Julian Tugaut

CONVERGENCE TO THE EQUILIBRIA FOR SELF-STABILIZING PROCESSES IN DOUBLE-WELL LANDSCAPE *

Keywords: AMS 2000 subject classifications: Primary 60H10, 35B40; secondary 35K55, 60J60, 60G10 McKean-Vlasov stochastic differential equations, Stationary measures, Double-well potential, Granular media equation, Self-interacting diffusion, Freeenergy 1

 about the monotonicity of the free-energy and combining this with a complete description of the set of the stationary measures, we prove the global convergence of the self-stabilizing processes.

Introduction. We investigate the weak convergence in long-time of the following so-called self-stabilizing process:

(I)

X t = X 0 + √ ǫB t - t 0 V ′ (X s ) ds - t 0 F ′ * u s (X s ) ds u s = L (X s )
.

Here, * denotes the convolution. Since the own law of the process intervenes in the drift, this equation is nonlinear -in the sense of McKean. We note that X t and u t depend on ǫ. We do not write ǫ for simplifying the reading. The motion of the process is generated by three concurrent forces. The first one is the derivative of a potential V -the confining potential. The second influence is a Brownian motion (B t ) t∈R + . It allows the particle to move upwards the potential V . The third term -the so-called self-stabilizing term -represents the attraction between all the others trajectories. Indeed, we remark: F ′ * u s (X s (ω 0 )) = ω∈Ω F ′ (X s (ω 0 ) -X s (ω)) dP (ω) where (Ω, F, P) is the underlying measurable space. This kind of processes were introduced by McKean, see [START_REF] Mckean | Propagation of chaos for a class of non-linear parabolic equations[END_REF] or [START_REF] Mckean | A class of Markov processes associated with nonlinear parabolic equations[END_REF].

Here, we will make some smoothness assumptions on the interaction potential F . Let just note that it is possible to consider non-smooth F . If F is the Heaviside step function and V := 0, (I) is the Burgers equation (see [START_REF] Stroock | Multidimensional diffusion processes[END_REF]).

If F := δ 0 and without confining potential, it is the Oelschläger equation, studied in [START_REF] Oelschläger | A law of large numbers for moderately interacting diffusion processes[END_REF].

The particle X t which verifies (I) can be seen as one particle in a continuous mean-field system of an infinite number of particles. The mean-field system that we will consider is a random dynamical system like (II)

                     dX 1 t = √ ǫdB 1 t -V ′ X 1 t dt -1 N N j=1 F ′ X 1 t -X j t dt . . . dX i t = √ ǫdB i t -V ′ X i t dt -1 N N j=1 F ′ X i t -X j t dt . . . dX N t = √ ǫdB N t -V ′ X N t dt -1 N N j=1 F ′ X N t -X j t dt
where the N brownian motions B i t t∈R + are independents. Mean-field systems are the subject of a rich literature: [START_REF] Dawson | Large deviations from the McKean-Vlasov limit for weakly interacting diffusions[END_REF] about the large deviations for N → +∞, [START_REF] Méléard | Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models[END_REF] under weak assumptions on V and F . For applications, see [START_REF] Collet | A simple mean field model for social interactions: dynamics, fluctuations, criticality[END_REF] about social interactions or [START_REF] Crisan | Approximate McKean-Vlasov representations for a class of SPDEs[END_REF] about the stochastic partial differential equations. The link between the self-stabilizing processes and the mean-field system when N tends to +∞ is called the propagation of chaos, see [START_REF] Sznitman | Topics in propagation of chaos[END_REF] under Lipschitz properties ; [START_REF] Benachour | Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos[END_REF] if V is a constant ; [START_REF] Malrieu | Logarithmic Sobolev inequalities for some nonlinear PDE's[END_REF] or [START_REF] Malrieu | Convergence to equilibrium for granular media equations and their Euler schemes[END_REF] when both potentials are convex ; [START_REF] Ben Arous | Increasing propagation of chaos for mean field models[END_REF] for a more precise result ; [START_REF] Bolley | Quantitative concentration inequalities for empirical measures on non-compact spaces[END_REF], [START_REF] Dai | McKean-Vlasov limit for interacting random processes in random media[END_REF] or [START_REF] Dawson | Large deviations from the McKean-Vlasov limit for weakly interacting diffusions[END_REF] for a sharp estimate ; [START_REF] Cattiaux | Probabilistic approach for granular media equations in the non-uniformly convex case[END_REF] for a uniform result in time in the non-uniformly convex case or [START_REF] Tugaut | Processus autostabilisants dans un paysage multi-puits[END_REF] for a half-uniform propagation of chaos. Equation (II) can be rewritten:

(II) dX t = √ ǫB t -N ∇Υ N (X t ) dt
where the i-th coordinate of X t (resp. B t ) is X i t (resp. B i t ) and

Υ N (X ) := 1 N N j=1 V (X j ) + 1 2N 2 N i=1 N j=1 F (X i -X j )
for all X ∈ R N . As noted in [START_REF] Tugaut | Processus autostabilisants dans un paysage multi-puits[END_REF], the potential Υ N converges towards a meta-potential Υ acting on the measures. Some perturbation (proportional to ǫ) of this meta-potential will play the central role in the article.

As observed in [START_REF] Dawson | Large deviations from the McKean-Vlasov limit for weakly interacting diffusions[END_REF], the empirical law of the mean-field system can be seen as a perturbation of the law of the diffusion (I). Consequently, the long-time behavior of L (X t ) that we study in this paper provides some consequences on the exit time for the particle system (II). Also, the convergence plays an important role in the exit problem for the self-stabilizing process since the exit time is strongly linked to the drift (according to the Kramers' law, see [START_REF] Dembo | Large deviations techniques and applications[END_REF] or [START_REF] Herrmann | Large deviations and a Kramers' type law for self-stabilizing diffusions[END_REF]) which converges towards a homogeneous function if the law of the process converges towards a stationary measure. Let us recall briefly some of the previous results on diffusions like (I). The existence problem has been investigated by two different methods. The first one consists in the application of a fixed point theorem, see [START_REF] Mckean | Propagation of chaos for a class of non-linear parabolic equations[END_REF], [START_REF] Benachour | Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos[END_REF], [START_REF] Cattiaux | Probabilistic approach for granular media equations in the non-uniformly convex case[END_REF] or [START_REF] Herrmann | Large deviations and a Kramers' type law for self-stabilizing diffusions[END_REF] in the non-convex case. The other consists in a propagation of chaos, see for example [START_REF] Méléard | Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models[END_REF]. In [START_REF] Mckean | Propagation of chaos for a class of non-linear parabolic equations[END_REF], the author proved -by using Weyl lemma -that the law of the (strong) solution du t admits a C ∞ -continuous density u t with respect to the Lebesgue measure for all t > 0. Furthermore, this density satisfies a nonlinear partial differential equation of the following type:

(III) ∂ ∂t u t (x) = ∂ ∂x ǫ 2 ∂ ∂x u t (x) + u t (x) V ′ (x) + F ′ * u t (x) .
It is then possible to study equations like (III) by probabilistic methods which involve the diffusions (I) or (II), see [START_REF] Cattiaux | Probabilistic approach for granular media equations in the non-uniformly convex case[END_REF], [START_REF] Funaki | A certain class of diffusion processes associated with nonlinear parabolic equations[END_REF], [START_REF] Malrieu | Convergence to equilibrium for granular media equations and their Euler schemes[END_REF]. Reciprocally, equation (I) is a useful tool for characterizing the stationary measure(s) and the long-time behavior, see [START_REF] Benachour | Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos[END_REF], [START_REF] Benachour | Nonlinear self-stabilizing processes. II. Convergence to invariant probability[END_REF], [START_REF] Tamura | On asymptotic behaviors of the solution of a nonlinear diffusion equation[END_REF], [START_REF] Tamura | Free energy and the convergence of distributions of diffusion processes of McKean type[END_REF] or [START_REF] Yu | On ergodic measures for McKean-Vlasov stochastic equations[END_REF].

In [HT10a], in the non-convex case, by using (III), it has been proved that the diffusion (I) admits at least three stationary measures under assumptions easy to verify. One is symmetric and the two others are not. Moreover, Theorem 3.2 in [HT10a] states the thirdness of the stationary measures if V ′′ is convex and F ′ is linear. This non-uniqueness prevents the long-time behavior to be as intuitive as in the case of unique stationary measure. The work in [START_REF] Herrmann | Stationary measures for self-stabilizing processes: asymptotic analysis in the small noise limit[END_REF] and [START_REF] Herrmann | Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small noise limit[END_REF] provides some estimates of the small-noise asymptotic of these three stationary measures. In particular, the convergence towards Dirac measures and its rate of convergence are investigated. This will be one of the two main tools for obtaining the convergence.

Convergence for (I) is not a new subject. In [START_REF] Benachour | Nonlinear self-stabilizing processes. II. Convergence to invariant probability[END_REF], if V is identically equal to 0, the authors proved the convergence towards the stationary measure by using an ultracontractivity property, a Poincaré inequality and a comparison lemma for stochastic processes. The ultracontractivity property still holds if V is not convex by using the results in [START_REF] Kavian | Quelques remarques sur l'ultracontractivité[END_REF]. It is possible to conserve the Poincaré inequality by using the theorem of Muckenhoupt (see [ABC + 00]) instead of the Bakry-Emery theorem. But, the comparison lemma needs some convexity properties. However, it is possible to apply these results if the initial law is symmetric in the synchronized case (V ′′ (0) + F ′′ (0) ≥ 0), see Theorem 7.10 in [START_REF] Tugaut | Processus autostabilisants dans un paysage multi-puits[END_REF].

Another method consists in using the propagation of chaos in order to derive the convergence of the self-stabilizing process from the one of the mean-field system. However, we shall use it independently of the time and the classical result which is on a finite interval of time is not sufficiently strong. Cattiaux, Guillin and Malrieu proceeded a uniform propagation of chaos in [START_REF] Cattiaux | Probabilistic approach for granular media equations in the non-uniformly convex case[END_REF] and obtained the convergence in the convex case, including the non-uniformly convex case. See also [START_REF] Malrieu | Convergence to equilibrium for granular media equations and their Euler schemes[END_REF]. Nevertheless, according to Proposition 5.17 and Remark 5.18 in [START_REF] Tugaut | Processus autostabilisants dans un paysage multi-puits[END_REF], it is impossible to find a general result of uniform propagation of chaos. In the synchronized case, if the initial law is symmetric, it is possible to find such a uniform propagation of chaos (see Theorem 7.11 and 7.12 in [START_REF] Tugaut | Processus autostabilisants dans un paysage multi-puits[END_REF]).

The method that we will use in this paper is based on the one of [START_REF] Benedetto | A non-Maxwellian steady distribution for one-dimensional granular media[END_REF].

See also [START_REF] Malrieu | Convergence to equilibrium for granular media equations and their Euler schemes[END_REF], [START_REF] Tamura | On asymptotic behaviors of the solution of a nonlinear diffusion equation[END_REF], [START_REF] Malrieu | Logarithmic Sobolev inequalities for some nonlinear PDE's[END_REF], [START_REF] Holley | Logarithmic Sobolev inequalities and stochastic Ising models[END_REF], [START_REF] Arnold | On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations[END_REF] for the convex case.

In the non-convex case, Carrillo, McCann and Villani provided the convergence in [START_REF] José | Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates[END_REF] under two restrictions: the center of mass is fixed and V ′′ (0) + F ′′ (0) > 0 (that means it is the synchronized case). However, by combining the results in [HT10a], [START_REF] Herrmann | Stationary measures for self-stabilizing processes: asymptotic analysis in the small noise limit[END_REF] and [START_REF] Herrmann | Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small noise limit[END_REF] with the work of [START_REF] Benedetto | A non-Maxwellian steady distribution for one-dimensional granular media[END_REF] (and the more rigorous proofs in [START_REF] José | Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates[END_REF] about the freeenergy), we will be able to prove the convergence in a more general setting.

The principal tool of the paper is the monotonicity of the free-energy along the trajectories of (III). Firstly, we introduce the following functional that we call the meta-potential:

(IV) Υ(u) := R V (x)du(x) + 1 2 R 2 F (x -y)du(x)du(y) .
This meta-potential appears intuitively as the limit of the potential in (II) for N → +∞. We consider now the free-energy of the self-stabilizing process (I):

Υ ǫ (u) := ǫ 2 R u(x) log(u(x))dx + Υ(u)
for all measures du which are absolutely continuous with respect to the Lebesgue measure. We can note that du t satisfies this hypothesis.

The paper is organized as follows. After presenting the assumptions, we will state the first results, in particular the convergence of a subsequence (u t k ) k . This subconvergence will be used for improving the results about the thirdness of the stationary measures. Then, we will give the main statement which is the convergence towards a stationary measure, briefly discuss the assumptions of the theorem and give the proof. Subsequently, we will study the basins of attraction by two different methods and prove that these basins are not reduced to a single point. Finally, we postpone four results in the appendix including Proposition A.2 which extends the classical higher-bound for the moments of the self-stabilizing processes.

Assumptions. We assume the following properties of the confining potential V :

(V-1) V is an even polynomial function with deg(V ) =: 2m ≥ 4.
(V-2) The equation V ′ (x) = 0 admits exactly three solutions: a, -a and 0 with a > 0. Furthermore, V ′′ (a) > 0 and V ′′ (0) < 0. Then, the bottoms of the wells are located in x = a and x = -a.

(V-3) V (x) ≥ C 4 x 4 -C 2 x 2 for all x ∈ R with C 2 , C 4 > 0. (V-4) lim x→±∞ V ′′ (x) = +∞ and V ′′ (x) > 0 for all x ≥ a. (V-5) V ′′ is convex. (V-6) Initialization: V (0) = 0. V -a a Fig 1. Potential V
The simplest and most studied example is V (x) := x 4 4 -x 2 2 . Also, we would like to stress that weaker assumptions could be considered but all the mathematical difficulties are present in the polynomial case and it permits to avoid some technical and tedious computations. Let us present now the assumptions on the interaction potential F :

(F-1) F is an even polynomial function with deg(F ) =: 2n ≥ 2. (F-2) F and F ′′ are convex. (F-3) Initialization: F (0) = 0.
Under these assumptions, we know by [HT10a] that (I) admits at least one symmetric stationary measure. And, if 2n-2 p=0

|F (p+2) (a)| p! a p < F ′′ (0) + V ′′ (a)
, there are at least three stationary measures: u 0 ǫ is symmetric and u + ǫ and u - ǫ are asymmetric. Furthermore, we know by [START_REF] Herrmann | Stationary measures for self-stabilizing processes: asymptotic analysis in the small noise limit[END_REF] that there is a unique nonnegative real

x 0 such that V ′ (x 0 ) + 1 2 F ′ (2x 0 ) = 0 and V ′′ (x 0 ) + F ′′ (0)+F ′′ (2x 0 ) 2 > 0.
The same paper provides that u 0 ǫ converges weakly towards 1 2 δ x 0 + 1 2 δ -x 0 and u ± ǫ converges weakly towards δ ±a ; in the small noise limit. We present now the assumptions on the initial law du 0 :

(ES) The 8q 2 -th moment of the measure du 0 is finite with q := max {m, n}. (FE) The probability measure du 0 admits a C ∞ -continuous density u 0 with respect to the Lebesgue measure. And, the entropy R u 0 log(u 0 ) is finite.

Under (ES), we know by Theorem 2.12 in [START_REF] Herrmann | Large deviations and a Kramers' type law for self-stabilizing diffusions[END_REF] that (I) admits a strong solution. Moreover, we have the following inequality:

(V) max 1≤j≤8q 2 sup t∈R + E |X t | j ≤ M 0 .
We deduce immediately that the family (u t ) t∈R + is tight. The assumption (FE) ensures that the free-energy is finite. In the following, we shall use occasionnaly one of the following three additional properties concerning the two potentials V and F and the initial law du 0 :

(LIN) F ′ is linear. (SYN) V ′′ (0) + F ′′ (0) > 0. (FM) For all N ∈ N, we have R |x| N du 0 (x) < +∞.
In the following, three important properties linked to the enumeration of the stationary measures for the self-stabilizing process (I) will be helpful for proving the convergence:

(M3) The process (I) admits exactly three stationary measures. One is symmetric: u 0 ǫ and the other ones are asymmetric: u + ǫ and u - ǫ . Furthermore,

Υ ǫ (u + ǫ ) = Υ ǫ (u - ǫ ) < Υ ǫ (u 0 ǫ ). ( M3
)' There exists M > 0 such that the diffusion (I) admits exactly three stationary measures with free-energy less than M . Furthermore, we have

Υ ǫ (u + ǫ ) = Υ ǫ (u - ǫ ) < Υ ǫ (u 0 ǫ ) ; u 0
ǫ is symmetric and u + ǫ and u - ǫ are asymmetric. (0M1) The process (I) admits only one symmetric stationary measure u 0 ǫ . In the following, we will give some simple conditions such that (M3), (M3)' or (0M1) are true. Finally, we recall the assumption (H) introduced in [HT10b]:

(H) A family of measures (v ǫ ) ǫ verifies the assumption (H) if the family of positive reals R x 2n u ǫ (x)dx ǫ is bounded.

The aim of the weaker assumption (M3)' is to get the convergence even if there exists a family of stationary measures which does not verify the assumption (H).

For concluding the introduction, we write the statement of the main theorem: Theorem: Set a probability measure du 0 which verifies (FE) and (FM). Under (M3), u t converges weakly towards a stationary measure.

1. First results. This section is devoted to present the tools that we will use for getting the main result of the paper. Furthermore, we provide some new results about the thirdness of the stationary measures for the selfstabilizing processes. We introduce the following functionnal:

Υ - ǫ (u) := ǫ 2 R u(x) log (u(x)) 1 {u(x)<1} dx + R V (x)u(x)dx .
This new functionnal does neither contain nor the interaction potential term nor the positive contribution of u in the entropy term. Due to the positivity of the interaction potential F , we get directly the inequality Υ ǫ (u) ≥ Υ - ǫ (u) for all the measures u which verify the previous assumptions. In the following, we will need two particular functions (the free-energy of the system and a function η t such that d dt u t (x) = d dx η t (x)).

Definition 1.1. For all t ∈ R + , we introduce the functions:

ξ(t) := Υ ǫ (u t ) and η t (x) := ǫ 2 ∂ ∂x u t (x) + u t (x) V ′ (x) + F ′ * u t (x) .
According to (III), we remark that if η t is identically equal to 0 then u t is a stationary measure for (I). We recall the following well-known entropy dissipation: Proposition 1.2. Let a probability measure du 0 which verifies (FE) and (ES). Then, for all t, s ≥ 0, we have

ξ(t + s) ≤ ξ(t) ≤ ξ(0) < +∞ .
Furthermore, we have:

ξ ′ (t) ≤ - R 1 u t (x) (η t (x)) 2 dx .
See [START_REF] José | Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates[END_REF] for a proof.

1.1. Preliminaries. Let's introduce the functionnal space

M 8q 2 := f ∈ C 2 0 (R, R + ) R f (x)dx = 1 .
We can remark that u t ∈ M 8q 2 for all t > 0, see [START_REF] Mckean | Propagation of chaos for a class of non-linear parabolic equations[END_REF]. The first tool is the Proposition 1.2 (that is to say the fact that the free-energy is decreasing along the potential lines). The second one is its lower-bound.

Lemma 1.3. There exists Ξ ǫ ∈ R such that inf

u∈M 8q 2 Υ ǫ (u) ≥ Ξ ǫ .
Proof. Let us recall Υ ǫ (u) ≥ Υ - ǫ (u). It suffices then to prove the inequality inf u∈M 8q 2 Υ - ǫ (u) ≥ Ξ ǫ . We proceed as in the first part of the proof of Theorem 2.1 in [START_REF] Benedetto | A non-Maxwellian steady distribution for one-dimensional granular media[END_REF]. We show that we can minorate the negative part of the entropy by a function of the second moment. Then a growth condition of V will provide the result. We split the negative part of the entropy into two integrals:

- R u(x) log (u(x)) 1 {u(x)<1} dx = -I + -I - with I + := R u(x) log (u(x)) 1 {e -|x| <u(x)<1} dx and I -:= R u(x) log (u(x)) 1 {u(x)≤e -|x| } dx .
By definition of I + , we have the following estimate:

I + ≥ R u(x) log e -|x| 1 {e -|x| <u(x)<1} dx ≥ - R |x|u(x)1 {e -|x| <u(x)<1} dx ≥ - R |x|u(x)dx ≥ - 1 2 - 1 2 R x 2 u(x)dx .
By putting γ(x) := √ x log(x)1 {x<1} , a simple computation provides γ(x) ≥ -2e -1 for all x < 1. We deduce:

I -= R u(x)γ(u(x))1 {u(x)≤e -|x| } dx ≥ -2e -1 R e -|x| 2 dx = -8e -1 .
Consequently, it yields:

- R u(x) log (u(x)) 1 {u(x)<1} dx ≤ 1 2 R x 2 u(x)dx + 1 2 + 8e -1 .
This implies:

Υ - ǫ (u) ≥ - ǫ 4 -4ǫe -1 + R V (x) - ǫ 4 x 2 u(x)dx . (1.1) By hypothesis, there exist C 2 , C 4 > 0 such that V (x) ≥ C 4 x 4 -C 2 x 2 so the function x → V (x) -ǫ 4
x 2 is lower-bounded by some negative constant that achieves the proof.

Let's note that the unique assumption we used is lim

x→±∞ V ′′ (x) = +∞.
Lemma 1.4. Let a probability measure du 0 which satisfies the assumptions (FE) and (ES). Then, there exists L 0 ∈ R such that Υ ǫ (u ǫ t ) converges towards L 0 as time elapses to infinity.

Proof.

The assumption (FE) implies ξ(0) = Υ ǫ (u 0 ) < ∞. As ξ is nonincreasing by Lemma 1.2 and lower-bounded by a constant Ξ ǫ according to Lemma 1.3, we deduce that the function ξ converges towards a real L 0 .

Lemma 1.5. If and only if ξ ′ (t) = 0, the following is true:

u t is a sta- tionary measure u ǫ . Proof. If u t is a stationary measure u ǫ , then ξ(t) = Υ ǫ (u t ) = Υ ǫ (u ǫ ) is a constant that provides ξ ′ (t) = 0. Reciprocally, if ξ ′ (t) = 0, Proposition 1.2 implies R 1 u t (x) (η t (x)) 2 dx = 0 .
We deduce η t (x) = 0 for all x ∈ R ; that means u t is a stationary measure.

1.2. Subconvergence.

Theorem 1.6. Let a probability measure du 0 which satisfies the assumptions (FE) and (ES). Then, there exists a stationary measure u ǫ and a sequence (t k ) k which converges to infinity such that u t k converges weakly towards u ǫ .

Proof. Plan: First, we use the convergence of ∞ t ξ ′ (s)ds towards 0 when t tends to infinity and we deduce the existence of a sequence (t k ) k such that ξ ′ (tk) tends to 0 when k tends to infinity. Then, we extract a subsequence of (t k ) k for obtaining an adherence value. By using a test function, we prove that this adherence value is a stationary measure.

Step 1: Lemma 1.4 implies that ∞ t ξ ′ (s)ds collapses at infinity. According to Proposition 1.2, the sign of ξ ′ is a constant so we deduce the existence of an increasing sequence (t k ) k∈N which converges to infinity such that ξ ′ (t k ) -→ 0.

Step 2: The uniform boundedness of the first 8q 2 moments with respect to the time allows us to use Prohorov's theorem: we can extract a subsequence (we continue to write it (t k ) k for simplicity) such that u t k converges weakly towards a probability measure u ǫ .

Step 3: We consider now a compact function ϕ ∈ C ∞ (R, R) L 2 (u ǫ ) and we estimate the following quantity:

R ϕ(x) ǫ 2 ∂ ∂x u t k (x) + u t k (x) V ′ (x) + (F ′ * u t k )(x) dx = R ϕ(x)η t k (x)dx = R ϕ(x) u t k (x) |η t k (x)| u t k (x) dx ≤ R ϕ(x) 2 u t k (x)dx 1 2 × R 1 u t k (x) (η t k (x)) 2 dx 1 2 ≤ -ξ ′ (t k ) R ϕ(x) 2 u t k (x) -→ 0
when k tends to infinity ; by using the Cauchy-Schwarz inequality, the hypothesis about the sequence (t k ) k and the weak convergence of u t k towards u ǫ . Thanks to the compactness of ϕ, we can apply this integration by parts and we obtain

R ϕ(x) ǫ 2 ∂ ∂x u t k (x) + u t k (x) V ′ (x) + F ′ * u t k (x) dx = R ϕ(x) V ′ (x) + F ′ * u t k (x) u t k (x)dx - R ǫ 2 ϕ ′ (x)u t k (x)dx .
The weak convergence of u t k towards u ǫ implies that the previous term tends to

R ϕ(x) [V ′ (x) + (F ′ * u ǫ )(x)] u ǫ (x)dx -R ǫ 2 ϕ ′ (x)u ǫ (x)dx. It has already been proved that R ϕ(x) ǫ 2 ∂ ∂x u t k (x) + u t k (x) (V ′ (x) + F ′ * u t k (x)) dx is collapsing when k tends to ∞. We get the following statement: R ϕ(x) V ′ (x) + F ′ * u ǫ (x) u ǫ (x)dx - R ǫ 2 ϕ ′ (x)u ǫ (x)dx = 0 . (1.2)
Step 4: This means that u ǫ is a weak solution of the equation

ǫ 2 ∂ ∂x u(x) + V ′ (x) + F ′ * u(x) u(x) = 0 .
Now, we consider a smooth function ϕ with compact support [a, b]. We put:

ϕ(x) := exp 2 ǫ [V (x) + F * u ǫ (x)] ϕ ′ (x) .
ϕ is also a smooth function with compact support. Indeed, the application x → F * u ǫ (x) is a polynomial function parametrized by the moments of u ǫ and these moments are bounded.

Equality (1.2) becomes R ϕ ′′ (x) exp 2 ǫ [V (x) + F * u ǫ (x)] u ǫ (x)dx = 0 .
By applying Weyl lemma, we deduce

x → exp 2 ǫ (V (x) + F * u ǫ (x)) u ǫ (x) is smooth. Moreover, its second derivative is equal to 0. Then, there exists A, B ∈ R such that u ǫ (x) = (Ax + B) exp - 2 ǫ (V (x) + F * u ǫ (x)) for all x ∈ R. If A = 0, we get u ǫ (-Ax) < 0 for x big enough. This is impossible. Consequently, u ǫ (x) = Z -1 exp -2 ǫ (V (x) + F * u ǫ (x))
. This means u ǫ is a stationary measure.

Definition 1.7. From now, we call A the set of the adherence values of the family (u t ) t∈R + .

Proposition 1.8. With the assumptions and the notations of Theorem 1.6, we have the following limit:

L 0 := lim t-→+∞ Υ ǫ (u t ) = Υ ǫ (u ǫ ) .
Proof. First of all, we aim to prove that (u t k ) k is uniformly bounded in the space W 1,1 . For doing this, we will bound the integral on R of ∂ ∂x u t k (x). The triangular inequality provides:

R ∂ ∂x u t k (x) dx ≤ 2 ǫ R |η t (x)| dx + 2 ǫ R V ′ (x) + F ′ * u t k (x) u t k (x)dx
where η t is defined in Definition 1.1. By using (V) and the growth property of V ′ and F ′ , it yields

R V ′ (x) + F ′ * u t k (x) u t k (x)dx ≤ C 1 R 1 + |x 2q | u t k (x)dx ≤ C 2
where C 2 is a constant. By using Cauchy-Schwarz inequality like in the proof of Theorem 1.6, we obtain:

R |η t (x)| dx ≤ -ξ ′ (t k ) .
The quantity -ξ ′ (t k ) tends to 0 so it is bounded. We get finally

R ∂ ∂x u t k (x) dx ≤ C 3 where C 3 is a constant. Consequently, u t k ≤ u t k (0) + C for all x ∈ R.
And, since the sequence

(u t k (0)) k converges, it is bounded so there ex- ists a constant C 4 such that u t k (x) ≤ C 4 for all k ∈ N and x ∈ R. It is then easy to prove the convergence of R u t k (x) log (u t k (x)) dx towards R u ǫ (x) log (u ǫ (x)) dx. Indeed, the Lebesgue's theorem implies that R u t k (x) log (u t k (x)) 1 {|x|≤R} dx converges towards R u ǫ (x) log (u ǫ (x)) 1 {|x|≤R} dx for all R ≥ 0 because the applications x → u t k (x) log (u t k (x)
) are lower-bounded, uniformly with respect to k. The other integral is split into two terms. The first one is:

R u t k (x) log (u t k (x)) 1 {|x|>R ; ut k (x)≥1} dx ≤ log(C)u t k ([-R; R] c ) ≤ log(C) M 0 R 2 .
The second term is bounded as in the proof of Lemma 1.3:

- R u t k (x) log (u t k (x)) 1 {|x|>R ; ut k (x)<1} dx ≤ [-R;R] c |x|u t k (x) -γ(u t k (x))e -1 2 |x| dx ≤ M 0 R + 4e -R 2 .
Consequently, Υ ǫ (u ǫ t k ) converges to Υ ǫ (u ǫ ) then Υ ǫ (u ǫ t ) converges to Υ ǫ (u ǫ ) since the free-energy is monotonous. By taking R big enough then k big enough, we can make the following quantity arbitrarily small:

u t k log (u t k ) -u ǫ log (u ǫ ) .
1.3. Consequences. When V is symmetric, Proposition 3.1 (resp. Theorem 4.6) in [HT10a] states the existence of at least three stationary measures

for ǫ small enough if F ′ is linear (resp. if ∞ p=0 F (p+2) (a) p! a p < F ′′ (0)+V ′′ (a)).
Theorem 1.6 permits to extend these results.

Corollary 1.9. For ǫ small enough, the process (I) admits at least three stationary measures: one is symmetric (u 0 ǫ ) and two are asymmetric (u + ǫ and u - ǫ ). Furthermore, under some critical value of ǫ,

Υ ǫ (u + ǫ ) = Υ ǫ (u - ǫ ) < Υ ǫ (u 0 ǫ ).
Proof. We know by Theorem 4.5 in [HT10a] there exists a symmetric stationary measure u 0 ǫ . Theorem 5.4 in [START_REF] Herrmann | Stationary measures for self-stabilizing processes: asymptotic analysis in the small noise limit[END_REF] implies the weak convergence of u 0 ǫ towards 1 2 (δ x 0 + δ -x 0 ) in the small noise limit where x 0 ∈ [0; a[ is the unique solution of

V ′ (x 0 ) + 1 2 F ′ (2x 0 ) = 0 V ′′ (x 0 ) + F ′′ (0) 2 + F ′′ (2x 0 ) 2 ≥ 0 . Lemma A.3 provides lim ǫ→0 Υ ǫ u 0 ǫ = V (x 0 ) + 1 4 F (2x 0 ) and lim ǫ→0 Υ ǫ v + ǫ = V (a) with v + ǫ (x) := Z -1 exp - 2 ǫ (V (x) + F (x -a)) .
We note that V (x 0 ) + 1 4 F (2x 0 ) > V (a). Consequently, for ǫ small enough, we have Υ ǫ (v + ǫ ) < Υ ǫ u 0 ǫ . We consider now the process (I) starting by u 0 := v + ǫ . This is possible because the 8q 2 -th moment of v + ǫ is finite. Theorem 1.6 implies the existence of a sequence (t k ) k which satisfies t k -→ +∞ and u t k converges weakly towards a stationary measure

u ǫ satisfying Υ ǫ (u ǫ ) ≤ Υ ǫ (u 0 ) = Υ ǫ (v + ǫ ) < Υ ǫ u 0 ǫ . So u 0 ǫ = u ǫ .
We immediatly deduce there is at least two stationary measures. If V ′′ (0) + F ′′ (0) = 0, we know by Theorem 7.3 and Theorem 7.4 in [START_REF] Herrmann | Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small noise limit[END_REF] that there exists a unique symmetric stationary measure for ǫ small enough. Hence u ǫ is not symmetric If V ′′ (0) + F ′′ (0) = 0, by (1.1), we have:

Υ ǫ (u) ≥ - ǫ 4 -4ǫe -1 + R V (x) + α 2 x 2 - ǫx 2 4 u(x)dx
for all the probability measures satisfying R xu(x)dx = 0 ; in particular for the symmetric measures because F (x)-F ′′ (0) 2 x 2 ≥ 0 (due to the convexity of

F ′′ ). Then, for ǫ small enough, Υ ǫ (u) > V (a)
2 for all the symmetric measures. However,

Υ ǫ (v + ǫ ) < V (a) 2 (then Υ ǫ (u ǫ ) < V (a)
2 ) for ǫ small enough. Consequently, the process admits at least one asymmetric stationary measure that we call u + ǫ . The measure u

- ǫ (x) := u + ǫ (-x) is invariant too. By construction of u + ǫ and u - ǫ , Υ ǫ (u + ǫ ) = Υ - ǫ (u - ǫ ) < Υ ǫ (u 0 ǫ ).
Remark 1.10. By a similar method, we could also prove the existence of at least one stationary measure in the asymmetric-landscape case.

We know by Theorem 3.2 in [HT10a] that if V ′′ is convex, if F ′ is linear then there are exactly three stationary measures for ǫ small enough. We present a more general setting. In view of the convergence, we will prove that the number of "useful" stationary measures is exactly three even if it is a priori possible to imagine some others stationary measures.

Theorem 1.11. We assume F ′′ (0) + V ′′ (0) ≥ 0. Then, for all M > 0, there exists ǫ(M ) > 0 such that for all ǫ ≤ ǫ(M ), the number of measures u satisfying the two following conditions is exactly three:

1. u is a stationary measure for the diffusion (I).

2. Υ ǫ (u) ≤ M . Moreover, if deg(V ) = 2m > 2n = deg(F )
, the diffusion (I) admits exactly three stationary measures for ǫ small enough.

Proof. Plan: We will begin to prove the second statement (when m > n). For doing this, we will Corollary (1.9) and the results in [START_REF] Herrmann | Stationary measures for self-stabilizing processes: asymptotic analysis in the small noise limit[END_REF] and [START_REF] Herrmann | Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small noise limit[END_REF]. Then, we will prove the first statement by using the second statement and a minoration of the free-energy for a sequence of stationary measures which does not verify (H).

Step 1: Corollary 1.9 implies the existence of ǫ 0 > 0 such that the process (I) admits at least three stationary measures (one is symmetric and two are asymmetric) if ǫ < ǫ 0 : u + ǫ , u - ǫ and u 0 ǫ .

Step 2: First, we assume that deg(V ) > deg(F ).

Step 2.1: Proposition 3.1 in [START_REF] Herrmann | Stationary measures for self-stabilizing processes: asymptotic analysis in the small noise limit[END_REF] implies that each family of stationary measures for the self-stabilizing process (I) verifies Condition (H). It has also been shown that under (H), we can extract a subsequence which converges weakly from any family of stationary measures (u ǫ ) ǫ>0 for the diffusion (I).

Step 2.2: Since F ′′ (0) + V ′′ (0) > 0, there are three possible limiting values: δ 0 , δ a and δ -a according to Proposition 3.7 and Remark 3.8 in [START_REF] Herrmann | Stationary measures for self-stabilizing processes: asymptotic analysis in the small noise limit[END_REF].

Step 2.3: As F ′′ (0) + V ′′ (0) > 0 and V ′′ and F ′′ are convex, there is a unique stationary symmetric measure for ǫ small enough by Theorem 7.3 in [START_REF] Herrmann | Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small noise limit[END_REF]. Also, Theorem 7.2 in [START_REF] Herrmann | Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small noise limit[END_REF] implies there are exactly two asymmetric stationary measures for ǫ small enough. That achieves the proof of the statement.

Step 3: Now, we will prove the first statement. Firstly, if m > n, by applying the second statement, the result is obvious. We assume now m ≤ n.

Set M > 0. All the previous results still hold if we restrict the study to the families of stationary measures which verify Condition (H). Consequently, it is sufficient to get the following results in order to achieve the proof of the theorem:

1. sup Υ ǫ u 0 ǫ ; Υ ǫ (u + ǫ ) ; Υ ǫ (u - ǫ ) < M for ǫ small enough. 2. If (u ǫ k ) k is a sequence of stationary measures, R x 2n u ǫ k (x)dx → ∞ implies Υ ǫ k (u ǫ k ) → ∞.
Step 3.1: Lemma A.3 tells us that Υ ǫ u 0 ǫ (resp. Υ ǫ (u + ǫ )) tends towards 0 (resp. V (a) < 0) when ǫ tends to 0. Hence, the first point is obvious.

Step 3.2: We will prove the second point. We recall the lower-bound (1.1):

Υ - ǫ (u) ≥ - ǫ 4 -4ǫe -1 + R V (x) - ǫ 4 x 2 u(x)dx . As V (x) ≥ C 4 x 4 -C 2 x 2 and Υ - ǫ (u) ≤ Υ ǫ (u) for all smooth u, we get Υ ǫ (u) ≥ R x 2 u(x)dx -C where C is a constant. It is now suffficient to prove that R x 2n u ǫ k (x)dx → ∞ implies R x 2 u ǫ k (x)dx → ∞.
We will not write the index k for simplifying the reading. We proceed a reductio ad absurdum by assuming the existence of a sequence

(u ǫ ) ǫ which verifies R x 2n u ǫ (x)dx → ∞ and R x 2 u ǫ (x)dx → C + ∈ R + .
Step 3.2.1: By taking the notations of [START_REF] Herrmann | Stationary measures for self-stabilizing processes: asymptotic analysis in the small noise limit[END_REF], we have the equality

u ǫ (x) = Z -1 exp -2 ǫ (W ǫ (x)) with W ǫ (x) := V (x) + F * u ǫ (x) = 2n k=1 ω k (ǫ) x k , ω k (ǫ) := 1 k!      V (k) (0) + (-1) k 2n j≥ k 2 F (2j) (0) (2j -k)! m 2j-k (ǫ)      and m l (ǫ) := R x l u ǫ (x)dx ∀l ∈ N . We introduce ω(ǫ) := sup |ω k (ǫ)| 1 2n-k ; 1 ≤ k ≤ 2n .
Step 3.2.2: We note that ω 2n (ǫ) = V (2n) (0)+F (2n) (0) (2n)! > 0. Then, ω(ǫ) is uniformly lower-bounded. Consequently, we can divide by ω(ǫ).

Step 3.2.3: The change of variable x := ω(ǫ)y provides

m 2l (ǫ) ω(ǫ) 2l = R y 2l exp -2 ǫ W ǫ (y) dy R exp -2 ǫ W ǫ (y) dy with W ǫ (x) := 2n k=1 ω k (ǫ) ω(ǫ) 2n-k x k for all l ∈ N, with ǫ := ǫ ω(ǫ) 2n . Step 3.2.4: The 2n sequences ω k (ǫ) ω(ǫ) 2n-k ǫ
are bounded so we can extract a subsequence of ǫ (that we continue to write ǫ for simplicity) such that

ω k (ǫ) ω(ǫ) 2n-k converges towards ω k when ǫ → 0. We put W (x) := 2n k=1 ω k x k . We call A 1 , • • • , A r the r ≥ 1 location(s) of the global minimum of W .
Step 3.2.5: By applying the result of Lemma A.4, we can extract a subsequence (and we continue to denote it by ǫ)

such that R y 2l exp[-2 ǫ Wǫ(y)]dy R exp[-2
ǫ Wǫ(y)]dy converges towards r j=1 p j A 2l j where p 1 + • • • + p r = 1 and p j ≥ 0.

Step 3.2.6: If ω(ǫ) ǫ is bounded, since the quantity r j=1 p j A 2n j is finite, we deduce that (m 2n (ǫ)) ǫ is bounded too. Since m 2n (ǫ) tends towards infinity when ǫ converges towards 0, we deduce (ω(ǫ)) ǫ converges towards infinity. As m 2 (ǫ) is bounded, the quantity m 2 (ǫ) ω(ǫ) 2 vabishes when ǫ tends to 0. This means

r j=1 p j A 2 j = 0 then r j=1 p j A 2n j = 0. Consequently, m 2n (ǫ) = o ω(ǫ) 2n . The Jensen's inequality provides m k (ǫ) = o ω(ǫ) k .
Step 3.2.7: We recall the definition of ω k (ǫ):

ω k (ǫ) = 1 k!      V (k) (0) + (-1) k 2n j≥ k 2 F (2j) (0) (2j -k)! m 2j-k (ǫ)      . We deduce ω k (ǫ) = O {m 2n-k (ǫ)} = o ω(ǫ) 2n-k . So ω(ǫ) = sup |ω k (ǫ)| 1 2n-k ; 1 ≤ k ≤ 2n = o {ω(ǫ)} .
This is a contradiction. This achieves the proof.

This theorem means that -even if the diffusion (I) admits more than three stationary measures -there are only three stationary measures which play a role in the convergence. Indeed, if we take a measure u 0 with a finite freeenergy, we know that for ǫ small enough, there are only three (maybe less) stationary measures which can be adherence value of the family (u ǫ t ) t∈R + . The assumption (LIN) implies (M3) (and (M3)' because it is weaker) and (0M1) for ǫ small enough. The condition (SYN) implies (M3)' and (0M1) for ǫ small enough. Furthermore, if deg(V ) > deg(F ), (SYN) implies (M3) when ǫ is less than some threshold. This description of the stationary measures permits to obtain the principal result that is to say the long-time convergence of the process.

Global convergence.

2.1. Statement of the theorem. We write the main result of the paper:

Theorem 2.1. Set a probability measure du 0 which verifies (FE) and (FM). Under (M3), u t converges weakly towards a stationary measure.

The proof is postponed in Subsection 2.3. Before, we will discuss briefly about the assumptions.

Remarks on the assumptions.

du 0 is absolutely continuous with respect to the Lebesgue measure. We shall use Theorem 1.6 and prove that the family (u t ) t∈R + admits a unique adherence value. This theorem needs that the initial law is absolutely continuous with respect to the Lebesgue measure. However, it is possible to relax this hypothesis by using the following result (see Lemma 2.1 in [HT10a] for a proof):

Set a probability measure du 0 which verifies R x 8q 2 du 0 (x) < +∞. Then, for all t > 0, the probability du t is absolutely continuous with respect to the Lebesgue measure. Consequently, it is sufficient to apply Theorem 2.1 to the probability measure u 1 since there is a unique solution to the non-linear equation (I).

The entropy of du 0 is finite. An essential point of the proof is the convergence of the free-energy. For being sure of this, we assume that it is finite at time 0. The assumption about the moments implies Υ ǫ (u t ) < +∞ if and only if R u t (x) log (u t (x)) dx < +∞. If V was convex, a little adaptation of the theorem in [START_REF] Otto | Comment on: "Hypercontractivity of Hamilton-Jacobi equations[END_REF] (taking into account the fact that the drift is not homogeneous here) would provide the non-optimal following inequality:

Υ ǫ (u t ) ≤ 1 2t inf E |X -Y | 2 ; L (X) = u t ; L (Y ) = v t with v t (x) := Z -1 exp - 2 ǫ V (x) + F * u t (x)
for all t > 0. The second moment of u t is upper-bounded uniformly with respect to t. By using the convexity of V and F , we can prove the same thing for v t . Consequently, since t > 0, the free-energy is finite so the entropy is finite. However, in this paper, we deal with non-convex landscape so we will not relax this hypothesis.

All the moments are finite. Theorem 1.6 tells us we can extract a sequence from the family (u t ) t∈R + such that it converges towards a stationary measure. The last step in order to obtain the convergence is the uniqueness of the limiting value. The most difficult part will be to prove this uniqueness when the symmetric stationary measure u 0 ǫ is an adherence value and the only one of these adherence values to be stationary. For doing this, we will consider a function like this one:

Φ(u) := R ϕ(x)u(x)dx
where ϕ is an odd and smooth function with compact support such that ϕ(x) = x 2l+1 for all x in a compact subset of R. Then, we will proveby proceeding a reductio ad absurdum -there exists an integer l such that Φ u 0 ǫ = Φ (u ∞ ), where u ∞ would be an other limiting value. This inequality will permit to construct a stationary measure u ǫ such that Φ(u ǫ ) / ∈ Φ u 0 ǫ ; Φ (u + ǫ ) ; Φ (u - ǫ ) . This implies the existence of a stationary measure which does not belong to u 0 ǫ ; u + ǫ ; u - ǫ . Under (M3), it is impossible. We make the integration with an "almost-polynomial" function because we need the square of the derivative of such function to be uniformly bounded with respect to the time.

However, it is possible to relax the condition (FM). Indeed, according to Proposition A.2, if we assume that R x 8q 2 du 0 (x) < +∞ (the condition used for the existence of a strong solution), we have

R x 2l u ǫ t (x)dx < +∞ ∀t > 0, l ∈ N .

Hypothesis (M3).

As written before, the key for getting the uniqueness of the adherence value is to proceed a reductio ad absurdum and then to construct a stationary measure u ǫ such that Φ(u ǫ ) takes a forbidden value (a value different from Φ u 0 ǫ , Φ (u + ǫ ) and Φ (u - ǫ )). But, it is possible to deal with a less strong hypothesis. Indeed, by considering an initial law with finite free-energy and since the free-energy is decreasing, it is impossible for u t to converge towards a stationary measure with a higher energy. Consequently, we can consider (M3)' instead of (M3).

All of these remarks permit to obtain the following result: Theorem 2.2. Set a probability measure du 0 with finite entropy. If V and F are polynomial functions such that F ′′ (0) + V ′′ (0) > 0, u ǫ t converges weakly towards a stationary measure for ǫ small enough.

2.3.

Proof of the theorem. In order to get the statement of Theorem 2.1, we will provide two lemmas and one proposition about the free-energy. The lemmas state that a probability measure which verifies simple properties and with some level of energy is necessary a stationary measure for the selfstabilizing process (I). The third one permits to confine all the adherences values under some level of energy.

Lemma 2.3. Under (M3), if u is a probability measure which satisfies (FE) and (ES),

Υ ǫ (u) ≤ Υ ǫ (u ± ǫ ) implies u ∈ {u + ǫ ; u - ǫ }.
Proof. Set u such a measure. We consider the process (I) starting by the initial law u 0 := u. Theorem 1.6 implies that there exists a stationary measure u ǫ such that Υ ǫ (u t ) converges towards Υ ǫ (u ǫ ). However, according to Proposition 1.2 and Proposition 1.8,

Υ ǫ (u ǫ ) = lim t→+∞ Υ ǫ (u t ) ≤ Υ ǫ (u t ) ≤ Υ ǫ (u) ≤ Υ ǫ (u ± ǫ ) . Condition (M3) provides u ǫ ∈ u + ǫ ; u - ǫ ; u 0 ǫ . But, Υ ǫ (u ǫ ) ≤ Υ ǫ (u ± ǫ ) < Υ ǫ (u 0 ǫ ) so u ǫ ∈ {u + ǫ ; u - ǫ }.
Without loss of generality, we will assume u ǫ = u + ǫ . Consequently, the function ξ Definition 1.1) is constant. We deduce that ξ ′ (t) = 0 for all t ≥ 0. Lemma 1.5 implies that u t is a stationary measure in other words u = u 0 = u ǫ = u + ǫ .

We have a similar result with the symmetric measures:

Lemma 2.4. Under (0M1), if u is a symmetric probability measure satisfying (FE) and (ES),

Υ ǫ (u) ≤ Υ ǫ (u 0 ǫ ) implies u = u 0 ǫ .
The key-argument is the following: if the initial law is symmetric then the law at time t is still symmetric. The proof is similar to the previous so it is left to the reader's attention.

Before making the convergence, we need a last result on the adherence values: the free-energy of a limiting value is less than the limit value of the freeenergy.

Proposition 2.5. We assume that u ∞ is an adherence value of the family (u t ) t∈R + . We call

L 0 := lim t→+∞ Υ ǫ (u t ). Then Υ ǫ (u ∞ ) ≤ L 0 .
Proof. As u ∞ is an adherence value of the family (u t ) t∈R + , there exists an increasing sequence (t k ) k which tends to infinity such that u t k converges weakly towards u ∞ . We remark:

Υ (u t k ) =V (a) + R (V (x) -V (a)) u t k (x)dx + 1 2 R 2 F (x -y)u t k (x)u t k
where the meta-potential Υ is defined in (IV). As V (x) -V (a) ≥ 0 for all x ∈ R, the Fatou lemma implies Υ (u ∞ ) ≤ lim inf k→∞ Υ (u t k ). In the same way:

R u ∞ (x) log (u ∞ (x)) 1 {u∞(x)≥1} dx ≤ lim inf k→∞ R u t k (x) log (u t k (x)) 1 {ut k (x)≥1} dx . Set R > 0. By putting γ - k (x) := u t k (x) log (u t k (x)) 1 {ut k (x)<1} 1 {|x|≤R} , we not that γ - k (x) ≤ e -1
1 {|x|≤R} for all x ∈ R and k ∈ N. We can apply the Lebesgue theorem:

R u ∞ (x) log (u ∞ (x)) 1 {u∞(x)≥1} 1 {|x|≤R} = lim k→∞ R γ - k (x)dx .
We put γ + k (x) := u t k (x) log (u t k (x)) 1 {ut k (x)<1} 1 {|x|>R} . By proceeding as in the proof of Lemma 1.3, we have:

-γ + k (x) = -u t k (x) log (u t k (x)) 1 {e -|x| ≤ut k (x)<1} 1 {|x|>R} -u t k (x) log (u t k (x)) 1 {ut k (x)<e -|x| } 1 {|x|>R} ≤|x|u t k (x)1 {|x|>R} + 2e -1 e -|x| 2 1 {|x|>R} .
Consequently, we get the lower-bound

R u ǫ t k (x) log u ǫ t k (x) 1 {u ǫ t k (x)<1} 1 {|x|>R} dx ≥ - M 0 R -8e -1 e -R 2
where M 0 is defined in (V). By introducing Υ ǫ (u) := Υ ǫ (u) -ǫ 2 R u(x) log(u(x))1 {u(x)<1} 1 {|x|>R} dx, we obtain:

Υ ǫ (u ∞ ) ≤ Υ ǫ (u ∞ ) ≤ lim inf k→∞ Υ ǫ (u t k ) ≤ lim inf k→∞ Υ ǫ (u t k ) + M 0 ǫ 2R + 4e -1 exp - R 2 ǫ ≤ L 0 + M 0 ǫ 2R + 4e -1 ǫ exp - R 2 for all R > 0. Consequently, Υ ǫ (u ∞ ) ≤ L 0 .
Proof of the theorem. Plan: The first step of the proof consists in the application of the Prohorov's theorem since the family of measure is tight. We shall prove the uniqueness of the adherence value. We will proceed a reductio ad absurdum. The previous results provide A u 0 ǫ ; u + ǫ ; u - ǫ = ∅ where A is introduced in Definition 1.7. We will then study all the possible cases and we will prove that all of these case imply contradictions. The cases u 0

ǫ ; u + ǫ ⊂ A, u 0 ǫ ; u - ǫ ⊂ A and A u 0 ǫ ; u + ǫ ; u - ǫ = {u +
ǫ } imply contradiction by using the fact that u + ǫ and u - ǫ are the unique minimizers of the free-energy. The cases A u 0

ǫ ; u + ǫ ; u - ǫ = u 0 ǫ and A u 0 ǫ ; u + ǫ ; u - ǫ = {u + ǫ ; u - ǫ }
imply the existence of an other stationary measure which is an adherence value.

Step 1: The inequality (V) implies that the family of probability measures {u t ; t ∈ R + } is tight. Prohorov Theorem permits to conclude that each extracted sequence of this family is relatively compact with respect to the weak convergence. So, in order to prove the statement of the theorem, it is sufficient to prove that this family admits exactly one adherence value. We proceed a reductio ad absurdum. We assume in the following that the family admits at least two adherence values.

Step 2: As the condition (M3) is true, there are exactly three stationary measures : u 0 ǫ , u + ǫ and u - ǫ . By Theorem 1.6, we know that A u 0 ǫ ; u + ǫ ; u - ǫ = ∅. We split this step into four cases:

• u 0 ǫ ; u + ǫ ⊂ A. • A u 0 ǫ ; u + ǫ ; u - ǫ = u 0 ǫ . • A u 0 ǫ ; u + ǫ ; u - ǫ = {u + ǫ ; u - ǫ }. • A u 0 ǫ ; u + ǫ ; u - ǫ = {u + ǫ }.
By symmetry, we will not deal with the two following cases:

u 0 ǫ ; u - ǫ ⊂ A and A u 0 ǫ ; u + ǫ ; u - ǫ = {u - ǫ }. Step 2.1: First case: u 0 ǫ ; u + ǫ ⊂ A.
According to Proposition 1.8, we have the following equality:

Υ ǫ u 0 ǫ = L 0 = Υ ǫ u + ǫ with L 0 := lim t→+∞ Υ ǫ (u t ) .
The hypothesis (M3) implies Υ ǫ u 0 ǫ > Υ ǫ (u + ǫ ). This is impossible.

Step 2.2: We will now prove that the second case:

A u 0 ǫ ; u + ǫ ; u - ǫ = u 0
ǫ is impossible. It will be the core of the proof.

Step 2.2.1: Set u ∞ an other adherence value of the family

(u t ) t∈R + . Proposi- tion 2.5 tells us Υ ǫ (u ∞ ) ≤ Υ ǫ u 0 ǫ . Since u ∞ = u 0 ǫ , Lemma 2.
4 implies u ∞ is not symmetric. We deduce there exists l ∈ N such that R x 2l+1 u ∞ (x)dx = 0. Set R > 0. We introduce the following function:

ϕ(x) := x 2l+1 1 [-R;R] (x) + x 2l+1 1 [R;R+1] (x) Z -1 R+1 x exp - 1 (y -R) 2 - 1 (y -R -1) 2 dy + x 2l+1 1 [-R-1;-R] (x) Z -1 x -R-1 exp - 1 (y + R) 2 - 1 (y + R + 1) 2 dy with Z := 1 0 exp - 1 z 2 - 1 (z -1) 2 dz .
By construction, ϕ is an odd function so R ϕ(x)u 0 ǫ (x)dx = 0. Furthermore, |ϕ(x)| ≤ |x| 2l+1 By using the triangular inequality and (FM), we have:

R ϕ(x)u ∞ (x)dx ≥ R x 2l+1 u ∞ (x)dx - [-R;R] c |x| 2l+1 u ∞ (x)dx ≥ R x 2l+1 u ∞ (x)dx - 1 R 3 C 0 where C 0 := sup t∈R + R |x| 2l+4 u t (x)dx < +∞. Since R x 2l+1 u ∞ (x)dx = 0,
we deduce that R ϕ(x)u ∞ (x)dx = 0 for R big enough. Consequently, we obtain the existence of a smooth function ϕ with compact support such that

0 = R ϕ(x)u 0 ǫ (x)dx < R ϕ(x)u ∞ (x)dx =: 3κ
with κ > 0. Moreover, we can verify that ϕ ′ (x) 2 ≤ C(R)x 4l+2 for all x ∈ R. This implies:

sup t∈R + R ϕ ′ (x) 2 u t (x)dx < +∞.
Step 2.2.2: By definition of A, there exist two increasing sequences (t

(1) k ) k (resp. (t (2) k ) k ) such that u t (1) k (resp. u t (2) k
) converges weakly towards u 0 ǫ (resp. u ∞ ). We deduce there exist two increasing sequences (r k ) k and

(s k ) k such that R ϕ(x)u r k (x)dx = κ and R ϕ(x)u s k (x)dx = 2κ. Then, for all k ∈ N, we put r k := sup t ∈ [0; s k ] | R ϕ(x)u t (x)dx = κ then we define s k := inf s ∈ [ r k ; s k ] | R ϕ(x)u s (x)dx = 2κ .
For simplicity, we write r k (resp. s k ) instead of r k (resp. s k ). And, we have:

κ = R ϕ(x)u r k (x)dx ≤ R ϕ(x)u t (x)dx ≤ R ϕ(x)u s k (x)dx = 2κ for all t ∈ [r k ; s k ].
Step 2.2.3: By applying Proposition A.1, we deduce there exists an increasing sequence (q k ) k converging to +∞ such that (u q k ) k converges weakly towards a stationary measure

u ǫ verifying R ϕ(x)u ǫ (x)dx ∈ [κ; 2κ]. As the set A {u + ; u - ǫ } is empty, we deduce u ǫ = u 0 ǫ . This is impossible since R ϕ(x)u 0 ǫ (x)dx = 0 / ∈ [κ; 2κ].
Step 2.3: We deal now with the third case:

A u 0 ǫ ; u + ǫ ; u - ǫ = {u + ǫ ; u - ǫ }.
Step 2.3.1: By definition of u + ǫ and u - ǫ , we know that these measures are not symmetric. Consequently, there exists l ∈ N such that R x 2l+1 u + ǫ (x)dx = 0. As u - ǫ (x) = u + ǫ (-x), by proceeding as in Step 2.2.1 and Step 2.2.2 and after the application of Proposition A.1, we deduce there exists an increasing sequence (q k ) k∈N which converges to ∞ such that u q k converges weakly towards a stationary measure

u ǫ which verifies R ϕ(x)u ǫ (x)dx ∈ [κ; 2κ] where ϕ a smooth function with compact support such that R ϕ(x)u ± ǫ (x)dx / ∈ [κ; 2κ]. We deduce that u ǫ = u 0 ǫ which contradicts u 0 ǫ / ∈ A.
Step 2.4: We consider now the last case: A u 0 ǫ ; u + ǫ ; u - ǫ = {u + ǫ }. Proposition 1.8 implies that Υ ǫ (u t ) converges towards Υ ǫ (u + ǫ ). Set u ∞ a limit value of the family (u t ) t∈R + which is not u + ǫ . By Proposition 2.5, we know that

Υ ǫ (u ∞ ) ≤ Υ ǫ (u + ǫ ) = lim t-→+∞ Υ ǫ (u t ). Then, Lemma 2.3 implies u ∞ = u - ǫ / ∈ A.

Conclusion

The family (u t ) t∈R + admits only one adherence value with respect to the weak convergence. So u t converges weakly towards a stationary measure that achieves the proof.

3. Bassins of attraction. Now, we shall provide some condition in order to precise the limit.

Domain of u 0

ǫ .

Theorem 3.1. Set a symmetric probability measure du 0 which verifies (FE) and (ES). We assume that V ′′ (0)+F ′′ (0) = 0. Then, for ǫ small enough u t converges weakly towards u 0 ǫ .

Proof. V ′′ (0) + F ′′ (0) = 0 and both functions V ′′ and F ′′ are convex. Theorem 7.3 and 7.4 in [START_REF] Herrmann | Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small noise limit[END_REF] imply the existence and the uniqueness of a symmetric stationary measure u 0 ǫ for ǫ small enough. Theorem 1.6 provides the existence of a stationary measure u ǫ and an increasing sequence (t k ) k which converges to ∞ such that u t k converges weakly towards u ǫ and Υ ǫ (u t ) converges towards Υ ǫ (u ǫ ). As u t is symmetric for all t ≥ 0, we deduce u ǫ = u 0 ǫ , the unique symmetric stationary measure. We proceed a reductio ad absurdum by assuming exists an other sequence (s k ) k which converges to ∞ such that u s k does not converge weakly towards u 0 ǫ . The uniform boundedness of the second moment with respect to the time permits to extract a subsequence (that we continue to write (s k ) k for simplicity) such that u s k weakly towards

u ∞ = u 0 ǫ . Proposi- tion 2.5 implies Υ ǫ (u ∞ ) ≤ Υ ǫ (u 0 ǫ ). Lemma 2.4 implies u ∞ = u 0 ǫ . This is absurd.
Remark 3.2. We assume V ′′ (0) + F ′′ (0) = 0 in order to have a unique symmetric stationary measure for ǫ small enough. We can extend to the case V ′′ (0) + F ′′ (0) = 0 by using the half-uniform propagation of chaos, see Theorem 6.5 in [START_REF] Tugaut | Processus autostabilisants dans un paysage multi-puits[END_REF]. We can also assume that n = 2 that means deg(F ) = 4 by Subsection 4.2 in [HT10a].

Remark 3.3. In the previous theorem, if we have assumed (FM) instead of (ES), we could have applies directly Theorem 2.1.

Domain of u ±

ǫ . The principal tool of the previous theorem is the stability of the subset of all the symmetric stationary measures with finite 8q 2 -moment. If we could find an invariant subset which contains u + ǫ but neither u 0 ǫ nor u - ǫ we could apply the same method for obtaining the convergence towards u + ǫ . Instead of this, we will consider some inequality linked to the meta-potential and we will exhibit a simple subset included in the domain of attraction of u + ǫ . Let us first introduce the following hyperplan:

H := u ∈ C ∞ (R ; R + ) R x 8q 2 u(x)dx < ∞ and R xu(x)dx = 0 .
Theorem 3.4. Set a probability measure du 0 which verifies (FE) and (FM). We assume also

Υ ǫ (u 0 ) < inf u∈H Υ ǫ (u) and R xu 0 (x)dx > 0 .
Under (M3), u t converges weakly towards u + ǫ .

Proof. We know by Theorem 2.1 that there exists a stationary measure u ǫ such that (u t ) t converges weakly towards u ǫ . And, by Proposition 1.8, Υ ǫ (u t ) converges towards Υ ǫ (u ǫ ).

Step 1: As R xu 0 ǫ (x)dx = 0 and R x 8q 2 u 0 ǫ (x)dx < +∞, we have

Υ ǫ u 0 ǫ ≥ inf u∈H Υ ǫ (u) > Υ ǫ (u 0 ) . We deduce u = u 0 ǫ since t → ξ(t) = Υ ǫ (u t ) is nonincreasing.
Step 2. We proceed now a reductio ad absurdum by assuming u ǫ = u - ǫ . There exists

t 0 > 0 such that R xu t 0 (x)dx = 0. Υ ǫ (u t 0 ) ≥ inf u∈H Υ ǫ (u) > Υ ǫ (u 0 )
which contradicts the fact that ξ is non-increasing.

Step 3. Assumption (M3) implies the weak convergence towards u + ǫ .

We use now Theorem 3.4 in some particular cases.

Theorem 3.5. Set a probability measure du 0 which verifies (FE) and (FM). We assume also

Υ (u 0 ) < V (x 0 ) + 1 4 F (2x 0 ) and R xu 0 (x)dx > 0
where x 0 is defined in the introduction. Under either Condition (LIN) or Condition (SYN), for ǫ small enough u t converges weakly towards u + ǫ .

Proof.

Step 1. 3.2 in [HT10a] and Theorem 1.11 imply Condition (M3) under (LIN) or (SYN).

Step 2. Lemma A.3 provides the following limit:

lim ǫ-→0 Υ ǫ (u 0 ǫ ) = V (x 0 ) + 1 4 F (2x 0 ) .
Then, we deduce

lim ǫ-→0 inf u∈H Υ ǫ (u) ≤ V (x 0 ) + 1 4 F (2x 0 ) . (3.1)
Step 3. We prove now that

V (x 0 ) + 1 4 F (2x 0 ) = lim ǫ-→0 inf u∈H Υ ǫ (u). Indeed, if
u is a probability measure such that R xu(x)dx = 0, it verifies the following inequality:

Υ ǫ (u) ≥ Υ - ǫ (u) + F ′′ (0) 4 R 2 (x -y) 2 u(x)u(y)dxdy ≥ Υ - ǫ (u) + F ′′ (0) 2 R x 2 u(x)dx .
By using (1.1), it yields

Υ ǫ (u) ≥ - ǫ 4 - 4ǫ exp(1) + R V (x) + F ′′ (0) 2 x 2 - ǫx 2 4 u(x)dx . (3.2)
We split now the study depending on whether we use Condition (LIN) or Condition (SYN):

(LIN) If F ′ is linear, α 2 x 2 = 1 4 F (2x). So the minimum of x → V (x)+ 1 4 F (2x) is V (x 0 ) + 1 4 (2x 0 ).
We can easily prove that

min x∈R V (x) + α 2 x 2 - ǫ 4 x 2 = V (x 0 ) + 1 4 F (2x 0 ) - ǫ 4 x 2 0 + o(ǫ) .
Consequently:

Υ ǫ (u) ≥ - ǫ 4 - 4ǫ exp(1) + V (x 0 ) + 1 4 F (2x 0 ) - ǫ 4 x 2 0 + o(ǫ)
for all u ∈ H. Then, lim

ǫ-→0 min u∈H Υ ǫ (u) ≥ V (x 0 )+ 1 4 F (2x 0 ). The inequality (3.1) provides lim ǫ-→0 inf u∈H Υ ǫ (u) = V (x 0 ) + 1 4 F (2x 0 ). (SYN) If V ′′ (0)+F ′′ (0) > 0, (3.2) implies Υ ǫ (u) ≥ -ǫ 4 -4ǫ exp(1) for all u ∈ H since ǫ is less than 2 (V ′′ (0) + F ′′ (0)). We deduce that lim ǫ-→0 inf u∈H Υ ǫ (u) ≥ 0. However, as V ′′ (0) + F ′′ (0) > 0, Theorem 5.4 in [HT10b] implies x 0 = 0 so V (x 0 ) + 1 4 F (2x 0 ) = 0.
The inequality (3.1) provides the following limit: lim

ǫ-→0 inf u∈H Υ ǫ (u) = 0 = V (x 0 ) + 1 4 F (2x 0 ).
Step 4. Consequently, Υ ǫ (u 0 ) < inf u∈H Υ ǫ (u) for ǫ small enough. Then, we apply Theorem 3.4.

Remark 3.6. We can replace R xu 0 (x)dx > 0 by R xu 0 (x)dx < 0 in Theorem 3.4 and 3.5 then the same results holds with u - ǫ instead of u + ǫ .

APPENDIX A: APPENDIX

In this appendix, we present some results used previously in the proofs of the main theorems which were postponed here.

Proposition A.1 permits to ensure that even if the free-energy does not reach its global minimum on the stationary measure u 0 ǫ , if the unique symmetric stationary measure is an adherence value, then it is unique. Proposition A.2 is a general result on the self-stabilizing processes. Indeed, it is well-known that du t is absolutely continuous with respect to the Lebesgue measure. Proposition A.2 extends this instantaneous regularization to the finiteness of all the moments since t > 0. Lemma A.3 consists in asymptotic computation of the in the small-noise limit for some useful measures. Lemma A.4 is a Laplace method tedious computation necessary for avoiding to assume that any family of stationary measures verify Condition (H).

We present now the essential proposition for proving Theorem 2.1.

Proposition A.1. Set a probability measure du 0 which verifies (FE) and (FM). We assume the existence of two polynomial functions P and Q, a smooth function ϕ with compact support such that |ϕ(x)| ≤ P(x) and |ϕ ′ (x)| 2 ≤ Q(x), κ > 0 and two sequences (r k ) k and (s k ) k which converge to ∞ such that for all r k ≤ t ≤ s k < r k+1 :

κ = R ϕ(x)u r k (x)dx ≤ R ϕ(x)u t (x)dx ≤ R ϕ(x)u s k (x)dx = 2κ .
Then, there exists a stationary measure u ǫ which verifies R ϕ(x)u ǫ (x)dx ∈ [κ; 2κ] and an increasing sequence (q k ) k which converges to ∞ such that u q k converges weakly towards u ǫ .

Proof.

Step 1: We will prove that lim sup k-→+∞ (s kr k ) > 0. We introduce the function :

Φ(t) := R ϕ(x)u t (x)dx .
This function is well-defined since |ϕ| is bounded by a polynomial function. The derivation of Φ, the use of equation (III) and an integration by part lead to:

Φ ′ (t) = - R ϕ ′ (x) ǫ 2 ∂ ∂x u t (x) + u t (x) V ′ (x) + F ′ * u t (x) dx = - R ϕ ′ (x)η t (x)dx .
The Cauchy-Schwarz inequality implies

Φ ′ (t) ≤ |ξ ′ (t)| R (ϕ ′ (x)) 2 u t (x)dx
where we recall that ξ(t) = Υ ǫ (u t ). The function (ϕ ′ ) 2 is bounded by a polynomial function and R x 2N u t (x) is uniformly bounded with respect to t ∈ R + for all N ∈ N. So, there exists C > 0 such that R (ϕ ′ (x)) 2 u t (x)dx ≤ C 2 for all t ∈ R + . We deduce

Φ ′ (t) ≤ C |ξ ′ (t)| . (A.1)
By definition of the two sequences (r k ) k and (s k ) k , we have

Φ(s k ) -Φ(r k ) = κ .
Combining this identity with (A.1), it yields

C s k r k |ξ ′ (t)|dt ≥ κ .
We apply the Cauchy-Schwarz inequality and obtain:

C √ s k -r k ξ(r k ) -ξ(s k ) ≥ κ
since ξ is non-increasing (see Proposition 1.2). Moreover, ξ(t) converges as t converges to ∞ Lemma 1.4). It implies the convergence of ξ(r k )ξ(s k ) towards 0 when k tends to +∞. Consequently, s kr k converges to +∞ so lim sup

k-→+∞ s k -r k > 0.
Step 2: Lemma 1.4, Υ ǫ (u t ) -Υ ǫ (u ǫ ) = ∞ t ξ ′ (s)ds converges to 0. As ξ ′ is nonpositive, we deduce that ∞ k=N s k r k ξ ′ (s)ds converges also to 0 when N tends to +∞. As lim sup k-→+∞ s kr k > 0, we deduce there exists an increasing sequence q k ∈ [r k ; s k ] which converges to ∞ and such that ξ ′ (q k ) converges to 0 when k tends to ∞. Furthermore, R ϕ(x)u q k (x)dx ∈ [κ; 2κ] for all k ∈ N.

Step 3: By proceeding similarly as in the proof of Theorem 1.6, we extract a subsequence of (q k ) k (we continue to write it q k for simplifying the reading) such that u q k converges weakly towards a stationary measure u ǫ . Moreover,

u ǫ verifies R ϕ(x)u ǫ (x)dx ∈ [κ; 2κ].
We provide now a result that allows us to get the results of the main theorem (Theorem 2.1) with less strong condition: Proposition A.2. Let a probability measure du 0 which verifies (FE) and (ES). Then, for all t > 0, du t satisfies (FM).

Proof.

Step 1: If du 0 verifies (FM) then du t satisfies (FM) for all t > 0, see Theorem 2.12 in [START_REF] Herrmann | Large deviations and a Kramers' type law for self-stabilizing diffusions[END_REF]. We assume now that du 0 does not satisfy (FM). Let us introduce l 0 := min l ≥ 0 | E X 2l 0 = +∞. We know that E X 2l 0 t < for all t ≥ 0.

Step 2: Set t 0 > 0. We proceed a reduction ad absurdum by assuming that E X 2l 0 t 0 = +∞. This implies directly E X 2l 0 t = +∞ for all t ∈ [0, t 0 ]. We recall that 2m (resp. 2n) is the of the confining (resp. interaction) potential V (resp. F ). Also, q := max {m ; n}. For all t ∈ [0, t 0 ], the application x → F ′ * u t (x) is a polynomial function with parameters m 1 (t), • • • , m 2n-1 (t), where m j (t) is the j-th moment of the law du t . We recall the inequality (V):

sup 1≤j≤8q 2 sup t∈[0,t 0 ] m j (t) ≤ M 0 .
Consequently, the application x → V ′ (x)+F ′ * u t (x) is a polynomial function with degree 2q -1. Furthermore, the principal term does not depend of the moments of the law du t so we can write:

V ′ (x) + F ′ * u t (x) = κ 2q-1 x 2q-1 + P t (x)
where κ 2q-1 ∈ R * + is a constant, and P t is a polynomial function with degree at most 2q -2. Moreover, P t is parametrized by the 2n first moments only. Set l ∈ N. We introduce the function Q t (x) := 2lx 2q-1 P t (x)-l(2l -1)ǫx 2l-2 . As Q t is a polynomial function of degree less than 2l + 2q -3, we have the following inequality:

2lκ 2q-1 x 2l+2q-2 + Q t (x) ≥ C l x 2l+2q-2 -1 (A.2)
where C l is some positive constant. The application of Ito formula provides:

dX 2l t = 2lX 2l-1 t √ ǫdB t -2lκ 2q-1 X 2l+2q-2 t + Q t (X t ) dt .
After integration, we obtain:

X 2l t 0 = X 2l 0 + 2l √ ǫ t 0 0 X 2l-1 t dB t - t 0 0 2lκ 2q-1 X 2l+2q-2 t + Q t (X t ) dt .
We put Ω R := 1 {sup u∈[0,t 0 ] |Xu|≤R} for all R > 0. We take now the expectation after multiplying by Ω R :

E X 2l t 0 Ω R = E X 2l 0 Ω R - t 0 0 E 2lκ 2q-1 X 2l+2q-2 t + Q t (X t ) Ω R dt ≤ E X 2l 0 + C l t 0 -C l t 0 0 E X t Ω R dt
after using (A.2). We take l := l + 1q:

0 ≤ E X 2l 0 +2-2q t 0 Ω R ≤ C 1 -C 2 t 0 0 E X 2l 0 t Ω R dt
where C 1 and C 2 are positive constants. As R → Ω R is increasing and converges towards 1 almost surely when R tends to +∞, we deduce the pointwise convergence of E X 2l 0 t Ω R towards E X 2l 0 t = +∞ when R tends to +∞. Also, the application R → E X 2l 0 t Ω R being increasing for all t ∈ [0, t 0 ], it yields the convergence of

t 0 0 E X 2l 0 t Ω R dt towards +∞ when R tends towards +∞ which implies E X 2l 0 +2-2q t 0 Ω R < 0 for R big enough.
This contradicts the fact that E X 2l 0 +2-2q t 0 Ω R is positive for all R and increasing with respect to R. Consequently, for all t 0 > 0, E X 2l 0 t 0 < +∞.

Step 3: Set T > 0 and l 1 ∈ N such that l 1 ≥ l 0 where the integer l 0 is defined as previously: l 0 := min l ≥ 0 | E X 2l 0 = +∞ . If l 1 = l 0 , the application of Step 2 leads to E X 2l 1 T < +∞. If l 1 > l 0 , we put t i := i l 1 +1-l 0 T for all 1 ≤ i ≤ l 1 + 1l 0 . We apply Step 2 to t 1 and we deduce E X 2l 0 t 1 < +∞. By recurrence, we deduce E X 2l 0 +2i

t i < +∞ for all 1 ≤ i ≤ l 1 + 1 -l 0 , in particular E X 2l 0 +2(l 1 -l 0 ) t l 0 -l 1 < +∞ that means E X 2l 1 T < +∞.
This inequality holds for all l 1 ≥ l 0 the probability measure du T satisfies (FM).

In order to get the thirdness of the stationary measure (or a weaker result, see Theorem 1.11), we need to compute the small noise limit of the freeenergy for the stationary measures u + ǫ , u - ǫ and u 0 ǫ .

Lemma A.3. Set ǫ 0 such that there exist three families of stationary measures (u + ǫ ) ǫ∈]0;ǫ 0 ] , (u - ǫ ) ǫ∈]0;ǫ 0 ] and u 0 ǫ ǫ∈]0;ǫ 0 ] which verify

lim ǫ→0 u ± ǫ = δ ±a and lim ǫ→0 u 0 ǫ = 1 2 δ x 0 + 1 2 δ -x 0
where x 0 is defined in the introduction. Then, we have the following limits:

lim ǫ→0 Υ ǫ u ± ǫ = V (a) and lim ǫ→0 Υ ǫ u 0 ǫ = V (x 0 ) + 1 4 F (2x 0 ) .
Plus, by considering the measure v + ǫ (x) := Z -1 exp -2 ǫ (V (x) + F (xa)) , we have:

lim ǫ→0 Υ ǫ v + ǫ = V .
Proof. 1: We begin to prove the result for u 0 ǫ .

Step 1.1: We can write u 0

ǫ (x) = Z -1 exp -2 ǫ V (x) + F * u 0 ǫ (x) since it is a stationary measure. Hence Υ ǫ u 0 ǫ = - ǫ 2 log R exp - 2 ǫ V (x) + F * u 0 ǫ (x) dx - 1 2 R 2 F (x -y)u 0 ǫ (x)u 0 ǫ (y)dxdy . It has been proved in [HT09] (Theorem 3.1 if V ′′ (0) + F ′′ (0) > 0, Corollary 3.6 if V (0) + F ′′ (0) = 0 and Theorem 4.5 if V ′′ (0) + F ′′ (0) < 0 applied with f 2l (x) := x 2l ) that the 2l-th moment of u 0 ǫ tends towards x 2l 0 for all l ∈ N. Since F is a polynomial function, we get the convergence of R 2 F (x - y)u 0 ǫ (x)u 0 ǫ (y)dxdy towards F (2x 0 ) 2 .
Step 1.2: If V ′′ (0) + F ′′ (0) = 0, we can apply Lemma A.4 in [START_REF] Herrmann | Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small noise limit[END_REF] to f (x) := 1 and U ǫ (x) := V (x) + F * u 0 ǫ (x). This provides

R exp - 2 ǫ V (x) + F * u 0 ǫ (x) dx = C ǫ exp - 2 ǫ V (x 0 ) + F (2x 0 ) 2
where the constant C ǫ verifies ǫ log(C ǫ ) -→ 0 in the small noise limit. We deduce

- ǫ 2 log R exp - 2 ǫ V (x) + F * u 0 ǫ (x) dx -→ V (x 0 ) + F (2x 0 ) 2
when ǫ -→ 0. Consequently, we get the following limit:

Υ ǫ u 0 ǫ -→ V (x 0 ) + 1 4 F (2x 0 ) .
Step 1.3: We assume now V ′′ (0) + F ′′ (0) = 0. Then x 0 = 0 according to Proposition 3.7 and Remark 3.8 in [START_REF] Herrmann | Stationary measures for self-stabilizing processes: asymptotic analysis in the small noise limit[END_REF]. Proposition 3.2 and 3.3 in [START_REF] Herrmann | Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small noise limit[END_REF] imply

0 < lim inf ǫ→0 ǫ 1 2m 0 R exp - 2 ǫ V (x) + F * u 0 ǫ (x) dx and lim sup ǫ→0 ǫ 1 2m 0 R exp - 2 ǫ V (x) + F * u 0 ǫ (x) dx < +∞
where m 0 ∈ N * depends only on V and F . We deduce

- ǫ 2 log R exp - 2 ǫ V (x) + * u 0 ǫ (x) dx -→ 0
when ǫ -→ 0. we get the following limit:

Υ ǫ u 0 ǫ -→ 0 = V (x 0 ) + 1 4 F (2x 0 ) .
Step 2: We prove now the result for u + ǫ (the proof is similar for u - ǫ ).

Step 2.1: We can write u

+ ǫ (x) = Z -1 exp -2 ǫ (V (x) + F * u + ǫ (x)) since it a stationary measure. Hence Υ ǫ u + ǫ = - ǫ 2 log exp - 2 ǫ V (x) + F * u + ǫ (x) dx - 1 2 R 2 F (x -y)u + ǫ (x)u + ǫ (y)dxdy .
It has been proved in [START_REF] Herrmann | Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small noise limit[END_REF] (Theorem 6.3 applied with f l (x) := x l ) that the l-th moment of u + ǫ tends towards a l for all l ∈ N. Since F is a polynomial function, we obtain the convergence of R 2 F (x-y)u + ǫ (x)u + ǫ (y)dxdy towards 0.

Step 2.2: the second derivative of the application x → V (x)+F (x-a) in a is positive, we can apply Lemma A.4 in [START_REF] Herrmann | Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small noise limit[END_REF] to f (x) := 1 and U ǫ (x) :=

V (x) + F * u + ǫ (x) (after noting that U (i) ǫ (x) tends to V (i) (x) + F (i) (x -a) uniformly on each compact for all i ∈ N). This provides R exp - 2 ǫ V (x) + F * u + ǫ (x) dx = C ǫ exp - 2 ǫ V (a)
where the constant C ǫ verifies ǫ log(C ǫ ) -→ 0 in the small noise limit. We deduce

- ǫ 2 log R exp - 2 ǫ V (x) + F * u + ǫ (x) dx -→ V (a)
when ǫ -→ 0. Consequently, we get the following limit:

Υ ǫ u 0 ǫ -→ V (a) .
Step 3: We proceed similarly for v + ǫ .

We provide here a useful asymptotic result linked to the Laplace method.

Lemma A.4. Let U k and U ∈ C ∞ (R, R) such that for all i ∈ N, U Furthermore, for all N ∈ N, there exists p 1 , • • • , p r which verify p 1 +• • •+p r = 1 and p i ≥ 0 for all 1 ≤ i ≤ r such that we can extract a subsequence ψ(k) which satisfies

lim k→+∞ R x l exp -2 ǫ ψ(k) U ψ(k) dx R exp -2 ǫ ψ(k) U ψ(k) dx = r j=1 p j A l j for all 1 ≤ l ≤ N .
Proof. 1. The first point of the lemma is exactly the one of Lemma A.4 in [START_REF] Herrmann | Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small noise limit[END_REF].

2. Since U k (x) ≥ x 2 for x ≥ R and k > k c , we can confine each A (k) j in a compact subset. Then, the uniform convergence on all the compact subset implies the convergence of A (k) j towards A j when k tends to +∞. 3. Set ρ > 0 arbitrarily small such that [A jρ, A j + ρ] ⊂ I j . For obvious reasons, we can extract a subsequence such that

A i +ρ A i -ρ exp -2 ǫ ψ(k) U ψ(k) (x) dx r j=1 A j +ρ A j -ρ exp -2 ǫ ψ(k) U ψ(k) (x) dx -→ λ i (ρ)
with λ i (ρ) ≥ 0 for all 1 ≤ i ≤ r and r j=1 λ j (ρ) = 1. We can note that the generation of the sequence ψ(k) depends on the choice of ρ. Consequently, in the following, we can take ρ arbitrarily small then ǫ ψ(k) arbitrarily small. As the r families (λ j (ρ)) are bounded, we can extract a subsequence (ρ p ) p such that λ j (ρ p ) tends to λ j when p tends to +∞. Furthermore, λ j ≥ 0 for all 1 ≤ j ≤ r and r j=1 λ j = 1. For simplicity, we will write ρ (resp. k) instead of ρ p (resp. ψ(k)).

We introduce the function ζ Set τ > 0 arbitrarily small. We take R ≥ 2 such that

max z∈[A 1 -1;A 1 +1] U (z) + 2 < R 2 2
3.1: The convergence of λ j (ρ) towards λ j implies the existence of ρ 0 > 0 such that for all ρ < ρ 0 , we have:

T 1 (ρ) ≤ τ 5 (A.4) for all 1 ≤ l ≤ N .

3.2: By taking ρ < min ρ 0 ; min 1≤l≤N τ 5lR l-1 , we get:

T 2 (ρ, R) ≤ τ 5 (A.5)
for all 1 ≤ l ≤ N .

3.3:

We will prove that the third term tends to 0. It is sufficient to prove the following convergence: Let us prove the convergence towards 0 of the right hand term:

I j [A j -ρ,A j +ρ] c ζ (k) l (x)dx [A j -ρ,A j +ρ]
I j [A j -ρ,A j +ρ] c ζ (k) 0 (x)dx [A j -ρ,A j +ρ] ζ (k) 0 (x)dx ≤ R l+1 sup ζ (k) 0 (z) ; z ∈ I j [A j -ρ, A j + ρ] c A j + ρ 2 A j -ρ 2 ζ (k) 0 (x)dx ≤ R l+1 ρ sup ζ (k) 0 (z) ; z ∈ I j [A j -ρ, A j + ρ] c inf ζ (k) 0 (z) ; z ∈ A j -ρ 2 , A j + ρ 2 ≤ R l+1 ρ exp    - 2 ǫ k   inf z∈I j ∩[A j -ρ,A j +ρ] c U k (z) - sup z∈[A j -ρ 2 ,A j + ρ 2 ] U k (z)     
Let ρ 1 > 0 such that for all ρ < ρ 1 , we have:

min 1≤j≤r    inf z∈I j ∩[A j -ρ,A j +ρ] c U (z) - sup z∈[A j -ρ 2 ,A j + ρ 2 ] U (z)    ≥ δ > 0 .
We take ρ < min ρ 0 , ρ 1 , min 1≤l≤N τ 5lR l-1 . As U k converges uniformly towards U on all the compact subset, we deduce that for k ≥ k 0 , we have: where C(l) is a constant. We recall that we assume max

z∈[A 1 -1;A 1 +1] U (z) + 2 < R 2 2
. Since U k converges U uniformly on all the compact subset, we have max z∈[A 1 -1;A 1 +1] U k (z) + 1 < R 2 2 for k ≥ k 1 (independantly of ρ). Consequently:

T 4 (R, k) ≤ 2C(l)ǫ l+1 2 k exp -2 ǫ k max z∈[A 1 -1;A 1 +1] U k (z) + 1 A 1 +1 A 1 -1 exp -2 ǫ k U k (z) dx ≤ C(l)ǫ l+1 2 k exp - 2 ǫ k .
For k ≥ k 2 , we have the inequality for all 1 ≤ l ≤ N . This achieves the proof.

Remark A.5. This lemma seems weaker than Lemma A.4 in [START_REF] Herrmann | Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small noise limit[END_REF]. However, in Lemma A.4 we do not assume that the second derivative of U is positive in all the global minimum locations.

It is a great pleasure to thank Samuel Herrmann for his remarks concerning this work. The most of the ideas of this work have been found while I was at the Institut Élie Cartan in Nancy. And so I wanted to mention that I would not have been able to write it without the hospitality I received from the beginning. Finalement, un très grand merci à Manue et à Sandra pour tout.

  k converges uniformly on all compact subset when k tends to +∞. Set a sequence (ǫ k ) k which tends to 0 as k tends to +∞. If U has r global minimum locations A 1 • • • < A r and if there exist R > 0 and k c such that U k (x) > x 2 for all |x| > R and k > k c , then, for k big enough, we get: 1. U k has exactly one global minimum location A (k) j on each interval I j , where I j represents the Voronoï cells corresponding to the central points A j , with 1 ≤ j ≤ r.2. A (k) jtends to A j when k tends to +∞.

l

  (x) := x l exp -2 ǫ k U ǫ k (x) for all l ∈ N. By using classical analysis' inequality, we obtain:l j ≤ T 1 + T 2 + T 3 + T 4 + T 5 with T 1 (ρ) := r j=1 (λ jλ j (ρ)) A l j , T 2 (ρ, R) := ρlR l-1 , T 3 (ρ, k) := r j=1 I j [A j -ρ,A j +ρ] c ζ 3 + T 4 ) .

  ζ

  all 1 ≤ j ≤ r. Since I j ⊂ [-R, R], we haveI j [A j -ρ,A j +ρ] c ζ (k) l (x)dx [A j -ρ,A j +ρ] ζ (k) 0 (x)dx ≤ R l I j [A j -ρ,A j +ρ] c ζ (k) 0 (x)dx [A j -ρ,A j +ρ] ζ

T 3

 3 (ρ, k) ≤ τ 5 1 + max 1≤l≤N r j=1 |A j | l (A.6) for all 1 ≤ l ≤ N .3.4: By using the growth property on U k then the change of variable x := √ ǫ k y, it yields

  1≤l≤N C(l) × 1 + max 1≤l≤N r j=1 |A j | l .By taking k ≥ max {k 0 , k 1 , k 2 }, we get:T 4 (R, k) ≤ τ 5 1 + max 1≤l≤N r j=1 |A j | l (A.7)for all 1 ≤ ≤ N .3.5 By taking < min ρ 0 , ρ 1 , τ 5lR l-1 and k ≥ max {k 0 , k 1 k 2 }, Inequalities (A.3), (A.4), (A.5), (A.6) and (A.7) provide R ζ

* Supported by the DFG-funded CRC 701, Spectral