Equation verified by Tightness of

separable and by an argument of boundedness of the second marginal of any accumulation point, thanks to H Q µ , we show the tightness in D[0, T ], M F , w. The proof is complete when we prove the uniqueness of any accumulation point.

3.1 Proof of the tightness result

We want to show successively: 1. Tightness of L ν N , ω in D[0, T ], M F , v, 2. Equation verified by any accumulation point, 3. Characterization of the marginals of any limit, 4. Convergence in D[0, T ], M F , w.

3.1.1 Equation verified by

ν N , ω For f ∈ C 2 b S 1 × R, we denote by f ′ , f ′′ the first and second derivative of f with respect to the first variable. Moreover, if m ∈ M 1 S 1 × R, then m , f stands for R S 1 ×R f x , πm dx, dπ. Applying Ito’s formula to 11, we get, for all f ∈ C 2 b S 1 × R, D ν N , ω t , f E = D ν N , ω , f E + 1 2 Z t ds ¬ ν N , ω s , f ′′ ¶ + Z t ds ¬ ν N , ω s , f ′ · b[·, ν N , ω s ] + c ¶ + M N , f t, where M N , f t := 1 N P N j= 1 R t f ′ x N , ω j , ω j dB j s is a martingale f ′ bounded.

3.1.2 Tightness of

L ν N , ω in D[0, T ], M F , v C c S 1 × R is separable: let f k k ≥1 elements of C ∞ S 1 × R a dense sequence in C c S 1 × R, and let f ≡ 1. We define Ω := D[0, T ], M 1 , v and the applications Π f , f ∈ C c S 1 × R by: Π f : Ω → D[0, T ], R m 7→ m , f . Let P n n a sequence of probabilities on Ω and Π f P n = P n ◦ Π −1 f ∈ D[0, T ], R. We recall the following result: Lemma 3.1. If for all k ≥ 0, the sequence Π f k P n n is tight in M 1 D[0, T ], R, then the sequence P n n is tight in M 1 D[0, T ], M 1 , v. 805 Hence, it suffices to have a criterion for tightness in D[0, T ], R. Let X n t be a sequence of processes in D[0, T ], R and F n t a sequence of filtrations such that X n is F n -adapted. Let φ n = {stopping times for F n }. We have cf. Billingsley [4]: Lemma 3.2 Aldous’ criterion. If the following holds, 1. L € sup t ≤T X n t Š n is tight, 2. For all ǫ 0 and η 0, there exists δ 0 such that lim sup n sup S ,S ′ ∈φ n ;S ≤S ′ ≤S+δ∧T P € X n S − X n S ′ η Š ≤ ǫ, then L X n is tight. Proposition 3.3. The sequence L ν N , ω is tight in D[0, T ], M F , v. Proof. For all ǫ 0, for all k ≥ 1 the case k = 0 is straightforward, P ‚ sup t ≤T D ν N , ω t , f k E 1 ǫ Œ ≤ ǫ f k ∞ E     sup t ≤T D ν N , ω t , 1 E | {z } =1     , Markov Inequality. The tightness follows. For all k ≥ 1, we have the following decomposition: D ν N , ω t , f k E = D ν N , ω , f k E + A N , ω t f k + M N , ω t f k , where A N , ω t f k is a process of bounded variations, and M N , ω t f k is a square-integrable martin- gale. Then it suffices to verify Lemma 3.2, 2 for A and M separately. For all ǫ 0 and η 0, for all stopping times S, S ′ ∈ φ N ; S ≤ S ′ ≤ S + δ ∧ T , we have: a N := P  A N , ω S ′ f k − A N , ω S f k η ‹ , ≤ 1 η E   Z S ′ S ds ¬ ν N , ω s , f ′ k · b[·, ν N , ω s ] + c ¶   + 1 η E   1 2 Z S ′ S ds ¬ ν N , ω s , f ′′ k ¶   , ≤ C η E S ′ − S ≤ ǫ, for δ sufficiently small. we use here that f k are of compact support for k ≥ 1; in particular the function x, π 7→ f ′ k x, πcx, π is bounded. Furthermore, P  M N , ω S ′ f k − M N , ω S f k η ‹ = P M N , ω S ′ f k − M N , ω S f k 2 η 2 , ≤ 1 η 2 E M N , ω S ′ f k − M N , ω S f k 2 , ≤ 1 N η 2 E   N X i= 1 Z S ′ S f ′2 k x i s , ω i ds   ≤ C N η 2 δ. 806 At this point, L ν N , ω is tight in D[0, T ], M F , v. 3.1.3 Equation satisfied by any accumulation point in D[0, T ],