Identification of the limit

Furthermore, for η 2 0, 0 θ ≤ C 2 and τ N ≤ T a stopping time, for all 1 ≤ j ≤ J, P Z τ N +θ τ N D η N , ω s , L ν N s ϕ j E ds ≥ η 2 ≤ 1 η 2 2 E    Z τ N +θ τ N D η N , ω s , L ν N s ϕ j E ds 2    , ≤ C 2 η 2 2 E   Z τ N +θ τ N D η N , ω s , L ν N s ϕ j E 2 ds   , ≤ C 2 η 2 2 Z T E D η N , ω s , L ν N s ϕ j E 2 ds, ≤ C C 2 η 2 2 Z T E h η N s 2 −3,2α i ds, ≤ C T C 2 η 2 2 A N , cf. 25. And, P  M N τ N +θ ϕ j − M N τ N ϕ j η 2 ‹ ≤ C C 2 η 2 2 1 N N X i= 1 1 + |ω i | 4 α . So, for all j ≥ 1, by definition of C 2 , P  η N , ω τ N +θ ϕ j − η N , ω τ N ϕ j ≥ η 2 ‹ ≤ η 1 A ǫ A N + 1 N N X i= 1 1 + |ω i | 4 α . Consequently, Θ N K ǫ 2 φ 1 , . . . , ϕ J ≥ P A N + 1 N N X i= 1 1 + |ω i | 4 α Aǫ . Letting J → ∞, we get lim sup N Θ N €S J K ǫ 2 ϕ 1 , . . . , ϕ J c Š ≤ ǫ. Eq. 33 is proved.

4.3.3 Identification of the limit

The proof of the fluctuations result will be complete when we identify any possible limit. Proposition 4.13 Identification of the initial value. The random variable ω 7→ L η N , ω converges in law to the random variable ω 7→ L X ω, where for all ω, X ω = Cω + Y , with Y a centered Gaussian process with covariance Γ 1 . Moreover ω 7→ Cω is a Gaussian process with covariance Γ 2 , where Γ 1 and Γ 2 are defined in 8 and 9. Proof. For simplicity, we only identify here the law of D η N , ω , ϕ E for all ϕ. The same proof works for the law of finite-dimensional distributions D η N , ω , ϕ 1 E , . . . , D η N , ω , ϕ p E , p ≥ 1. We write 820 Γ i for Γ i ϕ, ϕ, i = 1, 2. One has: D η N , ω , ϕ E = 1 p N N X i= 1 ‚ ϕξ i , ω i − Z S 1 ϕx, ω i λ dx Œ + 1 p N N X i= 1 ‚Z S 1 ϕx, ω i λ dx − ν , ϕ Œ , =: A N , ω + B N , ω . It is easy to see that B N , ω converges in law to Z 2 ∼ N 0, Γ 2 . Moreover, for P-almost every ω, A N , ω converges in law to Z 1 ∼ N 0, Γ 1 see Billingsley [5], Th. 27.3 p. 362. That means that for all u ∈ R, ψ A N u := E λ e iuA N , ω converges to ψ Y u := e − u2 2Γ1 . But, then, for all F ∈ C b

R, E

F E λ e iu D η N , ω , ϕ E = E h F E λ h e iuA N , ω +B N , ω ii = E h F e iuB N , ω ψ N u i . Since ψ N u converges almost surely to a constant, the limit of the expression above exists Slutsky’s theorem and is equal to E F e iuZ 2 − u2 2Γ1 . Proposition 4.14 Identification of the martingale part. For P-almost every ω, the sequence M N , ω converges in law in C [0, T ], S ′ to a Gaussian process W with covariance defined in 7. Proof. For fixed ω, M N , ω is a sequence of uniformly square-integrable continuous martin- gales cf. Remark 4.7, which is tight in C [0, T ], S ′ . Let W 1 and W 2 be two accumulation points continuous square-integrable martingales which a priori depend on ω and M φN ,ω and M ψN ,ω be two subsequences converging to W 1 and W 2 , respectively. Note that we can suppose that φN ≤ ψN for all N. For all ϕ ∈ S , lim N →∞ ¬ M φN ,ω ϕ , M ψN ,ω ϕ ¶ t = W 1 ϕ , W 2 ϕ t , for all t, and ¬ M φN ,ω ϕ , M ψN ,ω ϕ ¶ t = Z t ¬ ν φN s , ϕ ′ 2 ¶ ds. We now have to identify the limit: we already know that for P-almost every realization of the disorder ω, ν N , ω converges in law to P. But, the latter expression, seen as a function of ν, is continuous. So W 1 ϕ , W 2 ϕ t = R t ¬ P s , ϕ ′ 2 ¶ . So W 1 − W 2 is a continuous square integrable martingale whose Doob-Meyer process is 0. So W 1 = W 2 and is characterized as the Gaussian process with covariance given in 7. The convergence follows. Proof of the independence of W and X . We prove more : the triple Y, C, W is independent. For sake of simplicity, we only consider the case of Y ϕ, Cϕ, W t ϕ for fixed t and ϕ. Let us first recall some notations: let A N , ω , B N , ω and M N , ω t ϕ be the random variables defined in the proof of Proposition 4.13 and 4.14 and let ψ A N u := E e iuA N , ω , ψ B N v := E e i vB N , ω , ψ M N w := E e iw M N , ω t ϕ be their characteristic functions u, v, w ∈ R. We know that, for almost 821 every ω, ψ A N u converges to ψ Y u = e − u2 2Γ1 and that ψ M N w converges to the deterministic function ψ W w := E € e iwW t ϕ Š . But, if ψ C v = E € e iwC Š , then, for all u, v, w ∈ R, using the independence of the Brownian with the initial conditions, E E e iuA N , ω +i vB N , ω +iw M N , ω t ϕ − e i vB N , ω ψ A N uψ M N w = 0. Using Slutsky’s theorem, we see that any limit couple Y, C, W satisfies E € E € e iuY +i vC+iwW t ϕ ŠŠ = ψ Y uψ C vψ W w. which is the desired result. We recall that the limit second order differential operator L s is defined by L s ϕ y, π := 1 2 ϕ ′′ y, π + ϕ ′ y, πb[ y, P s ] + c y, π + P s , ϕ ′ ·, ·b·, y, π . As in Lemma 4.5, we can prove the following: Lemma 4.15. Assume H b ,c . Then for every N , s ≤ T , ω, the operator L s and L ν N s are linear continuous from S to S and L s ϕ 6, α ≤ C ϕ 8, α , L ν N s ϕ 6, α ≤ C ϕ 8, α . We are now in position to prove Theorem 2.10: Proof of Theorem 2.10. Let Θ be an accumulation point of Θ N . Thus, for a certain subsequence which will be also denoted as N for notations purpose, the random variable ω 7→ H N , ω con- verges in law to a random variable H with values in M 1 C [0, T ], S ′ with law Θ. Applying Skorohod’s representation theorem, there exists some probability space Ω 1 , P 1 , F 1 and ran- dom variables defined on Ω 1 , ω 1 7→ H N ω 1 and ω 1 7→ Hω 1 such that H N has the same law as ω 7→ H N , ω , H has the same law as H , and for P 1 -almost every ω 1 ∈ Ω 1 , H N ω 1 converges to H ω 1 in M 1 C [0, T ], S ′ . An easy application of Proposition 4.8 and Borel-Cantelli’s Lemma shows that P 1 -almost surely, E sup t ≤T η ω 1 t −6,α ∞. Then we know from Lemma 4.15 that the integral term R t L ∗ s η ω 1 s ds makes sense as a Bochner’s integral in W −8,α ⊆ S ′ . Let η N , ω 1 with law H N ω 1 ; η N , ω 1 converges in law to some η ω 1 with law H ω 1 . By uniqueness in law convergence, using Propositions 4.13 and 4.14, we see that η ω 1 , W as the same law as X ω 1 , W . For fixed ϕ ∈ S , we define F ϕ from C [0, T ], S ′ into R by F ϕ γ := γ t , ϕ − γ , ϕ − R t γ s , L s ϕ ds. The function F ϕ is continuous and since η N , ω 1 converges in law to η ω 1 , the sequence F ϕ η N , ω 1 converges in law to F ϕ η ω 1 . To prove the theorem, it remains to show that the law of the term R t D η N , ω 1 s , L ν N s ϕ − L s ϕ E ds converges in law to 0. We show that there 822 is convergence in probability: For all ǫ 0, for all A 0, using Proposition 4.8, Lemma 4.15, and Cauchy-Schwarz’s inequality, U N , ǫ := P 1 ‚ E –Z t D η N , ω 1 s , L ν N s − L s ϕ E ds ™ ǫ Œ , = P ‚ E –Z t D η N , ω s , L ν N s − L s ϕ E ds ™ ǫ Œ , ≤ P ‚Z t E η N , ω s 2 −6,α 1 2 E L ν N s − L s ϕ 2 6, α 1 2 ds ǫ Œ , ≤ P ‚ C N ω 1 , . . . , ω N 1 2 Z t E L ν N s − L s ϕ 2 6, α 1 2 ds ǫ Œ cf. Prop 4.8, ≤ P ‚Z t E L ν N s − L s ϕ 2 6, α 1 2 ds ǫ p A Œ + P C N A . Using 4.8, it suffices to prove that, for all ǫ 0, lim sup N →∞ P ‚Z t E L ν N s − L s ϕ 2 6, α 1 2 ds ǫ Œ = 0. 34 Indeed, for every ϕ ∈ S , U N s ϕ y, π := L ν N s − L s ϕ y, π = ϕ ′ y, πb[ y, ν N s ] − b[ y, P s ]. An analogous calculation as in Lemma 4.5 shows that, using Lipschitz assumptions on b, and Propo- sition 4.3: E   sup s ≤t U N s ϕ 2 6, α   ≤ ϕ 2 8, α CN + D N ω 1 , . . . , ω N , with the property that lim A →∞ lim sup N PN D N A = 0. Equation 34 is a direct consequence. Since there is uniqueness in law in 10, Θ is perfectly defined, and thus, unique. The convergence follows. 5 Proofs for the fluctuations of the order parameters We end by the proofs of paragraph 2.3.3.

5.1 Proof of Proposition 2.13