/ε G λ Ω (y, x) e V (x)/ε (3.10)
A tool often used in differential geometry is given by the Green identities, which here take the form Proposition 3.4. First Green identity (3.11) and Second Green identity (3.12) where φ, ψ ∈ C 2 (Ω).

Ω e -V (x)/ε (ε∇φ(x) • ∇ψ(x) -ψ(x)(L ε φ)(x))dx = ε ∂Ω e -V (x)/ε ψ(x)∂ n(x) φ(x)dσ Ω (x),
Ω e -V (x)/ε (φ(x)(L ε -λ)ψ(x) -ψ(x)(L ε -λ)φ(x))dx = ε ∂Ω e -V (x)/ε (ψ(x)∂ n(x) φ(x) -φ(x)∂ n(x) ψ(x))dσ Ω (x),
We introduce another useful tool, the Poisson kernel.

Definition-Proposition 3.5 (Poisson kernel). Consider for λ ∈ C the boundary value problem:

(L ε -λ)f (x) = 0, x ∈ Ω, f (x) = φ(x), x ∈ Ω c . (3.13)
We denote by H λ Ω , the Poisson kernel, the solution operator associated to (3.13) which can be represented in the form

f (x) = (H λ Ω φ)(x) = -ε ∂Ω e -[V (y)-V (x)]/ε φ(y)∂ n(y) G λ Ω (y, x)dσ Ω (y), (3.14)
where dσ Ω (y) denotes the surface measure on ∂Ω and ∂ n(y) denotes the derivative in the direction of the exterior normal vector to the surface ∂Ω at y acting on the first argument of the function G λ Ω (y, x).

The last object we need to introduce to get the results are the following Definition 3.6. Let A, B ⊂ R d be two disjoints sets.

1. The equilibrium potential h λ A,B is defined as the solution of the Dirichlet problem:

(L ε -λ)h λ A,B (x) = 0, x ∈ (A ∪ B) c , h λ A,B (x) = 1, x ∈ A, h λ A,B (x) = 0, x ∈ B.
(3.15)

2. The equilibrium measure e λ A,B is defined as the unique measure on ∂A such that :

h λ A,B (x) = ∂A G λ A c (x, y)e λ A,B (dy), (3.16)
where G λ A c is the Green function (recall in Definition 3.2)

3. The capacity, for A, B ⊂ R d and λ ∈ R, is defined by :

cap λ A (B) = ∂A e -V (y)/ε e λ A,B (dy).
(3.17) Remark 3.7. If λ = 0 the equilibrium potential has the following probabilistic interpretation: Remark that we can express the capacity in another way. Indeed, if we consider L ε as a map from H n (R d ) to H n-2 (R d ) we can write the relation (3.16) as :

h A,B (x) ≡ h 0 A,B (x) = P x [τ A < τ B ] . ( 3
(L ε -λ)h λ A,B (x) = (L ε -λ) ∂A G λ A c (x, y)e λ A,B (dy) = ∂A (L ε -λ)G λ A c (x, y)e λ A,B (dy) = ∂A δ(x -y)e λ A,B (dy) = e λ A,B (dx) (3.19)
Using the second Green identity (3.12) and the representation of Poisson kernel (3.14) we have :

(L ε -λ)h λ A,B (x) = ε∂ n(x) h λ A,B (x)dσ A∪B (x) -λ1 A dx (3.20)
Using the second Green identity (3.12) we deduce that:

cap λ A (B) = ε (A∪B) c e -F (x)/ε ∇h λ A,B (x) 2 2 - λ ε (h λ A,B (x)) 2 dx ≡ Φ λ (A∪B) c (h λ A,B ), (3.21)
where Φ λ (A∪B) c is called the Dirichlet form associated to the operator L ε -λ. A fundamental consequence of the previous relation is the variational representation of the capacity if λ ∈ R -:

cap λ A (B) = inf h∈H A,B Φ λ (A∪B) c (h), (3.22)
where H A,B denotes the following set of functions

H A,B = {h ∈ W 1,2 (R d ) | h(x) = 0 for x ∈ B , h(x) = 1 for x ∈ A}. (3.23)
Moreover, for all λ and all x ∈ (A ∪ B) c , we have :

h A,B (x) = E x e λτ A 1 τ A <τ B , (3.24)
That implies

d dλ h λ=0 A,B (x) = E x τ A 1 τ A <τ B . (3.25)
From this derivative we deduce that the function

w A,B (x) = E x τ A 1 τ A <τ B , x ∈ (A ∪ B) c , 0, x ∈ (A ∪ B), (3.26)
is solution of the Dirichlet problem

L ε w A,B (x) = h A,B (x), x ∈ (A ∪ B) c , w A,B (x) = 0, x ∈ (A ∪ B). (3.27)
Remark that in the particular case where B = ∅ we obtain the Dirichlet problem with the representation

L ε w A (x) = 1, x ∈ A c , w A (x) = 0, x ∈ A, (3.28)
4 Capacity matrix

Definition and generalities

Before defining the capacity matrix we have to establish some estimates on the behaviour of eigenfunctions near local minima of the potential V . For the analysis of harmonic functions that are not necessarily positive, we need an estimate for sub-harmonic functions that allows us to compare the oscillation to the L 2 norm (a local maximum principle).

sup y∈B c √ ε (x) φ(y) - inf y∈B c √ ε (x) φ(y) ≤ Cε -d/4 B 2c √ ε (x) φ(y) 2 dy 1/2 (4.1)
Proof: See [START_REF] Gilbarg | Elliptic partial differential equations of second order, volume 224 of Grundlehren der Mathematischen Wissenschaften[END_REF], Theorem 9.20 .

Our goal is to show that, in the balls of radius √ ε centered on the local minima of the potential V , the eigenfunctions associated to the exponentially small eigenvalues of L ε are either of constant sign or exponentially small. This is suggested by the following result we have taken in [START_REF] Kolokoltsov | Semiclassical analysis for diffusions and stochastic processes[END_REF](Chapter 8, Proposition 2.2): This estimate does not allow to conclude that the normalized eigenfunction φ does not change sign in the neighbourhood of the local minima. We will show, for a given local minimum, that φ does not change sign if the contribution of φ is important. For D ⊂ R d define

f 2,D = D e -V (y)/ε f (y) 2 dy 1/2 , ( 4.3)
and for a given eigenfunction φ define Proof: Same proof as in [START_REF] Bovier | Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues[END_REF] with the change |c j | instead of c j because in the symmetric case eigenfunctions are not always positive.

J = {j ∈ M | φ 2,D j ≥ e -γ/2ε }. ( 4
Remark 4.4. The Corollary 9.4 in [START_REF] Gilbarg | Elliptic partial differential equations of second order, volume 224 of Grundlehren der Mathematischen Wissenschaften[END_REF] we used for the proof of the Lemma 4.3, is a consequence of Theorem 8.22 of [START_REF] Gilbarg | Elliptic partial differential equations of second order, volume 224 of Grundlehren der Mathematischen Wissenschaften[END_REF], and by analysing the proof we see that α ∈]0, 1[.

Lemma 4.5 ([4]

, Lemma 4.4). Let φ be an eigenfunction like in Proposition 4.2. If j ∈ J then there exists a positive constant C < ∞ independent of ε such that

sup x∈B √ ε (x j )
|φ(x)| ≤ Cε -d/4 e -γ/2ε e V (x j )/2ε . (4.6)

Proof: See [START_REF] Bovier | Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues[END_REF].

The analysis of the eigenvalues, in subsection 3.1, shows that the necessary and sufficient condition for such λ to exist is the existence of a nontrivial function φ λ on ∂S k such that the surface measure

e -V (y)/ε (L ε -λ)f λ (y)dy = e -V (y)/ε (∂ n(y) + ∂ -n(y) )f λ (y)dσ S k (y) (4.7)
vanishes. An obvious necessary condition for this to be verified is

∂B i e -V (y)/ε (∂ n(y) + ∂ -n(y) )f λ (y)dσ S k (y) = 0 (4.8) for all B i ⊂ S k .
Because of Lemmas 4.3 and 4.5, by defining c i = inf y∈B i φ λ (y) one has the alternative :

1. Either sup y∈B i φ λ (y) c i -1 ≤ Cε α/2 . ( 4.9)

Or sup

y∈B i |φ λ (y)| ≤ Cε -d/4
e -γ/2ε e V (x i )/2ε . (4.10)

We will now consider all possible cases. Let J ⊂ M be the set of indices where (4.9) holds and J c the set of indices where (4.10) holds. With this partition let

f λ = j∈J c j (h λ B j ,S k \B j + ψ λ j ) + j∈J c ψ λ j , (4.11)
where h λ B j ,S k \B j are the equilibrium potentials defined in (3.15) and the functions ψ λ j satisfy for j ∈ J the problems

(L ε -λ)ψ λ j (y) = 0, y ∈ R d \ S k , ψ λ j (y) = φ λ (y) c j -1, y ∈ ∂B j , ψ λ j (y) = 0, y ∈ ∂S k \ B j , (4.12)
and for j

∈ J c (L ε -λ)ψ λ j (y) = 0, y ∈ R d \ S k , ψ λ j (y) = φ λ (y), y ∈ ∂B j , ψ λ j (y) = 0, y ∈ ∂S k \ B j , (4.13)
In the sequel we will denote h i = h B i ,S k \B i for all i ∈ M. With the decomposition (4.11) we can decompose the necessary condition (4.8) as follows

0 = ∂B i e -V (y)/ε (∂ n(y) + ∂ -n(y) )f λ (y)dσ S k (y) = ∂B i e -V (y)/ε h i (y)(∂ n(y) + ∂ -n(y) )f λ (y)dσ S k (y) = ∂S k e -V (y)/ε ∂ n(y) h i (y)f λ (y)dσ S k (y) - λ ε R d \S k e -V (y)/ε h i (y)f λ (y)dy = j∈J c j ∂B j e -V (y)/ε ∂ n(y) h i (y)(1 + ψ λ j (y))dσ S k (y) - λ ε R d \S k e -V (y)/ε h i (y)(h λ j (y) + ψ λ j (y))dy + j∈J c ∂B j e -V (y)/ε ∂ n(y) h i (y)ψ λ j (y)dσ S k (y) - λ ε R d \S k e -V (y)/ε h i (y)ψ λ j (y)dy
C ij = ε ∂B j e -V (y)/ε h j (y)∂ n(y) h i (y)dσ B j (y), (4.15)
and its normalized version K with elements :

K ij = C ij h i 2 h j 2 . (4.16)
Define further the auxiliary matrices

A ij = ε h i 2 h j 2 ∂B j
e -V (y)/ε ∂ n(y) h i (y)ψ λ j (y)dσ ∂S k (y), (4.17)

B ij = 1 -δ ij h i 2 h j 2 R d \S k e -V (y)/ε h i (y)(h λ j (y) + ψ λ j (y))dy, j ∈ J, B ij = 1 -δ ij h i 2 h j 2 R d \S k e -V (y)/ε h i (y)ψ λ j (y)dy, j ∈ J c , (4.18)
D ij = δ ij h j 2 2 R d \S k e -V (y)/ε h i (y)(h λ j (y) -h j (y) + ψ λ j (y))dy, j ∈ J (4.
ĉj (K ij -λδ ij + A ij -λ(D ij + B ij )) + j∈J c h j 2 (A ij -λB ij ) = 0, (4.21)
where ĉj = h j 2 c j , then λ can be an eigenvalue of L ε .

Proof: We write (4.14) using the notations of definition 4.6, simplifying h i 2 ε and replacing h j 2 c j by ĉj , we obtain the result.

To continue our analysis we will need various estimates and properties of these matrices, starting with almost orthogonality on the functions h i . Lemma 4.9 (See [START_REF] Bovier | Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues[END_REF], Lemma 4.5). We have

1. for all i = j R d \S k e -V (y)/ε h i (y)h j (y)dy ≤ Cε -1/2 min e -V (x i ,S k \B i )/ε , e -V (x j ,S k \B j )/ε ≤ Cε -d-1 2 C ii C jj , (4.22) 2. for all j R d e -V (y)/ε h j (y) 2 dy = e -V (x j )/ε (2πε) d/2 det(∇ 2 V (x j )) (1 + O( √ ε| ln ε| 3/2 )). ( 4

.23)

Proof: See [START_REF] Bovier | Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues[END_REF].

Then the auxiliary matrices (A ij , B ij , D ij ) Lemma 4.10 (See [START_REF] Bovier | Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues[END_REF], Lemma 4.6). We have :

1. for all j ∈ J and all i

|A ij | ≤ Cε α/2 |K ij |, (4.24)
2. for all j ∈ J |D jj | ≤ Cε α/2 , (4.25)

3. for all j ∈ J and all i = j

|B ij | ≤ Cε -d-1 2 K ii K jj , (4.26)
4. for all j ∈ J c and all i = j

h j 2 |B ij | ≤ Cε -d-1 2 e -γ/2ε K ii K jj (4.27) Proof: See [4].
Finally the two capacity matrices. The first two points comes also from [START_REF] Bovier | Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues[END_REF] and the last point is also true in the asymmetric case but it will be very useful in our case. 1. for all i, j

C ij = R d \S k
e -V (y)/ε ε ∇h i (y), ∇h j (y) dy, (4.28)

2. for all i, j

|K ij | ≤ K ii K jj , (4.29)
3. for all i j C ij = 0. (4.30)

Proof:

1. As h j (y) = 0 for all y ∈ S k \ B j then

C ij = ε ∂S k e -V (y)/ε h j (y)∂ n(y) h i (y)dσ S k (y).
To use the first Green formula, as n(y) is the exterior normal vector, one must move on to complementary to get the interior normal vector

C ij = ε ∂S c k e -V (y)/ε h j (y)∂ -n(y) h i (y)dσ S c k (y) = R d \S k e -V (y)/ε (ε ∇h i (y), ∇h j (y) -h j (y)L ε h i (y))dy.
As L ε h i (y) = 0 for all y ∈ R d \ S k , we get the result.

2. Using the Cauchy-Schwarz inequality, we have

|C ij | = R d \S k e -V (y)/ε ε ∇h i (y), ∇h j (y) dy ≤ R d \S k e -V (y)/ε ε | ∇h i (y), ∇h j (y) | dy ≤ R d \S k e -V (y)/ε ε ∇h i (y), ∇h i (y) dy 1/2 R d \S k e -V (y)/ε ε ∇h j (y), ∇h j (y) dy 1/2 = C ii C jj .
Hence

|K ij | = |C ij | h i h j ≤ C ii C jj h i 2 h j 2 = K ii K jj 3.
Using the previous point, the linearity of the integral and of the gradient, we have

j C ij = R d \S k e -V (y)/ε ε ∇h i (y), ∇ j h j (y) dy.
Due to the probabilistic interpretation of the equilibrium potentials (3.18), we know that h j (y

) = P y [τ B j < τ S k \B j ]. So, for all y ∈ R d j h j (y) = 1, hence ∇ j h j (y) = 0. Then j C ij = R d \S k e -V (y)/ε ε ∇h i (y), 0 dy = 0.
By grouping all of these results we obtain Theorem 4.12 (See [START_REF] Bovier | Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues[END_REF], Theorem 4.7). Let S k be defined as in Assumption 2.7 and let λ k be the principal eigenvalue of the generator L ε . Then a λ < λ k can be an eigenvalue of the generator L ε if there exist a non-empty set J and constants ĉj such that j∈J ĉ2 j = 1 and for all i ∈ J the relation (4.21) holds.

Elements of the symmetric case

It is expected that the eigenvalues of the generator L ε are close to the eigenvalues of the matrix K. Since we are in the case where there may be symmetries, we cannot prove it as in [START_REF] Bovier | Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues[END_REF]. For now, we will give some properties on the equilibrium potential h j as well as the capacity matrices, due to the symmetries in the general case of several orbits. Lemma 4.13. For all g ∈ G we have

∀x ∈ R d , ∇V (g(x)) = g(∇V (x)).
(4.31)

Proof: Let x ∈ R d .
Using the chain rule formula we get

∇V • g(x) = g -1 (∇V (g(x))).
Let y = g(x), then using (2.9)

g(∇V (x)) = g(∇V (g -1 (y))) = ∇V • g -1 (y) = ∇V (g(x)).
Proposition 4.14. Let X u t be the solution of (1.1) with initial condition X t 0 = u. Then X g(x 0 ) t et g(X x 0 t ) have the same law.

Proof: X g(x 0 ) t and X x 0 t satisfy

X g(x 0 ) t = g(x 0 ) - t 0 ∇V (X g(x 0 ) u )du + t 0 √ 2εdW (u), X x 0 t = x 0 - t 0 ∇V (X x 0 u )du + t 0 √ 2εdW (u).
Since g is a linear isometry

g(X x 0 t ) = g(x 0 ) - t 0 g(∇V (X x 0 u ))du + t 0 √ 2εdW (u).
We know that Brownian motion is invariant under isometries, using Lemma 4.13, we get Proof: By Proposition 4.14, X g(x 0 ) t and g(X x 0 t ) have the same law so we deduce

g(X x 0 t ) = g(x 0 ) - t 0 ∇V (g(X x 0 u ))du + t 0 √ 2εdW (u).
P x 0 [τ A < τ B ] = P g(x 0 ) τ g(A) < τ g(B) , (4.34)
for all subsets A, B ⊂ R d , all x 0 ∈ R d and all g ∈ G. Let g ∈ G, i ∈ M and j = g(i). Using (3.18) and (4.34), we deduce

h i (y) = P g(y) [τ g(B i ) < τ g(S k \B i ) ].
Since g(B i ) = B j and g(S k \ B i ) = S k \ B j , then we get (4.32). By a change of variable, we have

h j 2 2 = R d | det J g (z)| e -V (g(z))/ε h j (g(z)) 2 dz.
All the eigenvalues of a linear isometry have module 1, thus | det J g (z)| = 1. Using (2.9) and (4.32) we get (4.33).

Proposition 4.16. For all g ∈ G and all i, j we have

C ij = C g(i)g(j) , (4.35)
K ij = K g(i)g(j) . (4.36)
Proof: Let i, j and g ∈ G. Using (4.28) and Corollary 4.15, we have

C g(i)g(j) = R d \S k e -V (y)/ε ε ∇(h i • g -1 )(y), ∇(h j • g -1 )(y) dy.
By a change of variable, we get

C g(i)g(j) = R d \S k | det J g (z)| e -V (g(z))/ε ε ∇(h i • g -1 )(g(z)), ∇(h j • g -1 )(g(z)) dz.
All the eigenvalues of a linear isometry have module 1, thus | det J g (z)| = 1. Using the chain rule formula we have

∇(h i • g -1 )(g(z)), ∇(h j • g -1 )(g(z)) = d α,β=1 ∂h i ∂x α (z) ∂h j ∂x β (z) d k=1 ∂g -1 α ∂x k (z) ∂g -1 β ∂x k (z).
As g is a linear isometry, g -1 is too, so

∇(h i • g -1 )(g(z)), ∇(h j • g -1 )(g(z)) = d α,β=1 ∂h i ∂x α (z) ∂h j ∂x α (z)δ αβ = ∇h i (z), ∇h j (z) .
Using (2.9), we deduce (4.35). Using Corollary 4.15 and (4.35) we get (4.36).

Proofs

Irreducible representations of dimension 1

Let π be an irreducible representation of dimension 1 of G, k ∈ J1, n G K such that A k is active with respect to the representation π and a ∈ A k . Let us consider the space

E π = {φ : R d → R | ∀g ∈ G, φ • g = π(g)φ}. (5.1)
Then we have Proposition 5.1. The space E π is stable under L ε .

Proof: Let φ ∈ E π , g ∈ G and x ∈ R d . Since φ ∈ E π and L ε is a linear operator then π(g)L ε φ(x) = L ε (φ • g)(x) = -ε∆(φ • g)(x) + ∇V (x), ∇(φ • g)(x) (5.2)
Using the fact that g is an isometry and the chain rule formula we obtain

∇V (x), ∇(φ • g)(x) = g -1 (∇V (g(x))), g -1 (∇φ(g(x))) = ∇V (g(x)), ∇φ(g(x)) (5.3)
Using again the fact that g is an isometry and the chain rule formula, for all i ∈ J1, dK we obtain

∂φ • g ∂x 2 i (x) = (J -1 g H φ (g(x))J g ) ii (5.4)
where H φ (g(x)) is the Hessian matrix of φ in g(x) and J g is the Jacobian matrix of g (remark that g(x)

= J g x). So ∆(φ • g)(x) = Tr(J -1 g H φ (g(x))J g ) = Tr(H φ (g(x))) = ∆φ(g((x)) (5.5)
Combining (5.3) and (5.5) we get the result.

We see, in Theorem 4.12, that we need to find a set J and constants ĉj such that for all i ∈ J the relation (4.21) holds. So let

J = A k and ∀g ∈ G/ Ga , ĉg(a) = ( h∈G/ Ga π(h) 2 ) -1/2 π(g). (5.6)
Remark that we voluntarily leaves J instead of A k in this subsection because in the next subsection the subset J is not equal to A k in general.

Recall Definition 2.14, as in the proof of the fact that stabilisers of a given orbit are conjugated, it is easy to remark that ∀k ∈ G , C k(a) = kC a .

(

Since E π is stable, we focus on finding eigenfunctions of L ε in E π and their eigenvalues. Let us start by a technical lemma.

Lemma 5.2. If H(C a , S k \ C a ) = H(A k , S k-1 ) then for all j ∈ A k \ C a K aj = cap(A k , S k-1 ) h a 2 2 O(e -δ/ε ) (5.8) If H(C a , S k \ C a ) < H(A k , S k-1 ) then for all j ∈ A k \ C a K aj = - cap(C a , S k \ C a ) N π k |C a | h a 2 2 (1 j M k ∼ a + O(e -δ/ε )) (5.9)
where

N π k = |{g ∈ G/ Ga |π(g) = 1 and a M k ∼ g(a)}|.
We can deduce an estimate of the sum of elements of K in J.

Proposition 5.3. For all i ∈ J we have

j∈J K ij = 1 |A k | h i 2 2 cap(A k , S k-1 )
(5.10)

Proof: First remark that j∈J h j = h A k ,M k-1 . Let i ∈ J then j∈J K ij = ε h i 2 2 ∂A k e -V (y)/ε h A k ,M k-1 (y)∂ n(y) h i (y)dσ A k (y). (5.11)
Moreover, let i ∈ J then there exist g ∈ G such that g(i ) = i, and using Proposition 4.16 we obtain

j∈J K i j = j∈J K g(i )g(j) = j∈J K ig(j) = j∈J K ij .
(5.12)

Hence, using Corollary 4.15 we get

j∈J K ij = 1 |A k | i ,j∈J K i j = 1 |A k | i ∈J ε h i 2 2 ∂A k e -V (y)/ε h A k ,M k-1 (y)∂ n(y) h i (y)dσ A k (y) = ε |A k | h i 2 2 ∂A k e -V (y)/ε h A k ,M k-1 (y)∂ n(y) h A k ,M k-1 (y)dσ A k (y) (5.13)
Then, we can give an estimate of the sum of elements of ĉK in J, which is the important point of the asymmetric case.

Proposition 5.4. If H(C a , S k \ C a ) = H(A k , S k-1 ) then for all i ∈ J j∈J ĉj K ij = ĉi cap(A k , S k-1 ) |A k | h a 2 2 (1 + O(e -δ/ε )) (5.14) If H(C a , S k \ C a ) < H(A k , S k-1 ) then for all i ∈ J j∈J ĉj K ij = ĉi cap(C a , S k \ C a ) N π k |C a | h a 2 2 g∈G/Ga g(a) M k ∼ a (1 -π(g))(1 + O(e -δ/ε )) (5.15)
where

N π k = |{g ∈ G/ Ga |π(g) = 1 and a M k ∼ g(a)}|.
Proof: Let i ∈ J and k ∈ G such that i = k(a). Using Proposition 4.16 we can write the sum as

j∈J ĉj K ij = π(k) j∈J ĉj K aj . If H(C a , S k \ C a ) = H(A k , S k-1
), using Proposition 5.3 and (5.8), we have

j∈J ĉj K ij = π(k)ĉ a j∈A k K aj + π(k) j∈A k \Ca (ĉ j -ĉa )K aj = ĉi cap(A k , S k-1 ) |A k | h a 2 2 + π(k) j∈A k \Ca (ĉ j -ĉa ) cap(A k , S k-1 ) h a 2 2 O(e -δ/ε ) = ĉi cap(A k , S k-1 ) |A k | h a 2 If H(C a , S k \ C a ) < H(A k , S k-1
), using the same argument as in Proposition 5.3 and (5.9), we have

j∈J ĉj K ij = π(k)ĉ a j∈Ca K aj + π(k) j∈J\Ca ĉj K aj = π(k)ĉ a cap(C a , S k \ C a ) |C a | h a 2 2 -π(k) j∈J\Ca ĉj cap(C a , S k \ C a ) N π k |C a | h a 2 2 (1 j M k ∼ a + O(e -δ/ε )) = π(k) cap(C a , S k \ C a ) N π k |C a | h a 2 2 (N π k ĉa - j∈J\Ca ĉj (1 j M k ∼ a + O(e -δ/ε ))) = π(k) cap(C a , S k \ C a ) N π k ( h∈G/ Ga π(h) 2 ) 1/2 |C a | h a 2 2 (N π k - g∈G/Ga π(g) =1 π(g)(1 g(a) M k ∼ a + O(e -δ/ε ))) = ĉi cap(C a , S k \ C a ) N π k |C a | h a 2 2 g∈G/Ga g(a) M k ∼ a (1 -π(g))(1 + O(e -δ/ε ))
Now, it remains to control all the error terms. We need to get an estimate of the sums over the elements of the matrix A to get the result. Let us start by a technical lemma Lemma 5.5. For all i ∈ J and all g ∈ G we have

ψ λ g(i) • g = ψ λ i , (5.16)
where ψ λ is the solution of (4.12).

Proof:

Let x ∈ ∂ B i . Then g(x) ∈ ∂ B g(i) so ψ λ g(i) (g(x)) = φ(g(x)) c g(i) -1 = φ(x) c i -1 = ψ λ i (x)
Let x ∈ ∂ B j with j = i. Then g(x) ∈ ∂ B g(j) and g(j) = g(i) so

ψ λ g(i) (g(x)) = 0 = ψ λ i (x) Let x ∈ R d \ S k .
Using the same argument as in Proposition 5.1 we get

(L ε -λ)(ψ λ g(i) • g)(x) = (L ε -λ)(ψ λ g(i) )(g(x)) = 0. So ψ λ g(i)
• g is a solution of a Dirichlet problem whose only solution is ψ λ i .

Then we get an estimate of the sum over the elements of the matrix A.

Proposition 5.6. For all i, i ∈ J we have

j∈J A ij = j∈J A i j (5.17)
For all i, i ∈ C a we have

j∈Ca A ij = j∈Ca A i j (5.

18)

Proof: As in the first point of Proposition 4.11 (using the first Green identity (3.11)) we have

A ij = ε h i 2 h j 2 R d \S k
e -V (y)/ε ∇h i (y), ∇ψ λ j (y) dy, (5.19) for all i, j ∈ M. Then as in Proposition 4.16, using Corollary 4.15 and Lemma 5.5, we have

A g(i)g(j) = A ij , (5.20)
for all i, j ∈ M and all g ∈ G. Hence we deduce the result.

So we get an estimate of the sum over the elements ĉA in J.

Corollary 5.7. If H(C a , S k \ C a ) = H(A k , S k-1
) then for all i ∈ J we have

j∈J ĉj A ij ≤ Cε α/2 cap(A k , S k-1 ) h a 2 2
(5.21)

If H(C a , S k \ C a ) < H(A k , S k-1
) then for all i ∈ J we have

j∈J ĉj A ij ≤ Cε α/2 cap(C a , S k \ C a ) h a 2 2 . ( 5
A ij = 1 |A k | i,j∈J A ij = 1 |A k | i∈J ε h i 2 2 ∂A k e -V (y)/ε ∂ n(y) h i (y) j∈J ψ λ j (y)dσ S k (y) = ε |A k | h a 2 2 ∂A k e -V (y)/ε ∂ n(y) h A k ,M k-1 (y)
∂ n(x) h A k ,M k-1 (x) ≥ 0. So we get j∈J A ij = ε |A k | h a 2 2 ∂A k e -V (y)/ε ∂ n(y) h A k ,M k-1 (y) j∈J ψ λ j (y)dσ S k (y) ≤ Cε α/2 cap(A k , S k-1 ) |A k | h a 2 2 .
(5.24)

By the same argument (5.28)

j∈Ca A ij = ε |C a | h a 2 2 ∂Ca e -V (y)/ε ∂ n(y) h Ca,M k-1 (y) j∈Ca ψ λ j (y)dσ S k (y) ≤ Cε α/2 cap(C a , S k \ C a ) |C a | h a 2 2 , ( 5
And an estimate of the sum over the elements ĉA in J c .

Proposition 5.8. If H(C a , S k \ C a ) = H(A k , S k-1
) then for all i ∈ J we have

j∈J c h j 2 A ij ≤ C e -γ/2ε cap(A k , S k-1 ) h a 2 2 .
(5.29)

If H(C a , S k \ C a ) < H(A k , S k-1
) then for all i ∈ J we have

j∈J c h j 2 A ij ≤ C e -γ/2ε cap(C a , S k \ C a ) h a 2 2 .
(5.30)

Proof: Let i ∈ J, using the fourth point of the Lemma 4.10 we obtain

j∈J c h j 2 A ij ≤ ε |A k | k-1 l=1 C e -γ/2ε h i 2 h j l 2 ∂A l e -V (y)/ε ∂ n(y) h A k ,S k-1 (y)dσ S k (y) , (5.31)
where j l is an element of A l . Now let split the sum over l which satisfies V (j l ) < V (i) or not. Remark that V (j 1 ) ≤ V (i) always hold. If V (j l ) ≤ V (i) then by (4.22) we obtain

h i 2 h j l 2 ≤ 1, (5.32)
Since ∂ n(y) h A k ,S k-1 (y) ≤ 0 for all y ∈ S k-1 , by using (4.30) we get

ε |A k | k-1 l=1 V (j 1 )≤V (i) C e -γ/2ε h i 2 h j l 2 ∂A l e -V (y)/ε ∂ n(y) h A k ,S k-1 (y)dσ S k (y) ≤ C e -γ/2ε ε |A k | h i 2 2     k-1 l=2 V (j 1 )>V (i) ∂A l e -V (y)/ε ∂ n(y) h A k ,S k-1 (y)dσ S k (y) + cap(A k , S k-1 )     (5.33) Hence j∈J c h j 2 A ij ≤ C e -γ/2ε ε |A k | h i 2 2     k-1 l=2 V (j 1 )>V (i) ∂A l e -V (y)/ε ∂ n(y) h A k ,S k-1 (y)dσ S k (y) h i 2 2 h j l 2 -1 + cap(A k , S k-1 )    
(5.34)

Using Corolary 4.8 of [START_REF] Bovier | Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times[END_REF] and the fact that limit preserve inequalities we get

ε ∂A l e -V (y)/ε ∂ n(y) h A k ,S k-1 (y)dσ S k (y) ≤ C √ ε e -V (A k ,A l )/ε (5.35)
Using estimate of the capacity we get

ε ∂A l e -V (y)/ε ∂ n(y) h A k ,S k-1 (y)dσ S k (y) ≤ C cap(A k , S k-1 ) e -δ/ε , (5.36) where δ < V (A k , A l ) -V (A k , S k-1 )
. By the Assumption 2.7 we have

V (i) -V (j l ) < V (A k , A l ) -V (A k , S k-1 ), (5.37)
so we can chose δ between this two values. Finally using (4.22) we get

ε ∂A l e -V (y)/ε ∂ n(y) h A k ,S k-1 (y)dσ S k (y) h i 2 2 h j l 2 -1 ≤ C cap(A k , S k-1
) e -δ /ε , (5.38)

where δ = δ -V (i) + V (j l ) > 0, which lead to (5.29). If H(C a , S k \ C a ) < H(A k , S k-1 ), we get (5.30) by using cap(A k , S k-1 ) < C cap(C a , S k \ C a ).

(5.39)

Combining these results we finally get Theorem 5.9. Let (H1) be the condition

H(C a , S k \ C a ) = H(A k , S k-1
) and (H2) be the condition H(C a , S k \ C a ) < H(A k , S k-1 ). For ε small enough, the spectrum of L ε contain n π (the number of active orbits with respect to the representation π) eigenvalues of geometric multiplicity 1 given by

λ π k =          cap(A k ,S k-1 ) |A k | ha 2 2 (1 + O(e -δ/ε , ε α/2 )) if (H1), cap(Ca,S k \Ca) n|Ca| ha 2 2 g∈G/Ga g(a) M k ∼ a (1 -π(g))(1 + O(e -δ/ε , ε α/2 )) if (H2). ( 5

.40)

Moreover the associated normalized eigenfunction to λ π k is

φ π k (y) = g∈G/Ga ĉg(a) h g(a) (y) h i 2 (1 + O(ε α/2 )) + j∈J c h j (y) h j 2
O(e -δ/ε ) (5.41)

Proof: Collecting Lemma 4.10, Proposition 5.3, Corollary 5.7 and Proposition 5.8 we have

• j∈J ĉj (K ij -λδ ij ) =                ĉi cap(A k ,S k-1 ) |A k | h i 2 2 -ĉi λ (1 + O(e -δ/ε )) if (H1), ĉi      cap(Ca,S k \Ca) n|Ca| ha 2 2 g∈G/Ga g(a) M k ∼ a (1 -π(g)) -λ      (1 + O(e -δ/ε )) if (H2), • j∈J ĉj A ij =    cap(A k ,S k-1 ) ha 2 2 O(ε α/2 ) if (H1), cap(Ca,S k \Ca) ha 2 2 O(ε α/2 ) if (H2), , • j∈J ĉj λ(D ij + B ij ) = λO(ε α/2 , e -δ/ε ), • j∈J c h j 2 A ij =    cap(A k ,S k-1 ) ha 2 2 O(e -δ/ε ) if (H1), cap(Ca,S k \Ca) ha 2 2 O(e -δ/ε ) if (H2), , • j∈J c h j 2 λB ij = λO(e -δ/ε ).
Put all theses estimates in (4.21) we get (5.40). To get (5.41) recall that φ π k satisfies (4.11) so let

g = φ π k - j∈J ĉj h j h j 2 = j∈J c j ψ λ j + j∈J c ψ λ j . (5.42)
Then the function g is the solution of the Dirichlet problem

(L ε -λ)f (y) = 0, y ∈ R d \ S k , f (
y) = φ π k (y) -c j , y ∈ ∂B j and j ∈ J, f (y) = φ π k (y), y ∈ ∂B j and j ∈ J c .

(5.43)

Using Poisson kernel of the functions g and h j , the fact that ∂ n G Ω\S k < 0, (4.9) and (4.10) we get Then using the same technique as in [START_REF] Bovier | Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues[END_REF], we can improve the error estimate of Theorem 5.9, which lead to replacing ε α/2 by e -δ/ε .

|g(y)| ≤ j∈J Cε α/2 ĉj h j (y) h j 2 + j∈J c Cε -d/4 e -(γ-V (j))/2ε h j 2 h j (y) h j 2 . ( 5

Irreducible representations of dimension larger than 1

Let π be an irreducible representation of G of dimension larger than 1, k ∈ J1, n G K such that A k is active with respect to the representation π. In this subsection χ(h) = Tr π(h) denotes the characters of the irreducible representation π. The proofs in this section are essentially the same as in Section 5.1 and they will be left to the reader. The eigenspace analogue to (5.1) is the following :

E π = {φ ∈ H 1 (R d ) | ∀h ∈ G, ∀a ∈ A k , φ • h = g∈Ga χ(gh)
g∈Ga χ(g) φ}.

(5.46)

Proposition 5.10. The space E π is stable under L ε .

Proof: Same argument as in the proof of Proposition 5.1.

Since E π is stable, we focus on finding eigenfunctions of L ε in E π and their eigenvalues. In [START_REF] Berglund | The Eyring-Kramers law for Markovian jump processes with symmetries[END_REF] we have seen that dim E π = dα where α = 1 |Ga| g∈Ga χ(g) ∈ J0, dK. So we need to choose dα elements of A k such that if we define

J l = {h(a l ) | h ∈ G/ Ga l and g∈Ga l χ(gh) = 0}, (5.47) ĉl h(a l ) = g∈Ga l χ(gh) ( k∈G/ Ga l ( g∈Ga l χ(gh)) 2 ) 1/2 , (5.48)
for l ∈ J1, dαK, the l vectors ĉl are linearly independent. Recalling (2.29), remark that ∀p ∈ G , C p(a) = pC a .

(5.49) Lemma 5.11. For all l ∈ J1, dαK, if

• H(C a l , S k \ C a l ) = H(A k , S k-1 ) then for all j ∈ A k \ C a l K a l j = cap(A k , S k-1 ) h a l 2 2
O(e -δ/ε ) (5.50)

• H(C a l , S k \ C a l ) < H(A k , S k-1 ) then for all j ∈ A k \ C a l K a l j = - cap(C a l , S k \ C a l ) N π k |C a l | h a l 2 2
(1

j M k ∼ a l + O(e -δ/ε )) (5.51) where N π k = |{p ∈ G/ Ga l | g∈Ga l χ(gp) = g∈Ga l χ(g) and a 1 M k ∼ p(a 1 )}|.
Proof: Same proof as the proof of Lemma 5.2.

Proposition 5.12.

If H(C a 1 , S k \ C a 1 ) = H(A k , S k-1
) then for all l ∈ J1, dαK and for all i ∈ J l j∈J l

ĉl j K ij = ĉl i cap(A k , S k-1 ) |A k | h a l 2 2 (1 + O(e -δ/ε )) (5.52) If H(C a 1 , S k \ C a 1 ) < H(A k , S k-1
) then for all l ∈ J1, dαK and for all i ∈ J l j∈J

ĉl j K ij = ĉl i cap(C a 1 , S k \ C a 1 ) N π k |C a 1 | h a 1 2 2 p∈G/Ga 1 1 - g∈Ga 1 χ(gp) g∈Ga 1 χ(g) (1 + O(e -δ/ε )) (5.53)
where

N π k = |{p ∈ G/ Ga l | g∈Ga l χ(gp) = g∈Ga l χ(g) and a 1 M k ∼ p(a 1 )}|.
Proof: Same arguments as in the proof of Proposition 5.4.

Corollary 5.13. For all l ∈ J1, dαK, if

• H(C a l , S k \ C a l ) = H(A k , S k-1
) then for all i ∈ J l we have

j∈J l ĉl j A ij ≤ Cε α/2 cap(A k , S k-1 ) h a l 2 2
(5.54)

• H(C a l , S k \ C a l ) < H(A k , S k-1
) then for all i ∈ J l we have

j∈J l ĉl j A ij ≤ Cε α/2 cap(C a l , S k \ C a l ) h a l 2 2
.

(5.55)

Proof: Same arguments as in the proof of Corollary 5.4.

Proposition 5.14. For all l ∈ J1, dαK, if

• H(C a l , S k \ C a l ) = H(A k , S k-1
) then for all i ∈ J l we have

j∈(J l ) c h j 2 A ij ≤ C e -γ/2ε cap(A k , S k-1 ) h a l 2 2
.

(5.56)

• H(C a l , S k \ C a l ) < H(A k , S k-1 ) then for all i ∈ J l we have j∈(J l ) c h j 2 A ij ≤ C e -γ/2ε cap(C a l , S k \ C a l ) h a l 2 2
.

(5.57)

Proof: Same arguments as in the proof of Proposition 5.8.

Theorem 5.15. Let (H1) be the condition

H(C a 1 , S k \ C a 1 ) = H(A k , S k-1 ) and (H2) be the condition H(C a 1 , S k \ C a 1 ) < H(A k , S k-1
). For ε small enough, the spectrum of L ε contain an eigenvalues of geometric multiplicity dα given by

λ π (p) k =              cap(A k ,S k-1 ) |A k | ha 1 2 2 (1 + O(e -δ/ε , ε α/2 )) if (H1), cap(Ca 1 ,S k \Ca 1 ) N k |Ca 1 | ha 1 2 2 p∈G/ Ga 1 p(a 1 ) M k ∼ a 1 1 - g∈Ga 1 χ (p) (gp) g∈Ga 1 χ (p) (g) (1 + O(e -δ/ε , ε α/2
)) if (H2), (5.58) where k ∈ J1, n G K such that A k is active and where δ > 0 and h i = h B i ,S k \B i as in (3.15).

N k = {p ∈ G/ Ga 1 | g∈Ga 1 χ (p) (gp) =
Proof: Same arguments as in the proof of Theorem 5.9.

Then using the same technique as in [START_REF] Bovier | Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues[END_REF], we can improve the error estimate of Theorem 5.15, which leads to replacing ε α/2 by e -δ/ε .

Capacity estimates

In the previous subsections we have obtained the expressions of all eigenvalues and eigenfunctions. To conclude it remains to compute the capacities cap(A k , S k-1 ) and cap(C a , S k \ C a ) of Theorems 5.9 and 5.15. Let us start by the first capacity. where k ∈ J2, n G K, ∇ 2 V (x) denotes the Hessian matrix of V at x, n k is the number of optimal saddles between any element a of A k and M k-1 , s any of these saddles and λ -(s) is the unique negative eigenvalue of ∇ 2 V (s).

Proof: By Assumption 2.8 the gate G * (A k , S k-1 ) = {s 1 , . . . , s l } is uniquely defined so, using Theorem 3.1 in [START_REF] Bovier | Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times[END_REF] we get cap(A k , S k-1 ) = e -H(A k ,S k-1 )/ε (2πε) To get an estimate of the second capacity in the case where the gate is not uniquely defined, we need an estimate of capacity when we have two saddle point in series Theorem 5.17. Let us consider the case of two saddles points in series (see picture 3 in Figure 2), then we have cap(A, B) = e -H(A,B)/ε (2πε) d/2 2π 1 4

2 i=1 |λ -(z l )| | det(∇ 2 V (z l ))| (1 + O( √ ε ln ε 3/2 )).
(5.65)

Proof: The proof is the same as the proof of Theorem 3.1 in [START_REF] Bovier | Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times[END_REF]. The key point is to take a functions h ∈ H A,B with boundary conditions 1 in D x 1 , 1 2 in D x 2 and 0 in D x 3 where x 1 is the local minimum A, x 3 is the local minimum B and x 2 is the last local minimum.

Remark 5.18. The generalisation to multiple parallel configurations of two saddle points in series can be easily obtained and it will be left to the reader.

The last required capacity estimate is the following : where k ∈ J1, n G K such that A k is active, ∇ 2 V (x) denotes the Hessian matrix of V at x, a is an element of A k , n k is the number of optimal saddles between a and M k \ C a , s any of these saddle, λ -(s) is the unique negative eigenvalue of ∇ 2 V (s) and

D = 1 if G * (C a , S k \ C a ) is unique, 1 4
otherwise.

(5.67)

Proof: The proof of the case when the gate G * (C a , S k \ C a ) = {s 1 , . . . , s l } is uniquely defined is the same as in Theorem 5.16. In the case when the gate G * (C a , S k \ C a ) is not uniquely defined, Assumption 2.8 ensures that all minimal paths between x ∈ C a and S k \ C a only cross two saddle points at the same height, so we are in the case of multiple parallel configurations of two saddle points in series. Using Theorem 5.17 and the same argument as in the proof of Theorem 5.16 we deduce the formula in the case where the gate is not uniquely defined.

Exit times

It remains to prove relation (2.19) on exit times. Let us recall a Proposition from [START_REF] Bovier | Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times[END_REF] Proposition 5.20 (Proposition 6.1, [START_REF] Bovier | Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times[END_REF]). Let x ∈ M \ S k-1 . Then there exists α > 0 such that

E x τ S k-1 = S c k-1
e -V (y)/ε h Bε(x),S k-1 (y)dy cap ( B ε (x), S k-1 ) (1 + O(ε α/2 )).

(5.68)

Proof: Same proof as in [START_REF] Bovier | Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times[END_REF].

Then, using this Proposition, Theorem 5.16, an adaptation of (4.23) and the same technique as in [START_REF] Bovier | Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues[END_REF] to improve the error estimate which leads to replacing ε α/2 by e -δ/ε we obtain the relation (2.19).

Figure 2 . 2 -

 22 Figure 2. Examples of potentials and gates. (1) G * (A, B) = {z 1 }. (2) G * (A, B) = {z 1 , z 2 }. (3) G * (A, B) = {z 1 } or {z 2 }.

  gG a → g(a) provides a bijection from the set G/G a of left cosets to the orbit O a of a, and thus |G|/|G a | = |O a |. • For any g ∈ G and any a ∈ M, one has G g(a) = gG a g -1 , i.e. stabilisers of a given orbit are conjugated. • Burnside's lemma: g∈G |M g | = |G||M/G|.

  and then Tr P (p) i = 1;• or h∈Ga π (p) (h) = 0, and then Tr P

Definition 3 . 2 (

 32 Green's function). Consider for λ ∈ C the Dirichlet problem:

.18) Remark 3 . 8 .

 38 The relation(3.14) comes from the two Green identities 3.11 and 3.12. The relation(3.16) comes from the representation of the Poisson kernel(3.14) with Ω = A c .

Lemma 4 . 1 (

 41 [START_REF] Bovier | Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues[END_REF],Lemma 4.1). Let φ be a strong solution of (L ε -λ)φ = 0 on a ball B c √ ε (x). Then there exists a constant C independent of ε such that

Proposition 4 . 2 (

 42 [START_REF] Bovier | Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues[END_REF], Proposition 4.2). Let φ be a normalized eigenfunction of L ε associated to one of the |M| eigenvalues. Let γ < γ = min x,y∈M V (x, y) -V (y) . Let D i , basin of attraction of x i , be the set of points y ∈ R d such that the solution of the differential equation d dt y(t) = -∇V (y(t)), with initial condition y(0) = y, converges to x i ∈ M. Then there exist a constant C < ∞ dependent of γ and constants c i such that φi c i 1 D i 2 ≤ C e -γ/ε (4.2)Proof: See[START_REF] Kolokoltsov | Semiclassical analysis for diffusions and stochastic processes[END_REF].

. 4 )

 4 Lemma 4.3 ([4],Lemma 4.3). Let φ be an eigenfunction like in Proposition 4.2. If j ∈ J then there exist a constant c j , a positive constant C < ∞ independent of ε and a constant α such that supx∈B √ ε (x j ) |φ(x) -c j | ≤ Cε α/2 |c j |.(4.5)

(4. 14 )

 14 This decomposition of the necessary condition then motivates the following definition Definition 4.6. Define the capacity matrix C as the matrix of size |M| × |M|with elements

Proposition 4 . 11 .

 411 The matrices C and K are symmetric and satisfy :

Corollary 4 . 15 .

 415 Let g ∈ G and i ∈ M. Let j = g(i), then ∀y ∈ R d , h i (y) = h j (g(y)).(4.32)Moreover h i 2 = h j 2 (4.33)

  dσ S k (y),(5.23) for all i ∈ J. By definition of J, for all x ∈ ∂A k , we have | j∈J ψ λ j (x)| ≤ Cε α/2 and moreover

g∈Ga 1 χ

 1 (p) (g) and a 1 M k ∼ p(a 1 )}.(5.59)Moreover the associated normalized eigenfunctions toλ π (p) k are φ π (p) k,l (y) = g∈G/Ga l ĉl g(a l ) h g(a l ) (y) h a l 2 (1 + O(ε α/2 )) + j∈S k \J k l h j (y) h j 2O(e -δ/ε ),(5.60)

Theorem 5 . 16 .

 516 We havecap(A k , S k-1 ) = n k |A k |(2πε) d/2 2π |λ * -(s)| | det(∇ 2 V (s))| e -H(A k ,S k-1 )/ε (1 + O( √ ε ln ε 3/2)),(5.61)

Theorem 5 .

 5 19. We havecap(C a , S k \ C a ) = n k |C a |D 2π |λ -(s)| | det(∇ 2 V (s))| e -H(Ca,M k \Ca)/ε (1 + O( √ ε ln ε 3/2)) (5.66)

2 . 2 2 . 3 2 . 4 3 13 3 . 1 14 4

 22232433114 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The trivial representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Other irreducible representations of dimension 1 . . . . . . . . . . . . . . . 8 Irreducible representations of dimension larger than 1 . . . . . . . . . . . . 9 2.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Strategy Boundary value problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Potential theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capacity matrix 17 4.1 Definition and generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 Elements of the symmetric case . . . . . . . . . . . . . . . . . . . . . . . . . 22 5 Proofs 23 5.1 Irreducible representations of dimension 1 . . . . . . . . . . . . . . . . . . . 23 5.2 Irreducible representations of dimension larger than 1 . . . . . . . . . . . . 31 5.3 Capacity estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.4 Exit times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

  .22) Proof: As in the proof of Proposition 5.3, using Corollary 4.15 and Proposition 5.6 we get

	j∈J

  G such that k(i) = a. If H(C a , S k \ C a ) = H(A k , S k-1), using (5.24), Lemma 4.10 and (5.8) we have

	where k ∈ j∈J	ĉj A ij ≤ ĉa	j∈J	A aj +	j∈J	(ĉ j -ĉa )A aj ≤	Cε α/2 cap(A k , S k-1 ) 2 h a 2	.	(5.27)
	If H(C a , S k \ C a ) < H(A k , S k-1 ), using (5.25), Lemma 4.10 and (5.9) we have
		j∈J	ĉj A ij ≤ ĉa	j∈Ca	A aj +	j∈J\Ca	ĉj A aj ≤	Cε α/2 cap(C a , S k \ C a ) 2 h a 2	.
										.25)
	for all i ∈ C a . Using (5.20) we can write
									ĉj A aj	(5.26)
									j∈J

j∈J ĉj A ij = π(k)

  J1, lK , ∃g ∈ G , g(s i ) = s i . (5.63) Since G is a group of isometries we get ∀i, i ∈ J1, lK , Sp ∇ 2 V (s i ) = Sp ∇ 2 V (s i ).(5.64)And by Assumption 2.8 we have l = |A k ||G * ({a}, S k-1 )| for any a ∈ A k which leads to claimed result.

	2π	d/2	l i=1	|λ -(s l )| | det(∇ 2 V (s l ))|	(1 + O( √	ε ln ε 3/2 )). (5.62)
	By Assumption 2.8 we have
	∀i, i ∈

(1 + O(e -δ/ε ))

Université d'Orléans, Laboratoire Mapmo CNRS, UMR 7349 Fédération Denis Poisson, FR 2964 Bâtiment de Mathématiques, B.P. 6759 45067 Orléans Cedex 2, France E-mail addresses: sebastien.dutercq@univ-orleans.fr