Homogenization of Semilinear Pdes with Discontinuous Averaged Coefficients

We study the asymptotic behavior of solutions of semilinear PDEs. Neither periodicity nor ergod-icity will be assumed. On the other hand, we assume that the coefficients have averages in the Cesaro sense. In such a case, the averaged coefficients could be discontinuous. We use a proba-bilistic approach based on weak convergence of the associated backward stochastic differential equation (BSDE) in the Jakubowski S-topology to derive the averaged PDE. However, since the averaged coefficients are discontinuous, the classical viscosity solution is not defined for the averaged PDE. We then use the notion of "L p −viscosity solution" introduced in [7]. The existence of L p −viscosity solution to the averaged PDE is proved here by using BSDEs techniques.


Introduction
Homogenization of a partial differential equation (PDE) is the process of replacing rapidly varying coefficients by new ones such that the solutions are close.Let for example a be a one dimensional periodic function which is positive and bounded away from zero.For ǫ > 0, we consider the operator For small ǫ, L ǫ can be replaced by where a is the averaged (or limit, or effective) coefficient associated to a.As ǫ is small, the solution of the parabolic equation is close to the corresponding solution with L ǫ replaced by L.
The probabilistic approach to homogenization is one way to prove such results in the periodic or ergodic case.It is based on the asymptotic analysis of the diffusion process associated to the operator L ǫ .The averaged coefficient a is then determined as a certain "mean" of a with respect to the invariant probability measure of the diffusion process associated to L.
There is a vast literature on the homogenization of PDEs with periodic coefficients, see for example the monographs [3; 12; 21] and the references therein.There also exists a considerable literature on the study of asymptotic analysis of stochastic differential equations (SDEs) with periodic structures and its connection with homogenization of second order partial differential equations (PDEs).In view of the connection between BSDEs and semilinear PDEs, this probabilistic tool has been used in order to prove homogenization results for certain classes of nonlinear PDEs, see in particular [4; 5; 6; 9; 11; 13; 19; 23; 24] and the references therein.The two classical situations which have been mainly studied are the cases of deterministic periodic and random stationary coefficients.This paper is concerned with a different situation, building upon earlier results of Khasminskii and Krylov.
In [15], Khasminskii & Krylov consider the averaging of the following family of diffusions process where for each ǫ > 0 small, U 1, ǫ t is a one-dimensional null-recurrent fast component and U 2, ǫ t is a d-dimensional slow component.The function ϕ (resp.σ (1) , resp.b (1) ) is IR-valued (resp.IR d×(k−1)valued, resp.IR d -valued).(W, W ) is a k-dimensional standard Brownian motion whose component W (resp. W ) is one dimensional (resp.(k-1)-dimensional).Define now (X 1,ǫ , X 2,ǫ ) = (ǫU 1,ǫ , U 2,ǫ ).The process They define the averaged coefficients as limits in the Cesaro sense.With the additional assumption that the presumed SDE limit is weakly unique, they prove that the process (X As a byproduct, they derive the limit behavior of the linear PDE associated to (X 1, ǫ t , X 2, ǫ t ), in the case where weak uniqueness of the limiting PDE holds in the Sobolev space W 1,2 d+1,loc (IR + × IR d ) of all funcions u(t, x) defined on IR + × IR d such that both u and all the generalized derivatives D t u, D x u, and D 2 x x u belong to L d+1 l oc (IR + × IR d ).In the present note, we extend the results of [15] to parabolic semilinear PDEs.Note that the limiting coefficients can be discontinuous.More precisely, we consider the following sequence of semi-linear PDEs, indexed by ǫ > 0, (1.3) where ϕ, σ (1) and b (1) are those defined above in equation (1.1), and the real valued measurable functions f and H are defined on IR d+1 × IR and IR d+1 respectively.
Note that Y ǫ 0 does depend upon the pair (t, x) where x is the initial condition of the forward SDE part of (1.4), and t is the final time of the BSDE part of (1.4).It follows from e. g.Remark 2.6 in [22] that under suitable conditions upon the coefficients {v ǫ (t, The aim of the present paper is 1. to show that for each t > 0, x ∈ IR d+1 , the sequence of processes where M X is the martingale part of X and σ, b and f are respectively the average of σ, b and f , in a sense which will be made precise below; 2. deduce from the first result that for each (t, x), v ǫ (t, , where v solves the following averaged PDE in the L p -viscosity sense the averaged operator. The method used to derive the averaged BSDE is based on weak convergence in the S-topology and is close to that used in [23] and [24].In our framework, we show that the limiting system of SDE -BSDE (1.5) has a unique solution.However, due to the discontinuity of the coefficients, the classical viscosity solution is not defined for the averaged PDE (1.6).We then use the notion of "L p −viscosity solution".We use BSDE techniques to establish the existence of L p −viscosity solution for the averaged PDE.The notion of L p -viscosity solution has been introduced by Caffarelli et al. in [7] to study fully nonlinear PDEs with measurable coefficients.Note however that although the notion of a L p -viscosity solution is available for PDEs with merely measurable coefficients, continuity of the solution is required.In our situation, the lack of L 2 -continuity property for the flow X x := (X 1, x , X 2, x ) transfers the difficulty to the backward one and hence we cannot prove the L 2continuity of the process Y .To overcome this difficulty, we establish weak continuity for the flow x → (X 1, x , X 2, x ) and using the fact that Y x 0 is deterministic, we derive the continuity property for Y x 0 .
The paper is organized as follows: In section 2, we make precise some notations and formulate our assumptions.Our main results are stated in section 3. Section 4 and 5 are devoted to the proofs.

Notations and assumptions 2.1 Notations
For a given function g(x 1 , x 2 ), we define The average, in Cesaro sense, of g is defined by ) and denote by b(x 1 , x 2 ), ā(x 1 , x 2 ) and f (x 1 , x 2 , y), the averaged coefficients defined by bi where ā(x 1 , x 2 ) denotes the matrix (ā i j (x 1 , x 2 )) i, j .
It is worth noting that b, ā and f may be discontinuous at x 1 = 0.

Assumptions.
We consider the following conditions.
(A2) For each x 1 , the first and second order derivatives with respect to x 2 of these functions are bounded continuous functions of x 2 .
(A3) a (1) ρ denote respectively the gradient vector and the matrix of second derivatives of ρ with respect to x 2 .We assume that uniformly with respect to x 2 1 as x 1 → ±∞, (B2) For every i and j, the coefficients (ρa i j ) have averages in the Cesaro sense. (2.1) The coefficient 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).
(ii) There exists positive constant K such that for every (iii) H is continuous and bounded.
(C2) ρ f has a limit in the Cesaro sense and there exists a bounded measurable function (C3) For each x 1 , ρ f has derivatives up to second order in (x 2 , y) and these derivatives are bounded and satisfy (C2).
We now define the notion of L p -viscosity solution of a parabolic PDE.This notion has been introduced by Caffarelli et al. in [7] to study PDEs with measurable coefficients.Presentations of this topic can be found in [7; 8].
Let g : IR d+1 × IR −→ IR be a measurable function and denote the second order PDE operator associated to the SDE (3.1).
We consider the parabolic equation Here, G(t, x, ϕ(s, x)) is merely assumed to be measurable upon the variable x =: (x 1 , x 2 ).
, IR is a L p -viscosity solution if it is both a L p -viscosity subsolution and super-solution.
Remark 3.2.Condition (a) means that for every ǫ > 0, r > 0, there exists a set A ⊂ B r ( t, x) of positive measure such that, for every (s, x) ∈ A, The main results are (the S-topology is explained in the Appendix below) Theorem 3.3.Assume (A), (B), (C) hold.Then, for any (t, x) ∈ IR + × IR d+1 , there exists a process (X s , Y s , Z s ) 0≤s≤t such that, (i) the sequence of process X ǫ converges in law to the continuous process X, which is the unique weak solution to SDE (1.5), in C([0, t]; IR d+1 ) equipped with the uniform topology.
(ii) the sequence of processes (Y ǫ s , , where M X is the martingale part of X , equipped with the Stopology.) s be the unique solution of the BSDE (1.5).Then

Proof of Theorem 3.3.
In all of this section, (t, x) ∈ IR + × IR d+1 is arbitrarily fixed with t > 0.
Assertion (i) follows from [15] and [18].Assertion (iii) can be established as in [23; 24].We shall prove (ii).We first deduce from our assumptions (see in particular (A3) which says that the coefficients of the forward SDE part of (1.4) are bounded with respect to their first variable, and grow at most linearly in their second variable) Lemma 4.1.For all p ≥ 1, there exists constant C p such that for all ǫ > 0,

Tightness and convergence for the BSDE. Proposition 4.2.
There exists a positive constant C such that for all ǫ > 0 Proof.We deduce from Itô's formula (here and below X 1, ǫ r = X 1, ǫ r /ǫ) It follows from well known results on BSDEs that we can take the expectation in the above identity (see e. g. [22]; note that introducing stopping times as usual and using Fatou's Lemma would yield (4.1) below).We then deduce from Gronwall's lemma that there exists a positive constant C which does not depend on ǫ, such that for every s ∈ [0, t], Combining the last two estimates and the Burkhölder-Davis-Gundy inequality, we get In view of condition (C1) and Lemma 4.1, the proof is complete.
We deduce immediately from Proposition 4.2 Proof.Since M ǫ is a martingale, then by [20] or [14], the Meyer-Zheng tightness criteria is fulfilled whenever where the conditional variation C V is defined in appendix A.

Proposition 4.5.
There exists (Y, M ) and a countable subset D of [0, t] such that along a subsequence ǫ n → 0, , IR) endowed with the S-topology.(ii) follows from Theorem 3.1 in Jakubowski [14].
) equipped with the uniform topology.From assertion (i), (Y ǫ • ) ǫ>0 is tight in ([0, t], IR) equipped with the S-topology.Hence the subsequence ǫ n can be chosen in such a way that (iii) holds.

Identification of the limit finite variation process.
Proposition 4.6.Let (Y, M ) be any limit process as in Proposition 4. 5

. Then
(ii) M is a s -martingale, where s := σ X r , Y r , 0 ≤ r ≤ s augmented with the IP-null sets.
To prove this proposition, we need the following lemmas.

y), and the same is true with D
ǫ , x 2 , y), and the same is true with V ǫ replaced by D Proof.We will adapt the idea of [15] to our situation.For ǫ > 0 and (z, x 2 , y) ∈ IR d+2 we set where g(z, x 2 , y) := f (z, x 2 , y) − f (ǫz, x 2 , y).
We only treat the case where x 1 > 0. The same argument can be used in the case x 1 < 0. We successively use the definition of f and assumptions (C2), to obtain where α 1 ( Using assumptions (B1) and (C1-ii), one can show that α 1 is a bounded function which satisfies (2.2).Since ǫ , x 2 , y), we derive the result for D x 1 V ǫ (x 1 , x 2 , y).Further, by integrating it, we get where Clearly, β 2 (

2). The result for the other quantities can be deduced by similar arguments from assumptions (B1), (C1), (C2) and (C3). Lemma 4.8. As
Proof.We shall show that for every s ∈ [0, t], r tends to zero in probability as ǫ tends to zero.Let V ǫ denote the solution of equation (4.4).Note that V ǫ has first and second derivatives in (x 1 , x 2 , y) which are possibly discontinuous only at x 1 = 0.Then, as in [15], since ϕ 2 is bounded away from zero, we can use the Itô-Krylov formula to get In view of Lemma 4.7 and Corollary 4.3, s |≥ ǫ} and Lemma 4.7, we obtain From Lemma 4.1 and Proposition 4.2, we deduce that Then, since β 2 satisfy respectively (2.2), the right hand side of the previous inequality tends to zero as ǫ −→ 0. Similarly, one can show that s 0 Tr ace a (1) converges to zero in probability.Let us give an explanation concerning the one but last term, which is the most delicate one.
is the increasing process associated to a martingale which is uniformly L p (IP)−integrable for each p ∈ IN, its L p (IP) norm is bounded, for all p ≥ 1. Finally the same argument as above shows that sup 0≤r≤s For the proof of this Lemma, we need the following two results.
Using Itô's formula, we get Since φ is lower bounded by C 1 , taking the expectation, we get The same argument, applies to D ǫ n , allows us to show the first estimate.Lemma 4.11.Consider a collection {Z ǫ , ǫ > 0} of real valued random variables, and a real valued random variable Z. Assume that for each n ≥ 1, we have the decompositions Then Z ǫ ⇒ Z, as ǫ → 0.
Proof.The above assumptions imply that the collection of random variables {Z ǫ , ǫ > 0} is tight.Hence the result will follow from the fact that IEΦ(Z ǫ ) → IEΦ(Z), as ǫ → 0 for all Φ ∈ C b (IR) which is uniformly Lipschitz.Let Φ be such a function, and denote by K its Lipschitz constant.Then for all n ≥ 1.The result follows.
Proof of Lemma 4.9.For each n ≥ 1, define a function θ n ∈ C(IR, [0, 1]) such that θ n (x) = 0 for |x| ≤ 1 2n , and θ n (x) = 1 for |x| ≥ 1 n .We have Note that the mapping ) equipped with the product of the sup-norm and the S topologies into IR.Hence from Proposition 4.5, Z 1,ǫ n =⇒ Z 1 n as ǫ → 0, for each fixed n ≥ 1.Moreover, from Lemma 4.10, the linear growth property of f , Lemma 4.1 and Proposition 4.2, we deduce that Lemma 4.9 now follows from Lemma 4.11.
Proof of Proposition 4.6 Passing to the limit in the backward component of the equation (1.4) and using Lemmas 4.8 and 4.9, we derive assertion (i).Assertion (ii) can be proved by using the same arguments as those in section 6 of [24].

Identification of the limit martingale.
Since f is uniformly Lipschitz in y and H is bounded, then standard arguments of BSDEs (see e. g. [23]) show that the BSDE (1.5) has a strongly unique solution and we have, Proposition 4.12.Let ( Ȳs , Zs , 0 ≤ s ≤ t) be the unique solution to BSDE (1.5).Then, for every s ∈ [0, t], Proof.For every s ∈ [0, t] \ D, we have Arguing as in [24], we show that M := .
s Zr d M X r is a s -martingale.Since f satisfies condition (C1), we get by Itô's formula, that Therefore, Gronwall's lemma yields that IE|Y s − Ȳs Since the SDE (3.1) is weakly unique ( [18]), the martingale problem associated to X = (X 1 , X 2 ) is well posed.We then have the following: Remark.The continuity of the map (t, x) −→ v(t, x) := Y t, x 0 , which is assumed in assertion (ii) of Propostion 5.1, will be established in Proposition 5.3 below.
Proof of Proposition 5.1.(i) Thanks to Remark 3.5 of [23], it is enough to prove existence and uniqueness for the BSDE Since f satisfies (C) and ρ is bounded, one can easily verify that f is uniformly Lipschitz in y uniformly with respect to (x 1 , x 2 ) and satisfies (C1)-(ii).Existence and uniqueness of solution follow then from standard results for BSDEs, see e. g. [22].Moreover, since H is uniformly bounded and f satisfies the linear growth condition (C1)-(ii), one can prove that the solution Y t, x is bounded, see e. g. [1].
is measurable with respect to a trivial σ−algebra and hence it is deterministic.
(ii) Assume that the function v(t, x) := Y t,x 0 belongs to [0, T ] × IR d+1 , IR .We only prove that v is a L p -viscosity sub-solution.The proof of the super-solution property can be done similarly.Since the coefficient of PDE under consideration are time homogeneous, then v(t, x) is solution to the initial value problem (1.6) if and only if the function u(t, x) := v(T − t, x) is solution to the terminal value problem. (5.1) Working with this backward PDE will simplify the details of the proofs below.
Let X t,x s be the unique weak solution to SDE (3.1).We will establish that the solution Y of the Markovian BSDE From the choice of τ, (τ, X t, x τ ) ∈ B α ( t, x).Therefore, u(τ, X t, x τ ) ≤ ϕ(τ, X t, x τ ).

A Appendix: S-topology
The S-topology has been introduced by Jakubowski ([14], 1997) as a topology defined on the Skorohod space of càdlàg functions: ([0, T ]; IR).This topology is weaker than the Skorohod topology but tightness criteria are easier to establish.These criteria are the same as the one used in Meyer-Zheng [20].Let Let Ω, , IP, ( t ) t≥0 be a stochastic basis.If (Y ) 0≤t≤T is a process in ([0, T ]; IR) such that Y t is integrable for any t, the conditional variation of Y is defined by The process Y is called a quasimar t ingale if C V (Y ) < +∞.When Y is a t -martingale, C V (Y ) = 0.A variation of Doob's inequality (cf.lemma 3, p. 359 in Meyer and Zheng [20], where it is assumed that Y T = 0) implies that IP sup

| 2 =
0, ∀s ∈ [0, t] − D. Since Ȳ is continuous, Y is càd-lag and D is countable, then Y s = Ȳs , IP-a.s, ∀s ∈ [0, t].Moreover, we deduce that, IE [M − N a, b (z) denotes the number of up-crossing of the function z ∈ ([0, T ]; IR) from level a to level b (a < b).We recall some facts about the S-topology.Proposition A.1.(A criteria for S-tight).A sequence (Y ǫ ) ǫ>0 is said to be S-tight if and only if it is relatively compact for the S-topology.Let (Y ǫ ) ǫ>0 be a family of stochastic processes in ([0, T ]; IR).Then this family is tight for the Stopology if and only if ( Y ǫ ∞ ) ǫ>0 and (N a, b (Y ǫ )) ǫ>0 are tight for each a < b.
K|t − t n |.Hence A 2 n tends to zero in probability.Denote by(Y ′ , M ′ ) the weak limit of {(Y t n , x n , ′ r )d r − (M ′ t − M ′ s ), s ∈ [0, t] ∩ D c .The uniqueness of the considered BSDE ensures that ∀s ∈ [0, t], Y ′ s = Y t, x follows from (ii) and the second statement of Proposition 5.1.
r )d r ≤ s IP-ps.Hence Y t n , x n l aw ⇒ Y t, x .As in (i), one derive that Y