Weak convergence for a subsequence of invariant measures

• Step 1. First we will prove that, under the boundedness of the family {µ 2n −1 ε, ǫ 0} with 2n = degF , 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 associated to some measure u . • Step 2. We shall describe the measure u : 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 V x = x 4 4 − x 2 2 . • Step 4. 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.

3.1 Weak convergence for a subsequence of invariant measures

Let u ε ε0 be a family of stationary measures. The main assumption in the subsequent develop- ments is: H We assume that the family µ 2n −1 ε, ε 0 is bounded for 2n = degF . 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 ε0 a family of invariant measures for the diffusion 1.2. We assume that V is a polynomial function whose degree satisfies 2m := degV degF = 2n. Then the family €R R€ x 2m u ε xd x Š ε0 is bounded. Proof. Let us assume the existence of some decreasing sequence ε k k ∈N which tends to 0 and such that the sequence of moments µ 2m k := R R€ x 2m u ε k xd x tends to +∞. By 3.1 and 3.2 we obtain: µ 2m k = R R€ x 2m exp h − 2 ε k P 2m −1 r=1 M r kx r + C 2m x 2m i d x R R€ exp h − 2 ε k P 2m −1 r=1 M r kx r + C 2m x 2m i d x , where M r k is a combination of the moments µ j k, with 0 ≤ j ≤ max0, 2n − r, and C 2m = V 2m 02m . Let us note that the coefficient of degree 0 in the polynomial expression disap- pears since the numerator and the denominator of the ratio contain the same expression: that leads to cancellation. Moreover M r k does not depend on ǫ k for all r s.t. 2n ≤ r ≤ 2m and that there exists some r such that M r k tends to +∞ or −∞ since µ 2m k is unbounded. Let us define φ k := sup 1 ≤r≤2m −1 M r k 1 2m0−r . Then the sequences η r k := M r k φ 2m0−r k , for 1 ≤ r ≤ 2m − 1, are bounded. Hence we extract a subsequence ε Ψk k ∈N such that η r Ψk converges towards some η r as k → ∞, for any 1 ≤ r ≤ 2098 2m −1. For simplicity, we shall conserve all notations: µ 2m k, η r k, φ k ... even for sub-sequences. The change of variable x := φ k y provides µ 2m k φ 2m k = R R€ y 2m exp − 2 φ 2m0 k ε k P 2n −1 r=1 η r k y r + C 2m y 2m d y R R€ exp − 2 φ 2m0 k ε k P 2n −1 r=1 η r k y r + C 2m y 2m d y . 3.3 The highest moment appearing in the expression M k is the moment of order 2n − k. By Jensen’s inequality, there exists a constant C 0 such that |M r | ≤ Cµ 2m k 2n −r 2m0 for all 1 ≤ r ≤ 2n − 1. The following upper-bound holds: φ 2m k ≤ sup 1 ≤r≤2m −1 C 2m0 2m0−r µ 2m k 2n −r 2m0−r = o ¦ µ 2m k © 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 := P 2n −1 r=1 η r y r + C 2m y 2m which does not satisfy a priori U ′′ y 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 ξ k := ε k φ 2m k and f y := y 2m . 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 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 ≤ degF − 1, µ l ε k converges towards some limit value denoted by µ l 0 with |µ l 0| ∞. 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 ∈ N ∗ , we write : ω k 0 = ∞ X l=0 −1 l l F l+k 0µ l and W x = V x + ∞ X k=0 x k k ω k 0. 3.4 Like in 3.2, there is a finite number of terms non equal to 0 in the two sums so W x is defined for all x ∈ R€. 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 ∈ N, W j ε k k ≥1 converges towards W j , uniformly on each compact subset of R, where the limit pseudo-potential W is defined by 3.4, and € u ε k Š k ≥1 converges weakly towards some probability measure u . 2099 Proof. By 3.4, we obtain, for any p ≥ 1, ω p ε − ω p ≤ ∞ X l=1 F l+p l µ l ε − µ l . 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 ∈ N, ω p ε k tends to ω p 0 when k tends to infinity. Let x ∈ R, since V is C ∞ -continuous, both derivatives W j ε k x and W j x are well defined. Moreover we get W j ε k x − W j x = ∞ X p= j x p − j p − j ω p ε k − ∞ X p= j x p − j p − j ω p ≤ ∞ X p= j |x| p − j p − j ω p ε k − ω p . Since there is just a finite number of terms, and since each term tends to 0, we obtain the pointwise convergence of W j ε k towards 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: |W j ǫ k y| ≤ |V j y| + ∞ X p= j | y| p − j p − j |w p ǫ k |. 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: u ε k [−R; R] ≥ 1 − m 2 R 2 . Consequently the family of probability measures ¦ u ε k ; k ∈ N ∗ © is tight. Prohorov’s Theorem allows us to conclude that ¦ u ε k ; k ∈ N ∗ © is relatively compact with respect to the weak convergence. Lemma 3.4. Each limit function W admits a finite number r ≥ 1 of global minima. Proof. According to 3.4, W ′ x = V ′ x + P ′ x where P is a polynomial function. Since V is equal to an analytic function V on [−a; a] see Condition V-6, we deduce that W ′ has a finite number of zeros on this interval. Indeed, if this assertion is false, then W ′′ a = 0 which contradicts the following limit W ′′ 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 R + , 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 x ≥ C 4 x 4 − C 2 x 2 for all x ∈ R. This lower-bound allows us to conclude that there exists some large interval [ −R; R] which contains all the global minima of W . It remains then to study the minima of W on the compact K := [ −R; −a] S[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 ′′ uniformly on K. Consequently W ′′ x 0 for all x ∈ K and so W ′ admits a finite number of zeros on K. 2100 Since W admits r global minima, we define A 1 · · · A r their location: W A 1 = · · · = W A r = inf x ∈R W x =: w . 3.5 We introduce a covering U δ of these points: U δ = r [ i=1 A i − δ; A i + δ with δ ∈ 0; 1 2 min i 6= j A i − A j . 3.6 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 defined in Proposition 3.3 is concentrated around these points. Proposition 3.5. Let W and ǫ k k ≥0 be defined by Proposition 3.3. Then the following convergence results hold: 1 For all x ∈ U δ , lim k →∞ u ε k x = 0. 2 For δ 0 small enough, lim k →∞ Z U δ c u ε k xd x = 0. Proof. 1 The limit pseudo-potential W is C ∞ and its r global minima are in U δ . Since lim x →±∞ W x = +∞, there exists η 0 such that W x ≥ w + 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 W ′′ ε k x ≥ V ′′ x + α α for all |x| ≥ a and k ≥ 0. 3.7 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 + 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 k ∈ N such that W ε k x ≥ w + η 2 for k ≥ k and x ∈ [−R, R]∩U δ c . To sum up, the lower bound W ε k x ≥ w + η 2 holds for any x ∈ U δ c provided that k ≥ k . Finally we obtain: exp − 2 ε k W ε k x ≤ exp − 2 ε k w exp − η ε k , ∀x ∈ U δ , k ≥ k . 3.8 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 admits a finite number of minima, there exists γ with δ γ 0 such that, for all x ∈ U γ , we have W x ≤ w + η 8 . By Proposition 3.3, W ε k converges uniformly on each compact subset of R to W . Therefore there exists k 1 such that for all k ≥ k 1 , for all x ∈ U γ , we have W ε k x ≤ w + η 4 . Finally the denominator in 3.1 can be lower bounded: D ǫ k ≥ Z U γ exp − 2 ε k W ε k x d x ≥ 2rγ exp − 2 ε k  w + η 4 ‹ , k ≥ k 1 . 3.9 For k ≥ k ∨ k 1 , according to 3.8 and 3.9, the following bound holds u ε k x ≤ 1 2r γ exp − η 2 ε k . 3.10 2101 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: U δ = I 1 ∪ I 2 with I 1 = [−K; K] c respectively I 2 = [−K; K] TU δ c . Bienayme-Tchebychev’s inequality enables us to bound the integral of u ǫ k on I 1 by m 2 K 2 , where m 2 satisfies sup k ≥0 µ 2 ε k ≤ m 2 . By 3.10, the integral on the second support I 2 is bounded by K r γ exp h − η 2 ε k i . It suffices then to choose K and k large enough in order to prove the convergence of R U δ c u ε k xd x. The sequence of measures € u ε k Š k ∈N ∗ converges to a measure u . Furthermore the open set € U δ Š c is less and less weighted by u ε k as k becomes large. Intuitively u should be a discrete measure whose support corresponds to the set {A 1 , . . . , A r }. Theorem 3.6. Let ǫ k k ≥1 , W , u 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 Š k ≥1 converges weakly, as k becomes large, to the discrete probability measure u = P r i=1 p i δ A i where p i = lim k →+∞ Z A i +δ A i −δ u ε k xd x, 1 ≤ i ≤ r, δ 0 small enough. 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 R A i +δ A i −δ u ε k xd x as k → ∞. 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 p i δ − p i δ ′ = lim k →∞ Z A i −δ ′ A i −δ u ε k xd x + Z A i +δ A i +δ ′ u ε k xd x . 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 is a discrete probability measure. Let f be a continuous and bounded function on R. 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: Z R f xu ε k xd x − r X i=1 p i f A i = R + r X i=1 ∆ i f , with ∆ i f = R A i +δ A i −δ f xu ε k xd x − p i f A i and R = R U δ c f xu ε k xd x. 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 : ∆ i f ≤ Z A i +δ A i −δ f x − f A i d u ε k x + | f A i | u ε k U i δ − p i ≤ sup z ∈U i δ f z − f A i u ε k U i δ + f A i u ε k U i δ − p i . 2102 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 P r i=1 p i δ A i which can finally be identified with u . 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.

3.2 Description of the limit measures