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Self-stabilizing diffusions are stochastic processes, solutions of nonlinear stochastic differential equation, which are attracted by their own law. This specific self-interaction leads to singular phenomenons like non uniqueness of associated stationary measures when the diffusion moves in some non convex environment (see [5]). The aim of this paper is to describe these invariant measures and especially their asymptotic behavior as the noise intensity in the nonlinear SDE becomes small. We prove in particular that the limit measures are discrete measures and point out some properties of their support which permit in several situations to describe explicitly the whole set of limit measures. This study requires essentially generalized Laplace's method approximations.


Introduction
Historically self-stabilizing processes were obtained as McKean-Vlasov limit in particle systems and were associated with nonlinear partial differential equations [6,7].The description of the huge system is classical: it suffices to consider N particles which form the solution of the stochastic differential system: where (W i t ) i is a family of independent one-dimensional Brownian motions, ǫ some positive parameter.In (1.1), the function V represents roughly the environment the Brownian particles move in and the interaction function F describes the attraction between one particle and the whole ensemble.As N becomes large, the law of each particle converges and the limit is the distribution u ǫ t (d x) of the so-called self-stabilizing diffusion (X ǫ t , t ≥ 0).This particular phenomenon is well-described in a survey written by A.S. Sznitman [8].The process (X ǫ t , t ≥ 0) is given by This process is of course nonlinear since solving the preceding SDE (1.2) consists in pointing out the couple (X ǫ t , u ǫ t ).By the way, let us note in order to emphasize the nonlinearity of the study that u ǫ t (x) satisfies: Here * stands for the convolution product.There is a relative extensive literature dealing with the questions of existence and uniqueness of solutions for (1.2) and (1.3), the existence and uniqueness of stationary measures, the propagation of chaos (convergence in the large system of particles)...The results depend of course on the assumptions concerning both the environment function V and the interaction function F .Let us just cite some key works: [4], [6], [7], [10], [9], [1] and [2], [3].The aim of this paper is to describe the ǫ-dependence of the stationary measures for self-stabilizing diffusions.S. Benachour, B. Roynette, D. Talay and P. Vallois [1] proved the existence and uniqueness of the invariant measure for self-stabilizing diffusions without the environment function V .Their study increased our motivation to analyze the general equation (1.2), which is why our assumptions concerning the interaction function F are close to theirs.In our previous paper [5], we considered some symmetric double-well potential function V and emphasized a particular phenomenon which is directly related to the nonlinearity of the dynamical system: under suitable conditions, there exist at least three invariant measures for the self-stabilizing diffusion (1.2).In particular, there exists a symmetric invariant measure and several so-called outlying measures which are concentrated around one bottom of the double-well potential V .Moreover, if V ′′ is some convex function and if the interaction is linear, that is F ′ (x) = αx with α > 0, then there exist exactly three stationary measures as ǫ is small enough, one of them being symmetric.What about the ǫ-dependence of these measures ?In the classical diffusion case, i.e. without interaction, the invariant measure converges in the small noise limit (ǫ → 0) to 1  2 δ −a + 1 2 δ a where δ represents the Dirac measure and, both a and −a stand for the localization of the double-well V bottoms.The aim of this work is to point out how strong the interaction function F shall influence the asymptotic behavior of the stationary measures.In [5], under some moment condition: the 8q 2 -th moment has to be bounded, the analysis of the stationary measures permits to prove that their density satisfies the following exponential expression: To prove the hypothetical convergence of u ε towards some u 0 , the natural framework is Laplace's method, already used in [5].Nevertheless, the nonlinearity of our situation does not allow us to use these classical results directly.
Main results: We shall describe all possible limit measures for the stationary laws.Under a weak moment condition satisfied for instance by symmetric invariant measures (Lemma 5.2) or in the particular situation when V is a polynomial function satisfying deg(V ) > deg(F ) (Proposition 3.1), a precise description of each limit measure u 0 is pointed out: u 0 is a discrete measure u 0 = r i=1 p i δ A i (Theorem 3.6).The support of the measure is directly related to the global minima of some potential W 0 which enables us to obtain the following properties (Proposition 3.7): for any 1 ≤ i, j ≤ r, we get We shall especially construct families of invariant measures which converge to δ a and δ −a where a and −a represents the bottom locations of the potential V (Proposition 4.1).For suitable functions F and V , these measures are the only possible asymmetric limit measures (Proposition 4.4 and 4.5).Concerning families of symmetric invariant measures, we prove the convergence, as ǫ → 0, under weak convexity conditions, towards the unique symmetric limit measure 1 2 δ −x 0 + 1 2 δ x 0 where 0 ≤ x 0 < a (Theorem 5.4).A natural bifurcation appears then for F ′′ (0) = sup z∈ −V ′′ (z) =: θ .If F ′′ (0) < θ , then x 0 > 0 and the support of the limiting measure contains two different points.If F ′′ (0) ≥ θ , then x 0 = 0 so the support contains only one point : 0. It states some competitive behavior between both functions V and F .We shall finally point out two examples of functions V and F which lead to the convergence of any sequence of symmetric self-stabilizing invariant measures towards a limit measure whose support contains at least three points (Proposition 5.6 and 5.7): the initial system is obviously deeply perturbed by the interaction function F .The convexity of V ′′ and the linearity of F ′ imply the existence of exactly three limit measures: one is symmetric and concentrated on either one or two points and the two others u ± 0 are outlying.Furthermore if V ′′ and F ′′ are convex, any symmetric stationary measures converge to 1  2 The material is organized as follows.After presenting the essential assumptions concerning the environment function V and the interaction function F , we start the asymptotic study by the simple linear case F ′ (x) = αx with α > 0 (Section 2) which permits explicit computation for both the symmetric measure (Section 2.1) and the outlying ones (Section 2.2).In Section 3 the authors handle the general interaction case proving the convergence of invariant measures subsequences towards finite combination of Dirac measures.The attention shall be focused on these limit measures (Section 3.2).To end the study, it suffices to consider assumptions which permit to deduce that there exist exactly three limit measures: one symmetric, δ −a and δ a .This essential result implies the convergence of both any asymmetric invariant measure (Section 4) and any symmetric one (Section 5).Some examples are presented.

Main assumptions
Let us first describe different assumptions concerning the environment function V and the interaction function F .The context is similar to our previous study [5] and is also weakly related to the work [1].We assume the following properties for the function V : (V-1) Regularity: V ∈ ∞ ( , ). ∞ denotes the Banach space of infinitely bounded continuously differentiable function.
(V-2) Symmetry: V is an even function.
(V-3) V is a double-well potential.The equation V ′ (x) = 0 admits exactly three solutions : a, −a and 0 with a > 0; V ′′ (a) > 0 and V ′′ (0) < 0. The bottoms of the wells are reached for x = a and x = −a.

(V-7)
The growth of the potential V is at most polynomial: there exist q ∈ * and C q > 0 such that Typically, V is a double-well polynomial function.We introduce the parameter θ which plays some important role in the following: Let us note that the simplest example (most famous in the literature) is V (x) = x 4 4 − x 2 2 whose bottoms are localized in −1 and 1 and with parameter θ = 1.Let us now present the assumptions concerning the attraction function F .
(F-1) F is an even polynomial function of degree 2n with F (0) = 0. Indeed we consider some classical situation: the attraction between two points x and y only depends on the distance (F-2) F is a convex function.
(F-3) F ′ is a convex function on + therefore for any x ≥ 0 and y ≥ 0 such that x ≥ y we get (F-4) The polynomial growth of the attraction function F is related to the growth condition (V-7): Let us define the parameter α ≥ 0:

The linear interaction case
First, we shall analyze the convergence of different invariant measures when the interaction function F ′ is linear: In [5], we proved that any invariant density satisfies some exponential expression given by (1.4) provided that its 8q 2 -th moment is finite.This expression can be easily simplified, the convolution product is determined in relation to the mean of the stationary law.The symmetric invariant measure denoted by u 0 ε becomes The asymptotic behavior of the preceding expression is directly related to classical Laplace's method for estimating integrals and is presented in Section 2.1.If ǫ is small, Proposition 3.1 in [5] emphasizes the existence of at least two asymmetric invariant densities u ± ε defined by Here m ± ε represents the average of the measure u ± ε (d x) which satisfies: for any δ ∈]0, 1[ there exists ǫ 0 > 0 such that 3) enable us to develop, in Section 2.2, the asymptotic analysis of the invariant law in the asymmetric case.

Convergence of the symmetric invariant measure.
First of all, let us determine the asymptotic behavior of the measure u 0 ǫ (d x) as ǫ → 0. By (2.1), the density is directly related to the function W 0 which admits a finite number of global minima.Indeed due to conditions (V-5) and (V-6), we know that W ′′ 0 ≥ 0 on [−a; a] c and V is equal to an analytic function on [−a; a].Hence W ′ 0 admits a finite number of zeros on € or W ′ 0 = 0 on the whole interval [−a, a] which implies immediately W ′′ 0 = 0.In the second case in particular we get W ′′ 0 (a) = 0 which contradicts W ′′ 0 (a) = V ′′ (a) + α > 0. We deduce that W ′ 0 admits a finite number of zeros on €.The measure u 0 ǫ can therefore be developed with respect to the minima of W 0 .
Theorem 2.1.Let A 1 < . . .< A r the r global minima of W 0 and ω 0 = min z∈ W 0 (z).For any i, we introduce k As ǫ → 0, the measure u 0 ε defined by (2.1) converges weakly to the following discrete measure Proof.Let f be a continuous and bounded function on .We define A 0 = −∞ and A r+1 = +∞.Then, for i ∈]1; r[, we apply Lemma A.1 to the function U = W 0 and to each integration support ].We obtain the asymptotic equivalence as ǫ → 0: This equivalence is also true for the supports J 1 and J r .Indeed, we can restrict the semiinfinite support of the integral to a compact one since f is bounded and since the function W 0 admits some particular growth property: W 0 (x) ≥ x 2 for |x| large enough.Hence denoting Applying the preceding asymptotic result (2.5) on one hand to the function f and on the other hand to the constant function 1, we estimate the ratio We deduce the announced weak convergence of u 0 ε towards u 0 0 .
By definition, V is a symmetric double-well potential: it admits exactly two global minima.We now focus our attention on W 0 and particularly on the number of global minima which represents the support cardinal (denoted by r in the preceding statement) of the limiting measure.
Proposition 2.2.If V ′′ is a convex function, then u 0 0 is concentrated on either one or two points, that is r = 1 or r = 2.
Proof.We shall proceed using reductio ad absurdum.We assume that the support of u 0 0 contains at least three elements.According to the Theorem 2.1, they correspond to minima of W 0 .Therefore there exist at least two local maxima: W ′ 0 admits then at least five zeros.Applying Rolle's Theorem, we deduce that W ′′ 0 (x) is vanishing at four distinct locations.This leads to a contradiction since V ′′ is a convex function so is W ′′ 0 .
Let us note that the condition of convexity for the function V ′′ has already appeared in [5] (Theorem 3.2).In that paper, we proved the existence of exactly three invariant measures for (1.2) as the interaction function F ′ is linear.What happens if V ′′ is not convex ?In particular, we can wonder if there exists some potential V whose associated measure u 0 0 is supported by three points or more.
Proposition 2.3.Let p 0 ∈ [0, 1[ and r ≥ 1.We introduce • some family There exists a potential function V which verifies all assumptions (V-1)-(V-8) and a positive constant α such that the measure u 0 0 , associated to V and the linear interaction F ′ (x) = αx, is given by Proof.
Step 1.Let us define a function denoted by W as follows: where C is a positive constant and ξ is a polynomial function.C and ξ will be specified in the following.Using (2.1), we introduce According to Theorem 2.1, the symmetric invariant measure associated to the functions V and F converges to a discrete measure whose support is either the set 1 also enables us to evaluate the weights (p i ) i .An obvious analysis leads to k 0 = 1.Moreover, the second derivative of W satisfies and By (2.4), we know that ) and 2p 0 Step 2. Let us determine the polynomial function ξ.
Step 2.1.First case: p 0 = 0. We choose ξ such that ξ(A 2 r ) = 1.Then (2.12) leads to the equation Finally, it suffices to choose the following polynomial function which solves in particular (2.14): Step 2.2.Second case: p 0 = 0. Using similar arguments as those presented in Step 2.1, we construct some polynomial function ξ satisfying (2.13).First we set ξ(0) = 1 and define We choose the following function In Step 2.1 and 2.2, the stationary symmetric measures associated to ξ converge to Step 3. It remains to prove that all conditions (V-1)-(V-8) are satisfied by the function V defined in Step 1.The only one which really needs to be carefully analyzed is the existence of three solutions to the equation V ′ (x) = 0. Since W defined by (2.7) and (2.8) is an even function, x is well defined and represents some even polynomial function which tends to +∞ as |x| becomes large.Hence, there exists some R > 0 large enough such that ρ is strictly decreasing on the interval ] − ∞; −R] and strictly increasing on Then, for any α > α ′ , the equation ρ(z) = α admits exactly two solutions.This implies the existence of exactly three solutions to V ′ (x) = 0 i.e. condition (V-3).

Convergence of the outlying measures
In the preceding section, we analyzed the convergence of the unique symmetric invariant measure u 0 ǫ as ǫ → 0. In [5], we proved that the set of invariant measures does not only contain u 0 ǫ .In particular, for ǫ small enough, there exist asymmetric ones.We suppose therefore that ε is less than the critical threshold below which the measures u + ε and u − ε defined by (2.2) and (2.3) exist.We shall focus our attention on their asymptotic behavior.Let us recall the main property concerning the mean m ± (ǫ) of these measures: for all δ > 0, there exists ε 0 > 0 small enough such that m ± (ε) − (±a) + V (3) (±a) Here a and −a are defined by (V-3).Let us note that we do not assume V ′′ to be a convex function nor u + ε (resp.u − ε ) to be unique.
Proof.We just present the proof for u + ε since the arguments used for u − ε are similar.Let us define
By Lemma A.4 in [5] (in fact a slight modification of the result: the function f m appearing in the statement needs just to be (3) -continuous in a small neighborhood of the particular point x µ ), the following asymptotic result (as ǫ → 0) yields: We can obtain the lower-bound by similar arguments, just replacing ξ + ( y) by ξ − ( y) where Therefore, for any η > 1, there exists some for ǫ ≤ ǫ 1 .In the same way, we obtain some ǫ 2 > 0, such that for ǫ ≤ ǫ 2 .Taking the ratio of (2.18) and (2.17), we immediately obtain: for ǫ ≤ min(ǫ 1 , ǫ 2 ).δ is arbitrarily small and η is arbitrary close to 1, so we deduce the convergence of

The set of limit measures
In the particular case where V ′′ is a convex function and ǫ is fixed, we can describe exactly the set of invariant measures associated with (1.2).The statement of Theorem 3.2 in [5] makes clear that this set contains exactly three elements.What happens as ǫ → 0 ?We shall describe in this section the set of all measures defined as a limit of stationary measures as ǫ → 0. We start with a preliminary result: Proof.Since V ′′ is a symmetric convex function, we get θ = −V ′′ (0) where θ is defined by (1.5).
2) If α < −V ′′ (0), then χ admits at most three zeros due to the convexity of its derivative.Since χ(0) = 0 and since χ is an odd function, there exists at most one positive zero.The inequality α + V ′′ (0) < 0 implies that χ is strictly decreasing at the right side of the origin.Moreover lim x→+∞ χ(x) = +∞ which permits to conclude the announced existence of one positive zero x 0 .We easily verify that α + V ′′ (x 0 ) ≥ 0.
Proposition 2.6.If V ′′ is a convex function then the family of invariant measures admits exactly three limit points, as ǫ → 0. Two of them are asymmetric: δ a and δ −a and the third one is symmetric Proof.Since V ′′ a is convex function, there exist exactly three invariant measures for (1.2) provided that ǫ is small enough (see Proposition 3.2 in [5]).These measures correspond to u 0 ε , u + ε and u − ε defined by (2.1) and (2.2). 1) Theorem 2.1 emphasizes the convergence of u 0 ε towards the discrete probability measure u 0 0 defined by (2.4).Due to the convexity of V ′′ , the support of this limit measure contains one or two real numbers (Proposition 2.2) which correspond to the global minima of U(x) = V (x) + α 2 x 2 .According to the Lemma 2.5, we know that U admits a unique global minimum on + denoted by According to the Theorem 2.4, u ± ε converges to δ ±a .

The general interaction case
In this section we shall analyze the asymptotic behavior of invariant measures for the self-stabilizing diffusion as ǫ → 0. Let us consider some stationary measure u ε .According to Lemma 2.2 of [5], the following exponential expression holds: Since F is a polynomial function of degree 2n, the function W ǫ just introduced can be developed as follows µ l (ε) being the l-order moment of the measure u ε .W ε is called the pseudo-potential.Since F is a polynomial function of degree 2n, the preceding sums in (3.2) are just composed with a finite number of terms.In order to study the behavior of u ǫ for small ε, we need to estimate precisely the pseudo-potential W ε .Let us note that, for some specific p ∈ , the L p -convergence of u ε towards some measure u 0 implies the convergence of the associated pseudo-potential W ε towards a limit pseudo-potential W 0 .
The study of the asymptotic behavior of u ǫ shall be organized as follows: • Step 1.First we will prove that, under the boundedness of the family {µ 2n−1 (ε), ǫ > 0} with 2n = deg(F ), we can find a sequence (ε k ) k≥0 satisfying lim k→∞ ǫ k = 0 such that W ε k converges to a limit function W 0 associated to some measure u 0 .
• Step 2. We shall describe the measure u 0 : it is a discrete measure and its support and the corresponding weights satisfy particular conditions.
• Step 3. We analyze the behavior of the outlying measures concentrated around a and −a and prove that these measures converge towards δ a and δ −a respectively.We show secondly that these Dirac measures are the only asymmetric limit measures.We present some example associated with the potential function Finally we focus our attention on symmetric measures.After proving the boundedness of the moments, we discuss non trivial examples (i.e.nonlinear interaction function F ′ ) where there exist at least three limit points.

Weak convergence for a subsequence of invariant measures
Let u ε ε>0 be a family of stationary measures.The main assumption in the subsequent developments is: We assume that the family This assumption is for instance satisfied if the degree of the environment potential V is larger than the degree of the interaction potential: Proposition 3.1.Let (u ε ) ε>0 a family of invariant measures for the diffusion (1.2).We assume that V is a polynomial function whose degree satisfies 2m 0 := deg(V ) > deg(F ) = 2n.Then the family Proof.Let us assume the existence of some decreasing sequence (ε k ) k∈ which tends to 0 and such that the sequence of moments µ 2m 0 (k) := € x 2m 0 u ε k (x)d x tends to +∞.By (3.1) and (3.2) we obtain: , where r (k) is a combination of the moments µ j (k), with 0 ≤ j ≤ max(0, 2n − r), and C 2m 0 = V (2m 0 ) (0)/(2m 0 )!. Let us note that the coefficient of degree 0 in the polynomial expression disappears since the numerator and the denominator of the ratio contain the same expression: that leads to cancellation.Moreover r (k) does not depend on ǫ k for all r s.t.2n ≤ r ≤ 2m 0 and that there exists some r such that r (k) tends to +∞ or −∞ since µ 2m 0 (k) is unbounded.Let us define Then the sequences η r (k , for 1 ≤ r ≤ 2m 0 − 1, are bounded.Hence we extract a subsequence (ε Ψ(k) ) k∈ such that η r (Ψ(k)) converges towards some η r as k → ∞, for any 1 ≤ r ≤ 2m 0 −1.For simplicity, we shall conserve all notations: µ 2m 0 (k), η r (k), φ k ... even for sub-sequences.The change of variable x := φ k y provides The highest moment appearing in the expression k is the moment of order 2n − k.By Jensen's inequality, there exists a constant The following upper-bound holds: We deduce from the previous estimate that the left hand side of (3.3) tends to +∞.Let us focus our attention on the right term.In order to estimate the ratio of integrals, we use asymptotic results developed in the annex of [5], typically generalizations of Laplace's method.An adaptation of Lemma A.4 enables us to prove that the right hand side of (3.3) is bounded: in fact it suffices to adapt the asymptotic result to the particular function U( y) := 2n−1 r=1 η r y r + C 2m 0 y 2m 0 which does not satisfy a priori U ′′ ( y 0 ) > 0. This generalization is obvious since the result we need is just the boundedness of the limit, so we do not need precise developments for the asymptotic estimation.In the lemma the small parameter ǫ will be the ratio and f ( y) := y 2m 0 .Since the right hand side of (3.3) is bounded and the left hand side tends to infinity, we obtain some contradiction.Finally we get that {µ 2m 0 (ǫ), ǫ > 0} is a bounded family.
From now on, we assume that (H) is satisfied.Therefore applying Bolzano-Weierstrass's theorem we obtain the following result: Lemma 3.2.Under the assumption (H), there exists a sequence (ε k ) k≥0 satisfying lim k→∞ ǫ k = 0 such that, for any 1 ≤ l ≤ deg(F ) − 1, µ l ε k converges towards some limit value denoted by µ l (0) with As presented in (3.2), the moments µ l (ǫ) characterize the pseudo-potential W ε .We obtained a sequence of measures whose moments are convergent so we can extract a subsequence such that the pseudo-potential converges.For r ∈ * , we write : Like in (3.2), there is a finite number of terms non equal to 0 in the two sums so W 0 (x) is defined for all x ∈ €.
Proposition 3.3.Under condition (H), there exists a sequence (ε k ) k≥0 satisfying lim k→∞ ǫ k = 0 such that, for all j ∈ , (W ( j) 0 , uniformly on each compact subset of , where the limit pseudo-potential W 0 is defined by (3.4), and u ε k k≥1 converges weakly towards some probability measure u 0 .Proof.By (3.4), we obtain, for any p ≥ 1, Let us note that the sum in the right hand side of the previous inequality is finite.Using Lemma 3.2, we obtain the existence of some subsequence which enables us the convergence towards 0 of each term.Therefore, for all p ∈ , ω p (ε k ) tends to ω p (0) when k tends to infinity.Let x ∈ , since V is ∞ -continuous, both derivatives W ( j) 0 (x) are well defined.Moreover we get Since there is just a finite number of terms, and since each term tends to 0, we obtain the pointwise convergence of W ( j) In order to get the uniform convergence, we introduce some compact K: it suffices then to prove that, for j ≥ 1, sup y∈K, k≥0 W ( j) ǫ k ( y) is bounded.This is just an obvious consequence of the regularity of V and of the following bound: Let us prove now the existence of a subsequence of (ǫ k ) k≥0 for which we can prove the weak convergence of the corresponding sequence of invariant measures.According to condition (H), µ 2 ε k ; k ≥ 1 is bounded; we denote m 2 the upper-bound.Using the Bienayme-Tchebychev's inequality, the following bound holds for all R > 0 and k ≥ 1: Consequently the family of probability measures u ε k ; k ∈ * is tight.Prohorov's Theorem allows us to conclude that u ε k ; k ∈ * is relatively compact with respect to the weak convergence.Lemma 3.4.Each limit function W 0 admits a finite number r ≥ 1 of global minima.
Proof.According to (3.4), W ′ 0 (x) = V ′ (x) + P ′ (x) where P is a polynomial function.Since V is equal to an analytic function on [−a; a] (see Condition (V-6)), we deduce that W ′ 0 has a finite number of zeros on this interval.Indeed, if this assertion is false, then W ′′ 0 (a) = 0 which contradicts the following limit W ′′ 0 (a) = V ′′ (a) + lim k→∞ F ′′ * u ǫ k (a) > 0 due to the convexity property of F (F-2).By (3.1) and condition (V-4) and since F is even and F ′ is convex on + , we deduce that W ε (x) ≥ V (x) ≥ C 4 x 4 − C 2 x 2 for any ǫ > 0. Taking the limit with respect to the parameter ǫ, we get W 0 (x) ≥ C 4 x 4 − C 2 x 2 for all x ∈ .This lower-bound allows us to conclude that there exists some large interval [−R; R] which contains all the global minima of W 0 .It remains then to study the minima of W 0 on the compact K := [−R; −a] [a; R].According to the property (V-5), V ′′ (x) > 0 for all a ∈ K. Moreover F is a convex function (condition (F-2)) therefore the definition (3.1) implies: W ′′ ε (x) ≥ min z∈[−R;−a] V ′′ (z) > 0 for all x ∈ K.By Proposition 3.3, there exists some sequence (ǫ k ) k≥0 such that W ′′ ε k converges towards W ′′ 0 uniformly on K. Consequently W ′′ 0 (x) > 0 for all x ∈ K and so W ′ 0 admits a finite number of zeros on K.
Since W 0 admits r global minima, we define A 1 < • • • < A r their location: We introduce a covering U δ of these points: The set {A 1 , . . ., A r } plays a central role in the asymptotic analysis of the measures (u ǫ ) ǫ .In particular we can prove that u 0 defined in Proposition 3.3 is concentrated around these points.
Proposition 3.5.Let W 0 and (ǫ k ) k≥0 be defined by Proposition 3.3.Then the following convergence results hold: 2) For δ > 0 small enough, lim Proof. 1) The limit pseudo-potential W 0 is ∞ and its r global minima are in U δ .Since lim x→±∞ W 0 (x) = +∞, there exists η > 0 such that W 0 (x) ≥ w 0 + η for all x / ∈ U δ .Let us now point out a lower bound for W ǫ k .Using (V-5), (F-3), (1.6) and the definition (3.1), we prove that Moreover, due to the boundedness of the moments (condition (H)), W ′ ǫ k (0) and W ′′ ǫ k (0) are bounded uniformly with respect to k ≥ 0. This property combined with (3.7) leads to the existence of some R > 0 (independent of k) such that W ε k (x) ≥ w 0 + η for |x| ≥ R. What happens on the compact set [−R, R] ∩ (U δ ) c ? Proposition 3.3 emphasizes the uniform convergence of W ε k on this compact set.As a consequence, there exists some In order to estimate u ǫ k (x), we need a lower bound for the denominator in the ratio (3.1).We denote this denominator D ǫ .Since the continuous function W 0 admits a finite number of minima, there exists γ with δ > γ > 0 such that, for all x ∈ U γ , we have W 0 (x) ≤ w 0 + η 8 .By Proposition 3.3, W ε k converges uniformly on each compact subset of to W 0 .Therefore there exists k 1 such that for all k ≥ k 1 , for all x ∈ U γ , we have W ε k (x) ≤ w 0 + η 4 .Finally the denominator in (3.1) can be lower bounded: For k ≥ k 0 ∨ k 1 , according to (3.8) and (3.9), the following bound holds The right hand side in (3.10) tends to 0 as k → ∞ which proves the first part of the statement.
2) Let δ > 0 and K > 0. We split the integral support into two parts: Moreover p i is independent of the parameter δ.
Proof. 1) First we shall prove that the coefficients p i are well defined.Let us fix some small positive constant δ.We define p i (δ) the limit of Of course this limit exists since, by Proposition 3.3, (u ǫ k ) k≥1 converges weakly.Furthermore this limit is independent of δ.Indeed let us choose δ ′ < δ.By definition, we obtain An obvious application of the statement 2) in Proposition 3.5 allows us to obtain p i (δ ′ ) = p i (δ) =: p i .2) Let us prove now that u 0 is a discrete probability measure.Let f be a continuous and bounded function on .Let δ > 0 be small enough such that the intervals U i (δ) := [A i − δ; A i + δ] are disjointed.The weak convergence is based on the following difference: The boundedness of the function f and the statement 2) of Proposition 3.5 imply that R tends to 0 as k → ∞.Let us now estimate each term ∆ i ( f ): Due to the continuity of f , sup x∈U i (δ) f (x) − f (A i ) is small as soon as δ is small enough.Moreover for some fixed δ, the definition of p i leads to the convergence of u ε k (U i (δ)) − p i towards 0 as k → ∞.Combining these two arguments allows us to obtain the weak convergence of u ǫ k towards the discrete measure r i=1 p i δ A i which can finally be identified with u 0 .
It is important to note that we did not prove the convergence of any sequence of stationary measures u ε as ε → 0 because we did not prove the convergence of the moments.However, we know in advance that each limit value is a discrete probability measure.

Description of the limit measures
We have just pointed out in the previous section that all limit measures are discrete probability measures.Each limit measure shall be denoted in a generic way u 0 and is associated with some limit function W 0 defined by (3.4).Therefore we have the following expression u 0 = r i=1 p i δ A i where the A i are the global minima of W 0 , rearranged in increasing order, and We will now refine this result by exhibiting properties of the points A i and the weights p i .Proposition 3.7.1) For all 1 ≤ i ≤ r and 1 ≤ j ≤ r, we have : 3) If α ≥ θ , where α (resp.θ ) has been defined by (1.6) (resp.by (1.5)), the support of any limit measure contains some unique point.4) If V ′′ and F ′′ are convex, the support of any limit measure contains at most two points.
Proof. 1) In the proof of Proposition 3.3, the tightness of the family (u ǫ ) ǫ was presented.This argument combined with the convergence of the moments (µ l (ǫ k )) k≥0 stated in Lemma 3.2 and equation (3.1), enables us to express W ′ 0 as follows: It suffices then to make it clear that u 0 is a discrete measure (Theorem 3.6).By definition, A i is a local minimum of W 0 therefore it vanishes W ′ 0 and in addition, verifies W ′′ 0 (A i ) ≥ 0: we directly deduce (3.11) and (3.13).Furthermore, by definition (3.5) of A i , W 0 (A i ) = W 0 (A j ) for all i and j which implies (3.12).2) By definition, A r is the largest location of the global minimum of W 0 and A 1 is the smallest one.According to (3.11), they verify Since A r − A j > 0 for j < r and A 1 − A j < 0 for j > 1 and since F ′ is non decreasing, we deduce the inequalities V ′ (A 1 ) ≥ 0 and V ′ (A r ) ≤ 0. Consequently we get A 1 ≥ −a and A r ≤ a. Besides, if r ≥ 2 then V ′ (A 1 ) > 0 and V ′ (A r ) < 0. However, V ′ is positive on ] − a; 0[ and negative on ]0; a[, therefore −a < A 1 < 0 < A r < a.
3) We assume that A r > A 1 .By (V-6) the equation V ′′ (x) = −θ admits just a finite number of solutions, therefore we get − V ′ (A r ) − V ′ (A 1 ) < θ A r − A 1 .According to (3.11) applied to A 1 and to A r , we obtain: Using the following properties: F is an even function, see (F-1), and moreover This inequality contradicts the hypothesis A r > A 1 since α ≥ θ .

4)
We can compute the second derivative of the limit potential W ′′ 0 = V ′′ + r j=1 p j F ′′ (. − A j ).Moreover V ′′ and F ′′ are convex functions so is W ′′ 0 .Let us assume that W 0 admits at least three global minima then it also admits two local maxima: at least five critical points in all.The application of Rolle's theorem implies at least four zeros for the function W ′′ 0 which is in fact nonsense since W ′′ 0 is a convex function.
Proposition 3.7 allows us in a suitable situation to describe precisely the set of limit measures.
Remark 3.8.1) In the particular case when F (x) = 0, for all x ∈ , that is without self-stabilization, the equations (3.11), (3.12) and (3.13) are satisfied if and only if the locations A i are the minima of V .Hence the limit stationary measure weighs the points −a and a.
2) Let us assume α ≥ θ .If (u ǫ k ) k≥1 represents some family of symmetric stationary measures with bounded moments of order 1, . . ., deg(F ) − 1 then the sequence converges weakly to the Dirac measure u 0 = δ 0 .Indeed, according to (3.12) and Proposition 3.7, the single point corresponding to the support of u 0 is a zero of V ′ that is −a, 0 or a (see hypothese (V-3)).Besides u 0 is symmetric, so we get u 0 = δ 0 .
3) Let us assume that α ≥ θ .If we consider a sequence of outlying stationary measures which moments of order 1, . . ., deg(F ) − 1 are bounded, Proposition 3.7 implies that there exist at most two limit measures δ a and δ −a .

Convergence of the outlying measures
In this section we shall make the convergence results obtained for subsequences of invariant measures precise.We shall in particular study the case of asymmetric invariant measures.First let us recall that F is a polynomial function of degree 2n.
Remark 4.2.The statement 3) in Proposition 3.7 and Remark 3.8 emphasize that, under the assumption α ≥ θ , δ a and δ −a are the only possible asymmetric limit measures for families of invariant self-stabilizing measures whose (2n − 1)-th moments are uniformly bounded.
Let us observe now the situation where δ a and δ −a are not the only asymmetric limit measures.Proposition 4.3.Let us assume that V ′′ and F ′′ are convex functions.If there exists some sequence of stationary measures (u ǫ k ) k∈ * , whose (2n − 1)-th moments are bounded uniformly w.r.t.k, and which converges weakly to an asymmetric non-extremal measure u 0 (i.e.u 0 is different from δ ±a ), then the limit measure satisfies u 0 Proof.Since (u ǫ k ) k∈ * admits uniformly bounded (2n − 1)-th moments and since this sequence converges weakly, we obtain the convergence of the moments.Therefore, we can apply the statement of Proposition 3.7: the convexity of F ′′ and V ′′ implies that the support of u 0 contains at most two elements.1) Let us first consider the case when u 0 = δ A 1 .Equation (3.11) implies that V ′ (A 1 ) = 0. Therefore A 1 = −a, a, or 0. Since we assume that u 0 is asymmetric and non extremal, no solution of the preceding equation leads to the construction of some u 0 . 2 7 gives some information on the parameters.First of all A 1 A 2 < 0. Moreover (3.11), (3.12) and (3.13) immediately lead to (4.3) and (4.4).If p = 1/2 then (4.3) and (4.4) imply that V (A 1 ) = V (A 2 ).Due to the symmetry of V and since A 1 ∈] − a; 0[ and A 2 ∈]0; a[, we obtain A 1 = −A 2 and therefore u 0 is symmetric which contradicts the hypothesis of the proposition.Finally p − 1 2 > 0. Furthermore, if A 1 = −A 2 , then we observe that p = 1/2.This case was just studied: we obtain finally that Let us now study particular potential functions V and F which allows us to obtain that the only asymmetric limit measures are the Dirac measures δ a and δ −a .
Proposition 4.4.Let us consider some sequence (u ǫ k ) k≥0 of invariant measures, which admits uniformly bounded (2n − 1)-th moments and which is associated to both the potential function V (x) = x 4   4 − x 2 2 and the interaction F (x) = β x 2n 2n , with n ≥ 1 and β > 0. Then the only possible asymmetric limit measures are δ 1 and δ −1 .
Proof.Since V (4)  [5] states that the system admits exactly three invariant measures: one is symmetric and the others are the outlying measures u + ε and u − ε which tend respectively to δ a and δ −a according to Theorem 2.4.Let us just note that a = 1 in this particular case.>From now on we assume: n ≥ 2.
Step 2. Let us consider an asymmetric limit measure of the sequence (u ǫ k ) k≥1 denoted by u 0 .Due to the convexity of both F ′′ and V ′′ , we can apply Proposition 4.3.We deduce that u 0 = Besides, the equations (4.3) and (4.4) are satisfied.
Step 3. Let us prove that we cannot solve both equations (4.3) and (4.4).Considering the particular form of the functions V and F , (4.3) becomes, after dividing by A 2 2 − A 2 1 : This expression is equivalent to For n = 2, we get A 1 A 2 = 1 which contradicts the property developed in the statement 2) of Proposition 3.7: A 1 A 2 < 0. Hence the support of u 0 can not contains two elements if n = 2.For n > 2 the arguments are similar.Using the bounds |A i | ≤ 1, for i = 1 and 2, we obtain the inequality: We deduce that A 1 A 2 ≥ 1 which also contradicts the property just mentioned A 1 A 2 < 0.
Let us just note that the previous proof is relatively simple since the function F ′ is a divisor of F .We shall present some other particular case where the coefficient α defined by (1.6) is between 0 and −V ′′ (0) = θ (this previous equality is related to the convexity of V ′′ ).Under these conditions, Remark 4.2 can not be applied.
Reducing to the same denominator, we obtain:

This equation is equivalent to
We effectively consider only asymmetric limit measures and A 1 = −A 2 implies that p = 1/2.In order to conclude it suffices to note that A 1 A 2 = 1 + α/β contradicts the property A 1 A 2 < 0 which comes from the statement 2) of Proposition 3.7.We deduce that the support of any limit measure contains some unique point which corresponds to some zero of V ′ .The only possible value are 1 and −1 since the value 0 corresponds to some symmetric measure.

Convergence for sequences of symmetric invariant measures
In this last section, we consider the limit measures for families of symmetric invariant measures associated to the self-stabilizing process and denoted by (u 0 ε ) ε>0 .Let us introduce the notation: µ 0 l (ǫ) represents the l-th order moments of u 0 ǫ .Since the functions V and F are even, the function W 0 ǫ defined by (3.1) and (3.2) satisfies Let us note that the series appearing in the previous equality just contain a finite number of terms.
Introducing the following normalization coefficient λ First of all we shall prove that the family of moments (µ 0 2l (ǫ), ǫ > 0) is bounded.The starting step consists in the estimation of the normalization coefficient λ 0 ε .This result is a refinement of several arguments introduced in the proof of Lemma 4.3 in [5] and is also linked to Lemma 4.10 in [1].Lemma 5.1.There exists C > 0 such that for ε small enough, we have 1 Proof.
Step 1.Using the symmetry of the functions V and F 0 (defined in (1.6)) and the measure u 0 ε , we get Then, by using the hypotheses (F − 4) and the definition of F 0 , the following upper-bounds hold: due to the boundedness of the function y → Step 2. Let K > 0. We split the 2q-th moment into two integral terms: By construction, u 0 ǫ is directly related to F * u 0 ǫ : see (3.1).According to Lemma 4.2 in [5], we have the following inequality : Therefore, using (V-4) and the previous inequality, we get, for all x: 2 z 4 for z ≥ K, and so where C(q) is a positive constant.Finally there exists some ǫ 0 > 0 such that ε ≤ ε 0 implies: By hypotheses (V-7) and (V-8), we get V ′ (x) ≤ C q (1+ x 2q ) and |V (x)| ≤ C q |x|(1+ x 2q ).Hence, for all |x| ≥ a, we have ). Moreover the property (V-6) allows us to obtain, on the interval [−a, a]: a , the following bound holds: ).The upper-bounds (5.3) and (5.4) and the symmetry property of the density u 0 ǫ immediately imply the existence of constants C ′ and C ′′ such that Step 3. We deduce, by (5.5) and the definition of the normalization factor λ 0 ε : Defining µ = λ 0 ε and introducing the change of variable x := µ ε 2 ξ allows to point out the inequality µ ≥ h(µ) with Let us study the equation µ ≥ h(µ).Obviously h ′ (z) ≤ 0. Now, we provide an upper bound for the function −h ′ (z).
For ε small and z ≥ 0, Let us note that µ ≥ 1 implies directly 1 λ 0 ε (u) ≤ 1.This observation leads to study essentially the case: z ≤ 1.After some computation, we obtain For 0 ≤ z ≤ 1, the previous estimation of the derivative leads to the existence of some constant Moreover h(0) = 2ε + e −C ′′ ξ 2 dξ.Hence h(0)/ ǫ is independent of ǫ and shall be denoted by C 3 . We For α = 0, we obtain directly the bound µ ≥ a 3 ε 1+a 2 ε .Finally we obtain the existence of C > 0 s.t.λ 0 ε ≥ C 2 ε for ǫ small enough.
Using the upper bound of the normalization term λ 0 ǫ , we analyze the moments of symmetric invariant measures.We recall that µ 0 l represents the l-th moment.
Let us note that this result is a generalization of Proposition 3.1 in the symmetric context.We do not need any condition on the degree of F and V and we do not even assume that V is polynomial.
. We split the moment into two different terms: According to Lemma 4.2 2) in [5], )).Hence, for any |x| > K, V (x) ≥ x 2 .We have therefore: where C is some constant.
Let us now present the main global result concerning sequences of symmectric invariant measures for self-stabilizing processes.
Proposition 5.3.Let (u 0 ǫ ) ǫ>0 be a family of symmetric invariant measures.Then there exist some sequence (ǫ k ) k≥0 satisfying lim k→∞ ǫ k = 0 and a discrete limit measure u 0 0 such that u 0 ε k k≥0 converges weakly towards u 0 0 .Moreover u 0 0 takes the following form: Proof.By Lemma 5.2, the moments are bounded.Therefore Theorem 3.6 and Proposition 3.7 can be directly applied.In order to conclude it suffices to note that u 0 0 is symmetric and both functions V and F are even.
First of all, let us note that it is sufficient to choose some function F satisfying F ′′ (0) < −V ′′ (0) = 1 in order to ensure δ 0 is not a limit measure: let us fix F ′′ (0) = 0. Let us assume that F is an even polynomial function of degree 8. Since lim x→∞ F (x) = +∞, we obtain F (8) (0) ≥ 0. Moreover F has to satisfy (F-3), that is F ′ is a convex function on + which implies F (4) (0) ≥ 0. Finally since V ′′ is a convex function, F ′′ is not convex that is to say that F (6) (0) is necessarily negative.We can therefore choose F of the following kind: 8 with α 4 , α 6 and α 8 > 0. (5.11) F ′ has to be convex on + (F-3) that is to say F (3) (x)/x =: Q 2 (x 2 ) ≥ 0, for any x ∈ , where Q 2 is a polynomial function of degree 2. Moreover F ′′ is not convex on i.e.F (4) (x) =: R 2 (x 2 ) < 0 for some x ∈ where R 2 is another polynomial function of degree 2. In other words we choose some function F such that the discriminant of Q 2 is negative and the discriminant of R 2 is positive.We deduce the following bounds: (5.12) Let us now prove that there exists such a function F which does not satisfy the system (5.10).It suffices to prove the following implication (5.14) We replace (5.14) by 2×(5.14)−7×(5.13).So (5.13) and (5.14) are equivalent to P 3 (x) = 0 ⇒ P 2 (x) < 0 for x ≥ 0 and P 2 (x) = 32 α 6 x 2 − (16 α 4 + 1)x + 5. Using (5.12) we can prove that the discriminant of P ′ 3 is negative.We deduce that there exists some unique x 0 satisfying P 3 (x 0 ) = 0. We observe moreover that x 0 > 0. To conclude the proof, it suffices to point out some coefficients α 4 , α 6 and α 8 satisfying (5.12) and such that x 0 ∈]x − , x + [ where x ± are the possible roots of P 2 .In other words, we prove that P 3 (x − )P 3 (x + ) < 0. Let us introduce a parameter η ∈]35/25, 63/25[ such that α 6 = ηα 4 α 8 .Therefore we can express x ± , P 3 (x − ) and P 3 (x + ) with respect to η, α We shall verify at the end of the proof that ∆ > 0 is effectively satisfied.After some tedious computation we obtain that P 3 (x − )P 3 (x + ) < 0 is equivalent to Ψ η, α 4 ( α 8 ) < 0 where Ψ u,s (t) is the following polynomial function: where R u is a polynomial function of order 2 whose coefficients depend on the u-variable.By (5.12), the variable u satisfies u 2 = η < 320 72 which implies the inequality 320s 2 + 95 − 72u 2 s 2 > 0 for all s ∈ .Let us then choose α 8 in order to minimize Ψ η, α 4 ( α 8 ).Hence We have studied the case when V ′′ is a convex function and pointed out the existence of some polynomial interaction function F with non convex second derivative such that the support of any limit measure contains at least three elements.Let us observe now what happens if the interaction function F ′′ is convex.Is it possible to find some environment function V in order to obtain the same conclusion ?
Proof.The arguments developed for the proof of Proposition 5.7 are similar to those presented in Proposition 5.6.Let us first choose V such that δ 0 cannot be a limit measure of {u 0 ǫ , ǫ > 0}.By (5.9), applied to r = 1 and A 1 = 0, it suffices to assume that −V ′′ (0) > α.
Let us now focus our attention to limit measures whose support contains two elements A 1 and −A 1 .Due to Proposition 5.3, A 1 satisfies (5.7) and (5.9).Let us choose V some polynomial function of degree 6.In order to get the limit lim x→∞ V (x) = +∞ which is part of the condition (V-5), we choose the coefficient of degree 6 positive.Moreover the coefficient of degree 4 is negative since we need that V ′′ is not convex.We deduce that V has the following form x 2 with α 4 , α 6 > 0 and α 2 > α. (5.17) The equations (5.7) and (5.9) become

A Annex
We shall present here a useful asymptotic result which is close to the classical Laplace's method and which contributes to the proof of Theorem 2.1.This result and its proof are slight modifications of those appearing in the annex of [5] which is why we shall omit the proof.where Γ represents the Euler function.

Proposition 4 . 5 . 4 + α x 2 2 ,
Let us consider some sequence (u ǫ k ) k≥0 of invariant measures, which admits uniformly bounded third moments and which is associated to the potential function V (x) = x 4 4 − x 2 2 and the interaction F (x) = β x 4 with α ∈ [0, 1[ and β > 0. Then the only possible asymmetric limit measures are δ 1 and δ −1 .Proof.The arguments developed for the proof of Proposition 4.5 are similar to those presented in Proposition 4.4: the second step can directly be applied.Therefore the equations (4.3) and (4.4) are satisfied.Let us develop equation (4.3) for the particular functions V and F .Hence (4.3) is equivalent to

Lemma A. 1 .
Set ε > 0. Let U a ∞ ( )-continuous functions.Let us introduce some interval [a, b] satisfying: U ′ (a) = 0, U ′ (b) = 0 and U(x) admits some unique global minimum on the interval [a, b] reached at x 0 ∈]a, b[.We assume that there exists some exponent k 0 such that 2k 0 = min r∈ * U (r) (x 0 ) = 0 .Let f be a 4 ( )-continuous function.Then taking the limit ε → 0 we get b a f (t)e − U(t) ε d t = f (x 0 ) Bienayme-Tchebychev's inequality enables us to bound the integral of u ǫ k on I 1 byThe sequence of measures u ε k k∈ * converges to a measure u 0 .Furthermore the open set U δ c is less and less weighted by u ε k as k becomes large.Intuitively u 0 should be a discrete measure whose support corresponds to the set {A 1 , ..., A r }.Let (ǫ k ) k≥1 , W 0 , u 0 and A 1 , ..., A r be defined in the statement of Proposition 3.3 and by(3.5).Then the sequence of measures u ε k k≥1 converges weakly, as k becomes large, to the discrete probability measure u 0 = r i=1 p i δ A i where