BSDEs and PDEs with discontinuous coecients Applications to homogenization K. Bahlali, A. Elouain, E. Pardoux. Jena, March 2009

Transcription

1 BSDEs and PDEs with discontinuous coecients Applications to homogenization K. Bahlali, A. Elouain, E. Pardoux. Jena, March

2 1) L p viscosity solution to 2nd order semilinear parabolic PDEs with mesurable coecients and the link with BSDEs. 2) Homogenization of L p viscosity solution to 2nd order semilinear parabolic PDEs. 2

3 L p -viscosity solution (L. A. Caarelli et al. (CPAM 1996)) Let a ij, b i and f be measurable functions. Let L be dened by, L := d i, j=1 2 a ij (x 1, x 2 ) + x i x j d i=1 b i (x 1, x 2 ) x i. Consider the PDE, v s (s, x) = Lv(s, x) + f(x, v(s, x)) v(0, x) = H(x) (1) 3

4 Denition. Let p > N := d + 2 and v be a continuous function. -(a)- v is a L p -viscosity sub-solution to PDE (1), if v(t, x) H(x), x IR d+1 and for every ϕ W 1, 2 ( p, loc [0, T ] IR d+1, IR ) and ( t, x) [0, T ] IR d+1 at which v ϕ has a local maximum, one has { ess lim inf ϕ } (s, x) [Lϕ(s, x) + f(s, x, v(s, x))] 0. (s, x) (b t, bx) s 4

5 -(b)- v is a L p -viscosity super-solution to PDE (1) if, v(t, x) H(x), x IR d+1 and for every ϕ W 1, 2 ( p, loc [0, T ] IR d+1, IR ) and ( t, x) [0, T ] IR d+1 at which v ϕ has a local minimum, one has ess lim sup (s, x) (b t, bx) { ϕ (s, x) [Lϕ(s, x) + f(s, x, v(s, x))] s } 0. -(c)- v is a L p -viscosity solution to PDE (1) if it is both a L p -viscosity sub-solution and super-solution. 5

6 Remarks 1) The notion of Lp viscosity solution has been introduced by Caarelli et all. to give a unied treatment for fully nonlinear elliptic PDEs with measurable coecients. 2) If ϕ C 1, 2, then v is called a C viscosity solution. If moreover, the coecients are continuous this notion coincides with the classical viscosity solution. In this case, a C viscosity solution is a Lp viscosity solution. 3) If a weak (Sobolev) solution belongs to W 1, 2 p, loc enough), then it is also a Lp viscosity solution. (with p large Others properties of Lp viscosity solutions are given in Caarelli et all. 6

7 The following example concerning uniqueness gives another motivation to the notion of Lp viscosity solution. Let A be a borel subset of the interval [ 1, 1] s.t. for every interval I [ 1, 1], λ(a I) > 0 and λ(a c I) > 0. Put f := 1 A 1 A c and consider the Dirichlet problem, u = f on ( 1, 1) (2) u( 1) = u(1) = 0 Since for every y, lim inf x y f(x) = 1 and lim sup x y f(x) = 1, then any C function v which satises v 1 is a C viscosity solution to the problem (2). However, problem (2) has a unique Lp viscosity solution. 7

8 we will prove, by using BSDEs techniques, that the PDE (1) has a Lp viscosity solution. Théorème 1 Assume that, (H1) the martingale problem is well posed for the operator L, (H2) the matrix a is uniformly elliptic, (H3) a, b and f are with sub-linear growth, and, H is bounded, (H4) the function f(x, y) is uniformly Lipschitz in y. Then, the PDE (1) has a Lp viscosity solution. 8

9 Proof. (based on Krylov'estimate). Assume that v(t, x) is continuous (this will be proved later). We only prove that v is L p viscosity subsolution. To simplify the details of the proof, we deal with the terminal value problem, u (t, x) = (Lu)(t, x) + f(x, u(t, x)) t [0, T ], t (3) u(t, x) = H(x). Let X t,x be the diusion process associated to L and M Xt,x martingale part of X t,x. Let Y be the unique solution of the Markovian BSDE, be the Y t, x s T = H(X t,x T ) + s f(x t,x r, Yr t, x )dr T s Zr t, x dmr Xt,x (4) We will establish that u(t, x) := Y t,x t denes a L p viscosity sub-solution to the terminal value problem (3). 9

10 Let ϕ W 1, 2 p, loc ( [0, T ] IR d+1, IR ). Let ( t, x) [0, T ] IR d+1 be a local maximum of u ϕ. And assume that, u( t, x) = ϕ( t, x) (5) Since p > d + 2, then ϕ has a continuous version, which we consider from now on. (We argue by contradiction). Assume that ε > 0, and α > 0 such that, ϕ s (s, x) + Lϕ(s, x) + f(x, u(s, x)) < ε, λa.e. in B α ( t, x) (6) Since ( t, x) is a local Max. of u ϕ and satises (5), then α > 0 (which we can assume equal to α) s.t., u(t, x) ϕ(t, x) 0 in B α ( t, x) (7) 10

11 Dene, τ := inf { s t, ; X b t, bx s } x > α ( t + α) Since X is a Markov diusion, then for every r [ t, t + α], Y b t, bx r = u(r, X b t, bx r ) Hence, the process (Ȳs, Z s ) := {(Y b t, bx s τ ), 1 [0, τ] (s)(z b t, bx s )} satises the following BSDE, on [ t, t + α], 11

12 Ȳ s = u(τ, X b t, bx τ ) + bt+α s bt+α s 1 [0, τ] f(r, X b t, bx r Z r dm X b t, bx r., u(r, X b t, bx r ))dr On another hand, the Itô-Krylov formula shows that the process, ( (Ŷs, Ẑs) := ϕ(s τ, X b ) t, bx s τ ), 1 [0, τ] (s) ϕ(s, X b t, bx s ) satises the following BSDE, on [ t, t + α], Ŷ s = ϕ(τ, X b t, bx τ ) bt+α s bt+α s 1 [0, τ] [( ϕ r + Lϕ)(r, Xb t, bx r )]dr Ẑ r dm X b t, bx r 12

13 Let A := {(t, x) B α ( t, x), [ ϕ s + Lϕ + f(., u(.))](t, x) < ε} and Ā := B α ( t, x) \ A the complement of A. By (6) we have, λ(ā) = 0. Since the diusion process {X ˆt,ˆx s } is nondegenerate, then Krylov's inequality shows that It follows that, [ 1 [0, τ] [( ϕ r + Lϕ)(r, X b t,bx r of dt dip positive measure. 1 Ā (r, X b t, bx r ) = 0 dr dip a.e. ) + f(r, X b t,bx r, u(r, X b t,bx r ))])] > 0 on a set Comparison theorem (e.g. Pardoux 99) shows that, Ȳb t < Ŷb t, that is u( t, x) < ϕ( t, x), which contradicts our assumption (5). 13

14 Homogenization of PDEs. Example : Let a(x) c > 0. be a periodic function. For ε > 0, consider the operator L ε = d dx (a(x ε ) d dx ) For small ε, L ε can be replaced by L = d dx (ā d dx ) where ā is the average of 1 a over a period. ā is called the averaged (or limit) coecient associated to a. As ε is small, the solution of the parabolic equation t u = L ε u, u(0, x) = f(x) is close to the corresponding solution with L ε replaced by L. 14

15 Here, we consider a non necessary periodic case. 15

16 R.Z. Khashmiskii and N.V. Krylov (SPA 01) have considered the system of SDEs, t ( ) X 1, ε X 1, ε s t = x 1 + ϕ, Xs 2, ε dw s, ε X 2, ε t = x t 0 ( X b (1) 1, ε s ε, X 2, ε s ) ds + t They have proved that : if the averaged system Xt 1 = x 1 + t 0 ϕ(x1 s, Xs 2 )dw s 0 ( X σ (1) 1, ε s ε X 2 t = x 2 + t 0 b (1) (X 1 s, X 2 s )ds + t 0 σ(1) (X 1 s, X 2 s )d W s ), Xs 2, ε d W s, has a weakly unique solution. Then, (X 1, ε, X 2, ε ) law = (X 1, X 2 ). where F denotes the Cesaro mean of the function F and, dened by, (8) (9) 16

17 F (x 1, x 2 ) := lim x1 + 1 x 1 x1 lim x1 + ρ(t, x 0 2 )F (t, x 2 )dt 1 x1 x 1 ρ(t, x 0 2 )dt 1 {x1 >0} + lim x 1 1 x1 x 1 ρ(t, x 0 2 )F (t, x 2 )dt lim x1 1 x1 x 1 ρ(t, x 0 2 )dt 1 {x1 0} with ρ(x 1, x 2 ) := 1 [ϕ(x 1, x 2 )] 2. 17

18 They have deduced that, if for any ψ(x 1, x 2 ) Cb PDE v s (s, x 1, x 2 ) = L v(s, x 1, x 2 ) v(0, x 1, x 2 ) = ψ(x 1, x 2 ), the averaged (10) has a unique bounded solution v(t, x 1, x 2 ) W 1,2 d+1,loc bounded solution v ε (t, x 1, x 2 ) W 1,2 d+1,loc of the problem v ε s (s, x 1, x 2 ) = L ε v ε (s, x 1, x 2 ) v ε (0, x 1, x 2 ) = ψ(x 1, x 2 ) we have, lim ε 0 v ε (t, x 1, x 2 ) = v(t, x 1, x 2 ) where L (X 1, X 2 ) and L ε (X 1, ε, X 2, ε )., then for any (11) 18

19 Krylov (SPA 04) has proved that the martingale problem is well posed for (9). Krylov (arxiv 2007) has established that the averaged PDE (10) has a unique bounded solution v W 1,2 d+1,loc. These two results complete the result of Khasmiskii-Krylov. Our goal consists to extend The result Khasmiskii-Krylov to a system of SDE-BSDE. 19

20 We put, B := (W, W ) := IR IR d Brownian motion, b := (0, b (1) ), a 00 := 1 2 ϕ2, a ij := 1 2 (σ(1) σ (1) ) ij, i, j = 1,..., d, and σ := One has σ IR (d+1) k with σ 00 = ϕ, σ 0j = 0, j = 1,..., k 1 σ i0 = 0, i = 1,..., d σ ij = σ (1) ij, i = 1,..., d, j = 1,..., k 1 ϕ 0 0 σ (1). If moreover we put X ε := (X 1, ε, X 2, ε ), then SDE (8) becomes, X ε s = X ε 0 + s 0 b( X1, u ε ε, X 2, ε u )du + s 0 σ( X1, u ε ε, Xu 2, ε )db u 20

21 We consider the sequence of systems of SDE-BSDE, Xs ε = X0 ε + s 0 b( X1, ε u ε, Xu 2, ε )du + s 0 σ( X1, ε u ε, Xu 2, ε )db u, Y ε s = H(Xt ε ) + t s f( X1, ε u ε, Xu 2, ε, Yu ε )du t s Zε u dmu Xε where M Xε denotes a martingale part of the process X ε. The PDE associated to eqt (12) is, v ε s (s, x 1, x 2 ) = L ε (x 1, x 2 )v ε (s, x 1, x 2 ) + f( x 1 ε, x 2, v ε (s, x 1, x 2 )) v ε (0, x 1, x 2 ) = H(x 1, x 2 ) where L ε = a 00 ( x 1 ε, x 2) a ij ( x 1 x 1 ε, x 2) 2 + b (1) i ( x 1 x 2i x 2j ε, x 2) Assuming that the averaged coes σ, b, f exist, we then establish, (12) x 2i (13) 21

22 Théorème 2 (Xt ε, Yt ε, t s Zε u dmu Xε = (X t, Y t, t s Z u dmu X ) which is the unique solution to the system of SDE-BSDE, ) law X s = x + s 0 b(x u )du + s 0 σ(x u)db u, 0 s t. Y s = H(X t ) + t s f(x u, Y u )du t s Z udm X u, 0 s t (14) where σ, b and f are the averages of σ, b and f. Théorème 3 i) v ε (x 1, x 2 ) converges to v(x 1, x 2 ), ii) v(x 1, x 2 ), is a unique L p -viscosity solution to the PDE v s (s, x 1, x 2 ) = Lv(s, x 1, x 2 ) + f(x 1, x 2, v(s, x 1, x 2 )) v(0, x 1, x 2 ) = H(x 1, x 2 ) (15) such that v(t, x) = Y (t,x) 0. 22

23 Averaged coefts. For a function g {b i, a ij, f}, we dene g + (x 2 ) := lim x1 + 1 x1 x 1 g(t, x 0 2 )dt, g (x 2 ) := lim x1 ( 1 x 1 ) x 1 g(t, x 0 2 )dt We put, ρ(x 1, x 2 ) := a 00 (x 1, x 2 ) 1 = [ 1 2 ϕ2 (x 1, x 2 )] 1 g ± (x 1, x 2 ) := g + (x 2 )1 {x1 >0} + g (x 2 )1 {x1 0} and ḡ(x 1, x 2 ) := (ρg)± (x 1, x 2 ) ρ ± (x 1, x 2 ) g ± is called the Cesaro limit (or mean) of g. 23

24 We have, ḡ(x 1, x 2 ) = lim x1 + lim x1 + 1 x 1 x1 ρ(t, x 0 2 )g(t, x 2 )dt 1 x1 x 1 ρ(t, x 0 2 )dt 1 {x1 >0} + lim x 1 1 x1 x 1 ρ(t, x 0 2 )g(t, x 2 )dt lim x1 1 x1 x 1 ρ(t, x 0 2 )dt b, ā and f may have discontinuity at x1 = 0. 1 {x1 0} 24

25 Assumptions. (A1) The function b (1), σ (1), ϕ are uniformly Lipschitz in x (A2) For each x 1, the rst and second order derivatives with respect to x 2, of b (1), σ (1), ϕ, are bounded continuous functions of x 2. (A3) a := (σ (1) σ (1) ) is uniformly elliptic, i.e : Λ > 0; x, ξ IR d, ξ a(x)ξ Λ ξ 2. Moreover, there exists positive constants C 1, C 2, C 3 such that (i) C 1 a 00 (x 1, x 2 ) C 2 (ii) d i=1 [a ii(x 1, x 2 ) + b 2 i (x 1, x 2 )] C 3 (1 + x 2 2 ) (B1) Let D x2 u and D 2 x 2 u denote respectively the gradient vector and the matrix of second derivatives of u with respect to x 2. The 25

26 following limits are uniform in x 2, 1 x 1 x1 0 ρ(t, x 2 )dt ρ ± (x 2 ) as x 1 ± 1 x 1 1 x 1 x1 0 x1 0 D x2 ρ(t, x 2 )dt D x2 ρ ± (x 2 ) as x 1 ± D 2 x 2 ρ(t, x 2 )dt D 2 x 2 ρ ± (x 2 ) as x 1 ± 26

27 (B2) For every i and j, the functions ρb i, D x2 (ρb i ), D 2 x 2 (ρb i ), ρa ij, D x2 (ρa ij ), D 2 x 2 (ρa ij ) have averages in Cesaro sense. (B3) For every function k {ρb i, D x2 (ρb i ), D 2 x 2 (ρb i ), ρa ij, D x2 (ρa ij ), D 2 x 2 (ρa ij )}, there exists a bounded function α such that 1 x 1 x1 0 k(t, x 2 )dt k ± (x 1, x 2 ) = (1 + x 2 2 )α(x 1, x 2 ), lim x1 sup x2 IR d α(x 1, x 2 ) = 0. (16) (C1) (i) The generator f is uniformly Lipschitz in (x 1, x 2, y) and, for each x 1 IR, its derivatives in (x 2, y) up to and including second order derivatives are bounded continuous functions of (x 2, y). 27

28 (ii) There exists positive constant K such that for every (x 1, x 2, y), f(x 1, x 2, y) K(1 + x 2 + y ). (iii) H is continuous and bounded. (C2) ρf has a limit in Cesaro sense and there exists a bounded measurable function β such that 1 x 1 x2 0 ρ(t, x 2 )f(t, x 2, y)dt (ρf) ± (x 1, x 2, y) = (1 + x y 2 )β(x 1, x 2, y) lim x1 sup (x2, y) IR d IR β(x 1, x 2, y) = 0, (17) where (ρf) ± (x 1, x 2, y) := (ρf) + (x 2, y)1 {x1 >0} + (ρf) (x 2, y)1 {x1 0}. (C3) For each x 1, ρf has a derivatives up to a second order in x 2 uniformly in y and these derivatives are bounded and satisfy (C2). In the sequel, (A) stands for conditions (A1), (A2), (A3) ; (B) for 28

29 conditions (B1), (B2), (B3) and (C) for (C1), (C2), (C3). 29

30 Proof of Theorem 2. Step 1. By Khasminskii & Krylov (SPA 01) and Krylov (SPA 04) : the sequence of processes X ε := (X 1, ε, X 2, ε ) law = X := (X 1, X 2 ). Moreover, the limit X is the unique weak solution to the forward component of the averaged system of SDE-BSDE (14). 30

31 Step 2. Arguing as in Pardoux (99) or Buckdahn-Engelbert-Rascanu (05), one can show that : There exists (Y, M) and a countable subset D of [0, t] such that along a subsequence of ε, (i) (X ε, Y ε, M ε ) law = (X, Y, M) on C D ([0, t], IR) D ([0, t], IR) The space D is endowed with the S-topology. (ii) (Y ε, M ε ) (Y, M) in nite-distribution on D c. (iii) Y s = H(X t ) + t s f(x 1 u, X 2 u, Y u )du (M t M s ) The strong uniqueness of the BSDE ( f, H(X t )) allows to show that M r = r 0 Z u dm X u. 31

32 Step 3 The function v(t, x) := Y (t,x) 0 is continuous and is a L p viscosity solution to PDE (15). Remark : The main diculty, in the proof of step 3, stays in two points : 1) The identication of the limit as f(...) 2) The continuity of Y (t,x) 0 in (t, x). Note that, by Theorem 1, Y (t,x) 0 is a L p viscosity solution to PDE (15). 32

33 To prove the point 1) we write, s [ X 1, ε f( r ε 0 s = + 0 s [ X 1,ε f( r ε 0, Xr 2, ε, Yr ε ) f(x r 1, Xr 2, Y r ) ] dr (18) [ f(x 1, ε r, X2, r ε, Yr ε ) 1,ε f(x r, Xr 2,ε, Yr ε ) ] dr, Xr 2, ε, Yr ε ) f(x r 1, Xr 2, Y r ) ] dr Using the almost sure version of Skorokhod's representation theorem (proved by Jakubowski 99), we show that,. 0 f(x 1, ε r, Xr 2, ε, Yr ε )dr law =. 0 f(x 1 r, X 2 r, Y r )dr, in C([0, t], IR) 33

34 Using the following lemma 1, which is an extension of the Khashminskii-Krylov result to FBSDEs, we prove that s (f( X1, r ε ), Xr 2, ε, Yr ε 1, ε ) f(x r, Xr 2, ε, Yr ε ) dr ε sup 0 s t 0 0 in probability. 34

35 Lemma 1 For y IR, let V ε (x 1, x 2, y) denote the solution of the following equation : a 00 ( x 1 ε, x 2 )Dx 2 1 u(x 1, x 2, y) = f( x 1 ε, x 2, y) f(x 1, x 2, y) (19) u(0, x 2, y) = D x1 u(0, x 2, y) = 0. Then, for some bounded functions β 1 and β 2 satisfying (C2), (i) D x1 V ε (x 1, x 2, y) = x 1 (1 + x y 2 )β 1 ( x 1 ε, x 2, y), and the same is true with D x1 V ε replaced by D x1 D x2 V ε and D x1 D y V ε. (ii) for any K ε { V ε, D x2 V ε, Dx 2 2 V ε, D x1 D x2 V ε, D y V ε, DyV 2 ε, D x1 D y V ε, D x2 D y V ε} it holds, K ε (x 1, x 2, y) = x 2 1(1 + x y 2 )β 2 ( x 1 ε, x 2, y) 35

36 Proof of the continuity of the map (t, x) Y (t,x) 0 It should be noted that, the lack of L 2 -continuity property for the ow X x := (X 1, x, X 2, x ) transfers the diculty to the backward component and hence we cannot prove the L 2 -continuity of the process Y as usually done. To overcome this diculty, we establish weak continuity for the ow x (X 1, x, X 2, x ), then arguing as in homogenization part and using the fact that Y (t,x) 0 is deterministic, we derive the continuity of Y (t,x) 0. 36

37 Let (t n, x n ) (t, x). We assume that t > t n > 0. We have, Y t n, x n s = H(X x n t n ) + tn s f(x x n u, Y t n, x n u )du tn s Z t n, x n u dm Xx n u (20) where X x n law = X x. Since H is a bounded continuous function and f satises (C1), then the sequence {(Y t n, x n,. 0 1 [s,t n ](u)z x n u dm Xx n u )} is tight in D([0, t] IR IR) endowed with the Jakubowski S-topologie. 37

38 Let us rewrite equation (20) as follows, Y t n, x n s = H(X x n t n ) + t t n t s f(x x n u, Y t n, x n u )du t s 1 [s,tn ](u)z t n, x n u dm Xx n u x f(x n u, Y t n, x n u )du (21) := A 1 n + A 2 n Convergence of A 2 n t One has IE x f(x n u, Y t n, x n u )du K( x ) t t n. t n Hence, A 2 n tends to zero in probability. The convergence of A 1 n can be performed as in the homogenization law = Y t, x part. On can derive that Y t n, x n 0 continuity of Y t, x 0 since Y t n, x n 0 and, Y t, x 0 which yields to the 0 are deteministic. 38

39 Convergence of A 1 n Denote by (Y, M ) the weak limit of {(Y t n, x n,. 0 1 [s,tn ](u)z x n u dm Xx n u )}. Arguing as previously, we show that t s f(x x n u, Y t n, x n u )du = law t s f(x x u, Y u)du. Passing to the limit in (21), we obtain that Y s = H(X x t ) + t s f(x x u, Y u)du (M t M s), s [0, t] D c. The uniqueness of the considered BSDE ensures that s [0, t], Y s = Ys t, x IP a.s. law = Y t, x. Finally, as in 1), one can derive that law = Y t, x 0 which yields to the continuity of Y t, x 0. Hence, Y t n, x n Y t n, x n 0 39

40 Proposition 1 (Continuity in law of the ow x X x. ) Assume (A), (B). Let Xs x (14), and X x n s := x n + s 0 b(x x n u )du + be the unique weak solution of the SDE s 0 σ(x x n u )db u, 0 s t Assume that x n tends to x = (x 1, x 2 ) IR 1+d. Then, X x n = law X x. Proposition 2 Assume (A), (B), (C). Let p > d + 2. Then, (i) lim IE Y 0 ε v(t, x) 2 = 0. ε 0 (ii) Y t, x 0 := v(t, x) C(IR + IR d+1 ), and it is a L p -viscosity solution of PDE (15). 40

41 Proof (i) We shall prove that lim ε 0 IE Y 0 ε Ȳ0 2 = 0. We have, Y0 ε = H(Xt ε ) + t 0 f( X u, ε Xu 2, ε, Yu ε )du Mt ε Ȳ 0 = H(X t ) + t 0 f(x u, Ȳu)du M t From Jakubowski (1997), the projection : y y t is continuous in the S-topology. We then deduce that Y0 ε converges towards Ȳ0 in distribution. Moreover, since Y0 ε and Y 0 are deterministic and bounded, we have lim ε 0 IE Y 0 ε Ȳ0 2 = 0. That is lim ε 0 IE v ε (t, x) v(t, x) 2 = 0. (ii) (t, x) IR + IR d+1 Y t,x is continuous in law and as in (i), we derive the result. 41

42 Possible extentions. Theorems 1 to the case where f is continuous in (x, y) by using the existence of weak solution, due to Buckdahn-Engelbert-Rascanu (2005)? Theorems 2 and 3 to the case where f is continuous in (x, y) and assuming the uniqueness of the BSDE component. Extension to the case where f depends in the variable Z (work in progress). Homogenization of the weak (Sobolev) solution by using BSDEs (work in progress). 42

43 References. Bahlali, K. Existence and uniqueness of solutions for BSDEs with locally Lipschitz coecient. Electron. Comm. Probab. 7, , (2002). R. Buckdahn, H. J. Engelbert, A. Rascanu, On weak solutions of Backward SDEs. Theory of proba. and its Appl. 49 (2005), Caarelli, L.A, Crandall, M.G., Kocan, M., wiech, A. On viscosity solutions of fully nonlinear equations with measurable ingredients. Comm. Pure Appl. Math. 49, ,

44 Khasminskii, R ; Krylov, N. V. On averaging principle for diusion processes with null-recurrent fast component. Stochastic Processes and their applications, 93, , Krylov, N. V. On weak uniqueness for some diusions with discontinuous coecients. Stochastic Processes and their applications, 113, 37-64, Meyer, P. A., Zheng, W. A. Tightness criteria for laws of semimartingales. Ann. Inst. H. Poincaré Probab. Statist. 20, (4), , Jakubowski, A. A non-skorohod topology on the Skorohod space. Electron. J. Probab. 2, paper no. 4, pp.1-21,

45 Pardoux, E. BSDEs, weak convergence and homogenization of semilinear PDEs in F. H Clarke and R. J. Stern (eds.), Nonlinear Analysis, Dierential Equations and Control, Kluwer Academic Publishers., Bensoussan, A. ; Lions, J.-L. ; Papanicolaou, G. Asymptotic analysis for periodic structures. Studies in Mathematics and Its Applications, 5. North-Holland, Amsterdam-New York, Freidlin M. Functional integration and partial dierential equations. Annals of Mathematics Studies, 109, Princeton University Press, Princeton, Jikov, V. V. ; Kozlov, S. M. ; Ole nik, O. A. Homogenization of dierential operators and integral functionals. Translated from the Russian by G. A. Yosian. Springer, Berlin,