Tightness of the fluctuations process

where G ϕω := R ϕ y, ω i P ω i t d y − P t , ϕ . If we apply the same Hilbertian argument as for S N , ω , we see E T N , ω t 2 −3,2α ≤ 2C N E   N X i= 1 1 + |ω i | 4 α   + C + φ 7→ 1 p N N X i= 1 G φω i 2 −3,2α , 29 It is easy to see that the last term in 29 can be reformulated as B N ω 1 , . . . , ω N , with the property that lim A →∞ lim sup N →∞ PB N A = 0. Combining 24, 26, 28 and 29, Proposition 4.4 is proved.

4.3.2 Tightness of the fluctuations process

Applying Ito’s formula to 11, we obtain, for all ϕ bounded function on S 1 × R, with two bounded derivatives w.r.t. x, for every sequence ω, for all t ≤ T : D η N , ω t , ϕ E = D η N , ω , ϕ E + Z t D η N , ω s , L ν N s ϕ E ds + M N , ω t ϕ, 30 where, for all y ∈ S 1 , π ∈ R, L ν N s ϕ y, π = 1 2 ϕ ′′ y, π + ϕ ′ y, π € b[ y , ν N s ] + c y, π Š + P s , ϕ ′ ·, ·b·, y, π , and M N , ω t ϕ is a real continuous martingale with quadratic variation process ¬ M N , ω ϕ ¶ t = Z t ¬ ν N , ω s , ϕ ′ y, π 2 ¶ ds. Lemma 4.5. For every N , the operator L ν N s defines a linear mapping from S into S and for all ϕ ∈ S , L ν N s ϕ 2 3,2 α ≤ C ϕ 2 6, α . Proof. The terms 1 2 ϕ ′′ y, π and ϕ ′ y, πb[ y, ν N s ] clearly satisfy the lemma. We study the two remaining terms: P s , ϕ ′ b ·, y, π 2 3,2 α = X k 1 +k 2 ≤3 Z S 1 ×R ¬ P s , ϕ ′ ∂ y k1 ∂ π k2 b ·, y, π ¶ 2 1 + |π| 4 α d y d π, ≤ C Z R 1 1 + |π| 4 α d π Z S 1 ×R ϕ ′ y, π 2 P s d y, dπ, ≤ C ϕ 2 C 3, α Z R 1 1 + |π| 4 α d π Z S 1 ×R 1 + |π| α 2 P s d y, dπ, ≤ C ϕ 2 6, α Z R 1 1 + |π| 4 α d π Z R 1 + |π| α 2 µ dπ. 814 And, ϕ ′ y, πc y, π 2 3,2 α = X k 1 +k 2 ≤3 Z S 1 ×R € ∂ y k1 ∂ π k2 ϕ ′ y, πc y, π Š 2 1 + |π| 4 α d y d π. It suffices to estimate, for every differential operator D i = ∂ y ui ∂ π vi , i = 1, 2 with u 1 +u 2 + v 1 + v 2 ≤ 3, the following term: Z S 1 ×R |D 1 ϕ ′ y, πD 2 c y , π| 2 1 + |π| 4 α d y d π ≤ Z S 1 ×R |D 1 ϕ ′ y, π| 2 1 + |π| α 2 |D 2 c y , π| 2 1 + |π| α 2 1 + |π| 4 α d y d π, ≤ C ϕ 2 6, α Z R sup y ∈S 1 |D 2 c y , π| 2 1 + |π| 2 α d π. The result follows from the assumptions made on c. For the tightness criterion used below, we need to ensure that the trajectories of the fluctuations process are almost surely continuous: in that purpose, we need some more precise evaluations than in Prop. 4.4. Proposition 4.6. The process M N , ω t satisfies, for every ω, and for every T 0, E – sup t ≤T M N , ω t 2 −3,2α ™ ≤ C N N X i= 1 € 1 + |ω i | 4 α Š . Remark 4.7. In particular, a consequence of H F µ is that, for P-almost every sequence ω, sup N E – sup t ≤T M N , ω t 2 −3,2α ™ ≤ sup N C N N X i= 1 € 1 + |ω i | 4 α Š ∞. 31 Proof. Let ϕ p p ≥1 a complete orthonormal system in W 3,2 α . For fixed N , by Doob’s inequality, P p ≥1 E h sup t ≤T M N , ω t ϕ p 2 i is bounded by C X p ≥1 E h M N , ω T ϕ p 2 i = C X p ≥1 E   Z T D ν N , ω s , ϕ ′ p y, π 2 E ds   , = 1 N N X i= 1 E   Z T X p ≥1 ϕ ′ p x i ,N s , ω i 2 ds   , = 1 N N X i= 1 E   Z T H x i ,N s , ω i 2 3,2 α ds   , ≤ C N N X i= 1 € 1 + |ω i | 4 α Š , using 21. 815 Proposition 4.8. For every N , every ω, E – sup t ≤T η N , ω t 2 −6,α ™ C N ω 1 , . . . , ω N , 32 with lim A →∞ lim sup N →∞ P C N A = 0. Proof. Let ψ p be a complete orthonormal system in W 6, α of C ∞ function on S 1 × R with compact support. We prove the stronger result: E   X p ≥1 sup t ≤T D η N , ω t , ψ p E 2   ∞. Indeed, D η N , ω t , ψ p E 2 ≤ C ‚ D η N , ω , ψ p E 2 + T Z t D η N , ω s , L ν N s ψ p E 2 ds + M N , ω t ψ p 2 Œ . By Doob’s inequality, E   X p ≥1 sup t ≤T D η N , ω t , ψ p E 2   ≤ C   E η N , ω 2 −6,α + E Z T X p ≥1 D η N , ω s , L ν N s ψ p E 2 ds + X p ≥1 E h M N , ω T ψ p 2 i   . By Lemma 4.5, we have: D η N , ω s , L ν N s ψ E ≤ C η N , ω s −3,2α ψ 6, α . Then, E   Z T X p ≥1 D η N , ω s , L ν N s ψ p E 2 ds   ≤ C 2 Z T E η N , ω s 2 −3,2α ds, ≤ C 2 T sup s ≤T E η N , ω s 2 −3,2α ≤ C 2 TA N , where A N is defined in Proposition 4.4. The result follows. Proposition 4.9. 1. For every N , for P-almost every ω, the trajectories of the fluctuations process η N , ω are almost surely continuous in S ′ , 2. For every N , for P-almost every ω, the trajectories of M N , ω are almost surely continuous in S ′ . 816 Proof. We only prove for M N , ω , since, using Proposition 4.8, the proof is the same for η N , ω . Let ϕ p be a complete orthonormal system in W −3,2α , then for every fixed N and ω, we know from the proof of Proposition 4.6, that for all ǫ 0, there exists some M 0 such that X p ≥M sup t ≤T M N , ω t ϕ p 2 ǫ 3 , a.s. Let t m be a sequence in [0, T ] such that t m → m →∞ t . M N , ω t m − M N , ω t 2 −3,2α = X p ≥1 M N , ω t m − M N , ω t 2 ϕ p , ≤ M X p= 1 M N , ω t m − M N , ω t 2 ϕ p + 2 ǫ 3 ≤ ǫ, if t m is sufficiently large. We are now in position to prove the tightness of the fluctuations process. Let us recall some nota- tions: for fixed N and ω H N , ω is the law of the process η N , ω . Hence, H N , ω is an element of M 1 C [0, T ], S ′ , endowed with the topology of weak convergence and with B ∗ , the smallest σ-algebra such that the evaluations Q 7→ Q , f are measurable, f being measurable and bounded. We will denote by Θ N the law of the random variable ω 7→ H N , ω . The main result of this part is the following: Theorem 4.10. 1. for P-almost every sequence ω, the law of the process M N , ω is tight in M 1 C [0, T ], S ′ , 2. The law of the sequence ω 7→ H N , ω is tight on M 1 M 1 C [0, T ], S ′ . Before proving Theorem 4.10, we recall the following result and notations cf. Mitoma [19], Th 3.1, p. 993: Proposition 4.11 Mitoma’s criterion. Let P N be a sequence of probability measures on C S ′ := C [0, T ], S ′ , B C S ′ . For each ϕ ∈ S , we denote by Π ϕ the mapping of C S ′ to C := C [0, T ], R defined by Π ϕ : ψ· ∈ C S ′ 7→ ψ· , ϕ ∈ C . Then, if for all ϕ ∈ S , the sequence P N Π −1 ϕ is tight in C , the sequence P N is tight in C S ′ . Remark 4.12. A closer look to the proof of Mitoma shows that it suffices to verify the tightness of P N Π −1 ϕ for ϕ in a countable dense subset of the nuclear Fréchet space S , k · k p , p ≥ 1. Thanks to Mitoma’s result, it suffices to have a tightness criterion in R. We recall here the usual result cf. Billingsley [4]: A sequence of Ω N , F N t -adapted processes Y N with paths in C [0, T ], R is tight if both of the following conditions hold: 817 • Condition [T]: for all t ≤ T and δ 0, there exists C 0 such that sup N P € |Y N t | C Š ≤ δ, T t , δ,C • Condition [A]: for all η 1 , η 2 0, there exists C 0 and N such that for all F N -stopping times τ N , sup N ≥N sup θ ≤C P  Y N τ N − Y N τ N +θ ≥ η 2 ‹ ≤ η 1 . A η 1 , η 2 ,C Proof of Theorem 4.10. 1. Tightness of M N , ω : for a fixed realization of the disorder ω, for fixed ϕ ∈ S , we have: • For all t ∈ [0, T ], for all δ 0, for all C 0, P  M N , ω t ϕ C ‹ ≤ E h sup t ≤T n M N , ω t ϕ 2 oi C 2 , ≤ E sup t ≤T M N , ω t 2 −3,2α ϕ 2 3,2 α C 2 , ≤ C ϕ 2 3,2 α a 2 sup N 1 N N X i= 1 € 1 + |ω i | 4 α Š , cf. 31, ≤ δ, for a suitable C 0 depending on ω. Condition [T] is proved. • Let us verify Condition [A]: For every ϕ ∈ S , for every δ, θ , η 1 , η 2 0, θ ≤ δ, for every stopping time τ N , u N := P  M N τ N +θ ϕ − M N τ N ϕ η 2 ‹ ≤ 1 η 2 2 E M N τ N +θ ϕ − M N τ N ϕ 2 , ≤ 1 η 2 2 E   Z τ N +θ τ N ¬ ν N s , ϕ ′ y, π 2 ¶ ds   , ≤ ϕ 2 6, α 1 η 2 2 E   Z τ N +θ τ N Z S 1 ×R H y , π 2 −6,α d ν N s ds   , ≤ ϕ 2 6, α C η 2 2 E   Z τ N +θ τ N 1 N N X i= 1 1 + |ω i | 4 α ds   , cf. 18 and 21, ≤ ϕ 2 6, α C δ η 2 2 sup N 1 N N X i= 1 1 + |ω i | 4 α . This last term is lower or equal than η 1 for δ sufficiently small depending on ω. 2. Tightness of Θ N : we need to be more careful here, since the tightness is in law w.r.t. the disorder . Let ϕ j j ≥1 be a countable family in the nuclear Fréchet space S . Without any 818 restriction, we can always suppose that φ j 6, α = 1, for every j ≥ 1. We define the following decreasing sequences indexed by J ≥ 1 of subsets of M 1 C [0, T ], S ′ : K ǫ 1 ϕ 1 , . . . , ϕ J := n P ; ∀t, ∀1 ≤ j ≤ J, PΠ −1 ϕ j satisfies T t , δ,C 1 o , K ǫ 2 ϕ 1 , . . . , ϕ J := n P ; ∀1 ≤ j ≤ J, ∀η 1 , η 2 0, PΠ −1 ϕ j satisfies A η 1 , η 2 ,C 2 o , where C 1 = C 1 ǫ, δ, C 2 = C 2 ǫ, η 1 , η 2 will be precised later. By construction and by Mitoma’s theorem cf. Remark 4.12, K ǫ := \ J K ǫ 1 ϕ 1 , . . . , ϕ J ∩ K ǫ 2 ϕ 1 , . . . , ϕ J is a relatively compact subset of M 1 C [0, T ], S ′ . In order to prove tightness of Θ N , it is sufficient to prove that, for all ǫ 0, ∀i = 1, 2, lim sup N Θ N [ J K ǫ i φ 1 , . . . , φ J c ≤ ǫ. 33 For ǫ 0, let A = Aǫ such that lim inf N →∞ P A N ≤ A ≥ 1 − ǫ, and lim inf N →∞ P 1 N N X i= 1 1 + |ω i | 4 α + A N ω 1 , . . . , ω N ≤ A ≥ 1 − ǫ, where A N is the random variable defined in Proposition 4.4. We define the corresponding constants for a sufficiently large constant C: C 1 ǫ, δ := r A ǫ δ , C 2 ǫ, η 1 , η 2 := η 1 η 2 2 CA ǫ . Then, Θ N K ǫ 1 φ 1 , . . . , φ J ≥ P      ω, ∀t, ∀1 ≤ j ≤ J, ∀δ, E D η N , ω t , φ j E 2 C 1 δ, ǫ 2 ≤ δ      , ≥ P ‚ ω, sup t ≤T E η N , ω t 2 −6,α ≤ A Œ , by definition of C 1 , ≥ P A N ≤ Aǫ , cf. 18 and 25. Letting J → ∞ in the latter inequality, we obtain: Θ N S J K ǫ 1 φ 1 , . . . , φ J c ≤ PA N A. Taking on both sides lim sup N →∞ , we get the result. 819 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