Homogenization of semi-linear PDEs with discontinuous effective coefficients
aa r X i v : . [ m a t h . P R ] A ug Homogenization of semilinear PDEs withdiscontinuous averaged coefficients
K. Bahlali A. Elouaflin E. Pardoux IMATH, UFR Sciences, USVT, B.P. 132, 83957 La Garde Cedex, France. e-mail: [email protected] UFRMI, Université de Cocody, 22 BP 582 Abidjan, Côte d’Ivoire e-mail: [email protected] LATP, CMI Université de Provence, 39 rue Joliot-Curie,13453 Marseille e-mail: [email protected]
Abstract
We study the asymptotic behavior of solutions of semilinear PDEs. Neither period-icity nor ergodicity will be assumed. On the other hand, we assume that the coefficientshave averages in the Cesaro sense. In such a case, the averaged coefficients could bediscontinuous. We use a probabilistic approach based on weak convergence of the asso-ciated backward stochastic differential equation (BSDE) in the Jakubowski S-topologyto derive the averaged PDE. However, since the averaged coefficients are discontinu-ous, the classical viscosity solution is not defined for the averaged PDE. We then usethe notion of " L p − viscosity solution" introduced in [7]. The existence of L p − viscositysolution to the averaged PDE is proved here by using BSDEs techniques. Keys words : Backward stochastic differential equations (BSDEs), L p -viscosity solution forPDEs, homogenization, Jakubowski S-topology, limit in the Cesaro sense. MSC 2000 subject classifications , 60H20, 60H30, 35K60.
Homogenization of a partial differential equation (PDE) is the process of replacing rapidlyvarying coefficients by new ones such that the solutions are close. Let for example a be aone dimensional periodic function which is positive and bounded away from zero. For ε > ,we consider the operator L ε = div ( a ( xε ) ∇ ) For small ε , L ε can be replaced by L = div ( a ∇ ) Partially supported by PHC Tassili 07MDU705 and Marie Curie ITN, no. 213841-2. Supported by AUF bourse post-doctorale 07-08, Réf.: PC-420/2460. a is the averaged (or limit, or effective) coefficient associated to a . As ε is small, thesolution 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 .The probabilistic approach to homogenization is one way to prove such results in theperiodic or ergodic case. It is based on the asymptotic analysis of the diffusion process asso-ciated to the operator L ε . 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, seefor example the monographs [3, 12, 21] and the references therein. There also exists a con-siderable literature on the study of asymptotic analysis of stochastic differential equations(SDEs) with periodic structures and its connection with homogenization of second orderpartial differential equations (PDEs). In view of the connection between BSDEs and semi-linear PDEs, this probabilistic tool has been used in order to prove homogenization resultsfor certain classes of nonlinear PDEs, see in particular [4, 5, 6, 9, 11, 13, 19, 23, 24] and thereferences therein. The two classical situations which have been mainly studied are the casesof deterministic periodic and random stationary coefficients. This paper is concerned witha different situation, building upon earlier results of Khasminskii and Krylov.In [15], Khasminskii & Krylov consider the averaging of the following family of diffusionsprocess U , εt = x ε + 1 ε Z t ϕ ( U , εs , U , εs ) dW s ,U , εt = x + Z t b (1) ( U , εs , U , εs ) ds + Z t σ (1) ( U , εs , U , εs ) d f W s , (1.1)where for each ε > small, U , εt is a one-dimensional null-recurrent fast component and U , εt is a d –dimensional slow component. The function ϕ (resp. σ (1) , resp. b (1) ) is IR-valued (resp.IR d × ( k − -valued, resp. IR d -valued). ( W, f W ) is a k -dimensional standard Brownian motionwhose component W (resp. f W ) is one dimensional (resp. ( k - )-dimensional). Define now ( X ,ε , X ,ε ) = ( εU ,ε , U ,ε ) . The process { X εt := ( X ,εt , X ,εt ) , t ≥ } solves the SDE X , εt = x + Z t ϕ (cid:18) X , εs ε , X , εs (cid:19) dW s ,X , εt = x + Z t b (1) (cid:18) X , εs ε , X , εs (cid:19) ds + Z t σ (1) (cid:18) X , εs ε , X , εs (cid:19) d f W s , (1.2)They define the averaged coefficients as limits in the Cesaro sense. With the additionalassumption that the presumed SDE limit is weakly unique, they prove that the process ( X , εt , X , εt ) converges in distribution towards a Markov diffusion ( X t , X t ) . As a byproduct,they derive the limit behavior of the linear PDE associated to ( X , εt , X , εt ) , in the case whereweak uniqueness of the limiting PDE holds in the Sobolev space W , d +1 , loc ( IR + × IR d ) of allfuncions 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 xx u belong to L d +1 loc ( IR + × IR d ) .In the present note, we extend the results of [15] to parabolic semilinear PDEs. Notethat the limiting coefficients can be discontinuous. More precisely, we consider the followingomogenization of semilinear PDEs with discontinuous coefficients 3sequence of semi-linear PDEs, indexed by ε > , ∂v ε ∂t ( t, x , x ) = ( L ε v ε )( t, x , x ) + f ( x ε , x , v ε ( t, x , x )) , t > v ε (0 , x , x ) = H ( x , x ); ( x , x ) ∈ IR × IR d . (1.3) L ε := a ( x ε , x ) ∂ ∂ x + d X i, j =1 a ij ( x ε , x ) ∂ ∂x i ∂x j + d X i =1 b (1) i ( x ε , x ) ∂∂x i , where ϕ , σ (1) and b (1) are those defined above in equation (1.1), a := 12 ϕ , a ij := 12 ( σ (1) σ (1) ∗ ) ij , i, j = 1 , ..., d, and the real valued measurable functions f and H are defined on IR d +1 × IR and IR d +1 respectively.We put b := (cid:18) b (1) (cid:19) , a ( x ) := 12 ( σσ ∗ )( x ) , with σ := (cid:18) ϕ σ (1) (cid:19) . We write B := (cid:18) W f W (cid:19) and X ε := (cid:18) X , ε X ,ε (cid:19) . The PDE (1.3) is then connected to the system of SDE – BSDE X εs = x + Z s b ( X , εr ε , X , εr ) dr + Z s σ ( X , εr ε , X , εr ) dB r ,Y εs = H ( X εt ) + Z ts f ( X , εr ε , X , εr , Y εr ) dr − Z ts Z εr dM X ε r , ∀ s ∈ [0 , t ] (1.4)where M X ε is the martingale part of the process X ε i. e. M X ε s = Z s σ ( X , εr ε , X , εr ) dB r , ≤ s ≤ t. Note that Y ε does depend upon the pair ( t, x ) where x is the initial condition of the forwardSDE 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 ε ( t, x ) := Y ε , t ≥ , x = ( x , x ) ∈ IR d +1 } solves the PDE (1.3).The aim of the present paper is1. to show that for each t > , x ∈ IR d +1 , the sequence of processes ( X εs , Y εs , R ts Z εr dM X ε r ) ≤ s ≤ t converges in law to the process ( X s , Y s , R ts Z r dM Xr ) ≤ s ≤ t which is the unique solutionto the system of SDE – BSDE X s = x + Z s ¯ b ( X r ) dr + Z s ¯ σ ( X r ) dB r , ≤ s ≤ t.Y s = H ( X t ) + Z ts ¯ f ( X r , Y r ) dr − Z ts Z r dM Xr , ≤ s ≤ t, (1.5)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;omogenization of semilinear PDEs with discontinuous coefficients 42. deduce from the first result that for each ( t, x ) , v ε ( t, x , x ) −→ v ( t, x , x ) , where v solves the following averaged PDE in the L p -viscosity sense ∂v∂t ( t, x , x ) = ( ¯ Lv )( t, x , x ) + ¯ f ( x , x , v ( t, x , x )) t > ,v (0 , x , x ) = H ( x , x ) , (1.6)with ¯ L = X i, j ¯ a ij ( x , x ) ∂ ∂x i ∂x j + X i ¯ b i ( x , x ) ∂∂x i 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 limitingsystem of SDE – BSDE (1.5) has a unique solution. However, due to the discontinuity of thecoefficients, the classical viscosity solution is not defined for the averaged PDE (1.6). Wethen use the notion of " L p − viscosity solution". We use BSDE techniques to establish theexistence of L p − viscosity solution for the averaged PDE. The notion of L p -viscosity solutionhas been introduced by Caffarelli et al. in [7] to study fully nonlinear PDEs with measurablecoefficients. Note however that although the notion of a L p -viscosity solution is availablefor PDEs with merely measurable coefficients, continuity of the solution is required. In oursituation, the lack of L -continuity property for the flow X x := ( X , x , X , x ) transfers thedifficulty to the backward one and hence we cannot prove the L -continuity of the process Y . To overcome this difficulty, we establish weak continuity for the flow x ( X , x , X , x ) and using the fact that Y x is deterministic, we derive the continuity property for Y x .The paper is organized as follows: In section , we make precise some notations andformulate our assumptions. Our main results are stated in section . Section and aredevoted to the proofs. For a given function g ( x , x ) , we define g + ( x ) := lim x → + ∞ x Z x g ( t, x ) dtg − ( x ) := lim x →−∞ x Z x g ( t, x ) dt The average, in Cesaro sense, of g is defined by g ± ( x , x ) := g + ( x )1 { x > } + g − ( x )1 { x ≤ } Let ρ ( x , x ) := a ( x , x ) − (= [ ϕ ( x , x )] − ) and denote by ¯ b ( x , x ) , ¯ a ( x , x ) andomogenization of semilinear PDEs with discontinuous coefficients 5 ¯ f ( x , x , y ) , the averaged coefficients defined by ¯ b i ( x , x ) = ( ρb i ) ± ( x , x ) ρ ± ( x , x ) , i = 1 , ..., d ¯ a ij ( x , x ) = ( ρa ij ) ± ( x , x ) ρ ± ( x , x ) , i, j = 0 , , ..., d ¯ f ( x , x , y ) = ( ρf ) ± ( x , x , y ) ρ ± ( x , x ) . ¯ σ ( x , x ) = (¯ a ( x , x )) where ¯ a ( x , x ) denotes the matrix (¯ a ij ( x , x )) i,j .It is worth noting that ¯ b, ¯ a and ¯ f may be discontinuous at x = 0 . We consider the following conditions. (A1)
The functions b (1) , σ (1) , ϕ are uniformly Lipschitz in the variables ( x , x ) . (A2) For each x , the first and second order derivatives with respect to x of these functionsare bounded continuous functions of x . (A3) a (1) := ( σ (1) σ (1) ∗ ) is uniformly elliptic, i. e. ∃ Λ > ∀ x, ξ ∈ IR d , ξ ∗ a (1) ( x ) ξ ≥ Λ | ξ | . Moreover, there exist positive constants C , C , C such that ( ( i ) C ≤ a ( x , x ) ≤ C ( ii ) | a (1) ( x , x ) | + | b ( x , x ) | ≤ C (1 + | x | ) . (B1) Let D x ρ and D x ρ denote respectively the gradient vector and the matrix of secondderivatives of ρ with respect to x . We assume that uniformly with respect to x x Z x ρ ( t, x ) dt −→ ρ ± ( x ) as x → ±∞ , x Z x D x ρ ( t, x ) dt −→ D x ρ ± ( x ) as x → ±∞ , x Z x D x ρ ( t, x ) dt −→ D x ρ ± ( x ) as x → ±∞ . (B2) For every i and j , the coefficients ρb i , D x ( ρb i ) , D x ( ρb i ) , ρa ij , D x ( ρa ij ) ,D x ( ρa ij ) have averages in the Cesaro sense. (B3) For every function k ∈ { ρb i , D x ( ρb i ) , D x ( ρb i ) , ρa ij , D x ( ρa ij ) , D x ( ρa ij ) } , thereexists a bounded function α : IR d +1 → IR such that x Z x k ( t, x ) dt − k ± ( x , x ) = (1 + | x | ) α ( x , x ) , lim | x |−→∞ sup x ∈ IR d | α ( x , x ) | = 0 . (2.1)omogenization of semilinear PDEs with discontinuous coefficients 6 (C1) (i) The coefficient f is uniformly Lipschitz in ( x , x , y ) and, for each x ∈ IR, its derivativesin ( x , y ) up to and including second order derivatives are bounded continuous functionsof ( x , y ) .(ii) There exists positive constant K such thatfor every ( x , x , y ) , | f ( x , x , y ) | ≤ K (1 + | x | + | y | ) . (iii) H is continuous and bounded. (C2) ρf has a limit in the Cesaro sense and there exists a bounded measurable function β : IR d +2 → IR such that x Z x ρ ( t, x ) f ( t, x , y ) dt − ( ρf ) ± ( x , x , y ) = (1 + | x | + | y | ) β ( x , x , y )lim | x |→∞ sup ( x , y ) ∈ IR d × IR | β ( x , x , y ) | = 0 , (2.2) (C3) For each x , ρf has derivatives up to second order in ( x , y ) and these derivatives arebounded and satisfy (C2).Throughout the paper, (A) stands for conditions (A1), (A2), (A3); (B) for conditions (B1),(B2), (B3) and (C) for (C1), (C2), (C3). Consider the equation X xt = x + Z t ¯ b ( X xs ) ds + Z t ¯ σ ( X xs ) dB s , t ≥ . (3.1)Assume that (A), (B) hold. Then, from Khasminskii & Krylov [15] and Krylov [18], wededuce that for each fixed, x ∈ IR d +1 the process X ε := ( X , ε , X , ε ) converges in distributionto the process X := ( X , X ) which is the unique weak solution to SDE (3.1).We now define the notion of L p -viscosity solution of a parabolic PDE. This notion hasbeen 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 ¯ L := X i, j ¯ a ij ( x ) ∂ ∂x i ∂x j + X i ¯ b i ( x ) ∂∂x i denote the second order PDE operator associated to the SDE (3.1).We consider the parabolic equation ∂v∂t ( t, x ) = ( ¯ Lv )( t, x ) + g ( x, v ( t, x )) , t ≥ v (0 , x ) = H ( x ) . (3.2)omogenization of semilinear PDEs with discontinuous coefficients 7 Definition 3.1.
Let p be an integer such that p > d + 2 .(a) A function v ∈ C (cid:0) [0 , T ] × IR d +1 , IR (cid:1) is a L p -viscosity sub-solution of the PDE (3.2),if for every x ∈ IR d +1 , v (0 , x ) ≤ H ( x ) and for every ϕ ∈ W , p, loc (cid:0) IR + × IR d +1 , IR (cid:1) and ( b t, b x ) ∈ (0 , T ] × IR d +1 at which v − ϕ has a local maximum, one hasess lim inf ( t, x ) → ( b t, b x ) (cid:26) ∂ϕ∂t ( t, x ) − ( ¯ Lϕ )( t, x ) − g ( x, v ( t, x )) (cid:27) ≤ . (b) A function v ∈ C (cid:0) [0 , T ] × IR d +1 , IR (cid:1) is a L p -viscosity super-solution of the PDE(3.2), if for every x ∈ IR d +1 , v (0 , x ) ≥ H ( x ) and for every ϕ ∈ W , p, loc (cid:0) IR + × IR d +1 , IR (cid:1) and ( b t, b x ) ∈ (0 , T ] × IR d +1 at which v − ϕ has a local minimum, one hasess lim sup ( t, x ) → ( b t, b x ) (cid:26) ∂ϕ∂t ( t, x ) − ( ¯ Lϕ )( t, x ) − g ( x, v ( t, x )) (cid:27) ≥ . Here, G ( t, x, ϕ ( s, x )) is merely assumed to be measurable upon the variable x =: ( x , x ) .(c) A function v ∈ C (cid:0) [0 , T ] × IR d +1 , IR (cid:1) is a L p -viscosity solution if it is both a L p -viscosity sub-solution and super-solution. Remark . Condition (a) means that for every ε > , r > , there exists a set A ⊂ B r ( b t, b x ) of positive measure such that, for every ( s, x ) ∈ A , ∂ϕ∂s ( s, x ) − ( ¯ Lϕ )( t, x ) − g ( x, v ( t, x )) ≤ ε. 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 existsa process ( X s , Y s , Z s ) ≤ s ≤ t such that,(i) the sequence of process X ε converges in law to the continuous process X, which is theunique weak solution to SDE (1.5), in C ([0 , t ]; IR d +1 ) equipped with the uniform topology.(ii) the sequence of processes ( Y εs , R ts Z εr dM X ε r ) ≤ s ≤ t converges in law to the process ( Y s , R ts Z r dM Xr ) ≤ s ≤ t in D ([0 , t ]; IR ) , where M X is the martingale part of X , equipped withthe S –topology.(iii) (Y,Z) is the unique solution to BSDE (1.5) such that,(a) (Y,Z) is F X − adapted and ( Y s , R ts Z r dM Xr ) ≤ s ≤ t is continuous.(b) IE (cid:0) sup ≤ s ≤ t | Y s | + R t | Z r σ ( X r ) | dr (cid:1) < ∞ The uniqueness means that, if ( Y , Z ) and ( Y , Z ) are two solutions of BSDE (1.5) sat-isfying (iii) (a)-(b) then, IE (cid:16) sup ≤ s ≤ t | Y s − Y s | + R t | Z r σ ( X r ) − Z r σ ( X r ) | dr (cid:17) = 0 , i. e.since σσ ∗ is elliptic (see (A3) ), Y s = Y s ∀ ≤ s ≤ t , IP a. s., and Z s = Z s ds × d IP a. e.
Theorem 3.4.
Assume (A), (B), (C) hold. For ε > , let v ε be the unique solution to theproblem (1.3). Let ( Y ( t,x ) s ) s be the unique solution of the BSDE (1.5). Then(i) Equation (1.6) has a unique L p -viscosity solution v such that v ( t, x ) = Y ( t,x )0 .(ii) For every ( t, x ) ∈ IR + × IR d +1 , v ε ( t, x ) → v ( t, x ) , as ε → . omogenization of semilinear PDEs with discontinuous coefficients 8 In all of this section, ( t, x ) ∈ IR + × IR d +1 is arbitrarily fixed with t > .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) whichsays that the coefficients of the forward SDE part of (1.4) are bounded with respect to theirfirst variable, and grow at most linearly in their second variable) Lemma 4.1.
For all p ≥ , there exists constant C p such that for all ε > ,IE (cid:18) sup ≤ s ≤ t [ | X ,εs | p + | X ,εs | p ] (cid:19) ≤ C p . Proposition 4.2.
There exists a positive constant C such that for all ε > IE (cid:18) sup ≤ s ≤ t | Y εs | + Z t | Z εr σ ( X εr ) | dr (cid:19) ≤ C. Proof.
We deduce from Itô’s formula (here and below ¯ X , εr = X , εr /ε ) | Y εs | + Z t | Z εr σ ( X εr ) | dr ) ≤ | H ( X εt ) | + K Z ts | Y εr | dr + Z ts | f ( ¯ X , εr , X , εr , | dr − Z ts h Y εr , Z εr dM X ε s i . It follows from well known results on BSDEs that we can take the expectation in the aboveidentity (see e. g. [22]; note that introducing stopping times as usual and using Fatou’sLemma would yield (4.1) below). We then deduce from Gronwall’s lemma that there existsa positive constant C which does not depend on ε , such that for every s ∈ [0 , t ] ,IE (cid:0) | Y εs | (cid:1) ≤ C IE (cid:18) | H ( X εt ) | + Z t | f ( ¯ X , εr , X , εr , | dr (cid:19) and IE (cid:18)Z t | Z εr σ ( X εr ) | dr (cid:19) ≤ C IE (cid:18) | H ( X εt ) | + Z t | f ( ¯ X , εr , X , εr , | dr (cid:19) . (4.1)Combining the last two estimates and the Burkhölder-Davis-Gundy inequality, we getIE (cid:18) sup ≤ s ≤ t | Y εs | + 12 Z t | Z εr σ ( X εr ) | dr (cid:19) ≤ C IE (cid:18) | H ( X εt ) | + Z t | f ( ¯ X , εr , X , εr , | dr (cid:19) In view of condition ( C and Lemma 4.1, the proof is complete.omogenization of semilinear PDEs with discontinuous coefficients 9We deduce immediately from Proposition 4.2 Corollary 4.3. sup ε> | Y ε | < ∞ . Proposition 4.4.
For ε > , let Y ε be the process defined by equation (1.4) and M ε be itsmartingale part. The sequence ( Y ε , M ε ) ε> is tight in the space D ([0 , t ] , IR ) × D ([0 , t ] , IR ) endowed with the S -topology. Proof.
Since M ε is a martingale, then by [20] or [14], the Meyer-Zheng tightness criteria isfulfilled whenever sup ε (cid:18) CV ( Y ε ) + IE (cid:18) sup ≤ s ≤ t | Y εs | + | M εs | (cid:19)(cid:19) < + ∞ . (4.2)where the conditional variation CV is defined in appendix A.>From [25], the conditional variation CV ( Y ε ) satisfies CV ( Y ε ) ≤ IE (cid:18)Z t | f ( ¯ X , εs , X , εs , Y εs ) | ds (cid:19) , Now clearly (4.2) follows from ( C , Lemma 4.1 and Proposition 4.2. Proposition 4.5.
There exists ( Y, M ) and a countable subset D of [0 , t ] such that along asubsequence ε n → ,(i) ( Y ε n , M ε n ) = ⇒ ( Y, M ) on D ([0 , t ] , IR ) × D ([0 , t ] , IR ) endowed with the S –topology.(ii) The finite dimensional distributions of ( Y sε n , M ε n s ) s ∈ D c converge to those of ( Y s , M s ) s ∈ D c .(iii) ( X ,ε n , X ,ε n , Y ε n ) = ⇒ ( X , X , Y ) , in the sense of weak convergence in C ([0 , t ] , IR d +1 ) × D ([0 , t ] , IR ) , equipped with the product of the uniform convergence and the S topology. Proof. (i) From Proposition 4.4, the family ( Y ε , M ε ) ε is tight in D ([0 , t ] , IR ) ×D ([0 , t ] , IR ) endowed with the S -topology. Hence along a subsequence (still denoted by ε ), ( Y ε , M ε ) ε converges in law on D ([0 , t ] , IR ) × D ([0 , t ] , IR ) towards a càd-làg process ( Y, M ) .(ii) follows from Theorem 3.1 in Jakubowski [14].(iii) According to Theorem 3.3 ( i ) , ( X ,ε , X ,ε ) = ⇒ ( X , X ) in C ([0 , t ] , IR d +1 ) equipped withthe uniform topology. From assertion (i), ( Y ε · ) ε> is tight in D ([0 , t ] , IR ) equipped with the S –topology. Hence the subsequence ε n can be chosen in such a way that (iii) holds. Proposition 4.6.
Let ( Y, M ) be any limit process as in Proposition 4.5. Then ( i ) for every s ∈ [0 , t ] \ D , Y s = H ( X t ) + Z ts ¯ f ( X r , X r , Y ) dr − ( M t − M s ) , IE (cid:0) sup ≤ s ≤ t (cid:2) | Y s | + | X s | + | X s | (cid:3)(cid:1) ≤ C ; (4.3) ( ii ) M is a F s -martingale, where F s := σ { X r , Y r , ≤ r ≤ s } augmented with the IP-nullsets. omogenization of semilinear PDEs with discontinuous coefficients 10To prove this proposition, we need the following lemmas. Lemma 4.7.
Assume (A), (B) , (C2) and (C3). For x ∈ IR d , y ∈ IR, let V ε ( x , x , y ) denote the solution of the following equation: a ( x ε , x ) D x V ε ( x , x , y ) = f ( x ε , x , y ) − ¯ f ( x , x , y ) , x ∈ IR ,V ε (0 , x , y ) = D x V ε (0 , x , y ) = 0 . (4.4) Then, for some bounded functions β and β satisfying (2.2),(i) D x V ε ( x , x , y ) = x (1 + | x | + | y | ) β ( x ε , x , y ) , and the same is true with D x V ε replaced by D x D x V ε and D x D y V ε ;(ii) V ε ( x , x , y ) = x (1 + | x | + | y | ) β ( x ε , x , y ) ,and the same is true with V ε replaced by D x V ε , D y V ε , D x V ε , D y V ε and D x D y V ε . Proof.
We will adapt the idea of [15] to our situation. For ε > and ( z, x , y ) ∈ IR d +2 weset F ε ( z, x , y ) := 1 εz Z εz ρ ( tε , x ) g ( tε , x , y ) dt where g ( z, x , y ) := f ( z, x , y ) − ¯ f ( εz, x , y ) .We only treat the case where x > . The same argument can be used in the case x < .We successively use the definition of ¯ f and assumptions (C2) , to obtain F ε ( x ε , x , y ) = 1 x Z x ρ ( tε , x ) f ( tε , x , y ) dt − ( ρf ) + ( x , y )+ ( ρf ) + ( x , y ) − ( ρf ) + ( x , y ) ρ + ( x ) 1 x Z x ρ ( tε , x ) dt = (1 + | x | + | y | ) β ( x ε , x , y )+ ( ρf ) + ( x , y ) ρ + ( x ) (cid:2) ρ + ( x ) − x Z x ρ ( tε , x ) dt (cid:3) = (1 + | x | + | y | ) β ( x ε , x , y )+ (1 + | x | + | y | ) α ( x ε , x , y ) where α ( x ε , x , y ) := ( ρf ) + ( x , y )(1+ | x | + | y | ) ρ + ( x ) (cid:2) ρ + ( x ) − x R x ρ ( tε , x ) dt (cid:3) .Using assumptions (B1) and (C1-ii) , one can show that α is a bounded function which sat-isfies (2.2). Since D x V ε ( x , x , y ) = x F ε ( x ε , x , y ) , we derive the result for D x V ε ( x , x , y ) .Further, by integrating it, we get V ε ( x , x , y ) = x (1 + | x | + | y | ) (cid:0) ( εx ) Z x ε tβ ( t, x , y ) dt (cid:1) , where β = α + β .Clearly, β ( x ε , x , y ) := ( εx ) R x ε tβ ( t, x , y ) dt is bounded function which satisfies (2.2).The result for the other quantities can be deduced by similar arguments from assumptions (B1) , (C1) , (C2) and (C3) .omogenization of semilinear PDEs with discontinuous coefficients 11 Lemma 4.8. As ε −→ , sup ≤ s ≤ t (cid:12)(cid:12)(cid:12)(cid:12) Z s (cid:18) f ( X , εr ε , X , εr , Y εr ) − ¯ f ( X , εr , X , εr , Y εr ) (cid:19) dr (cid:12)(cid:12)(cid:12)(cid:12) → in probability . Proof.
We shall show that for every s ∈ [0 , t ] , (cid:12)(cid:12)(cid:12)R s (cid:2) f ( X , εr ε , X , εr , Y εr ) − ¯ f ( X , εr , X , εr , Y εr ) (cid:3) dr (cid:12)(cid:12)(cid:12) tends to zero in probability as ε tends to zero. Let V ε denote the solution of equation (4.4).Note that V ε has first and second derivatives in ( x , x , y ) which are possibly discontinu-ous only at x = 0 . Then, as in [15], since ϕ is bounded away from zero, we can use theItô-Krylov formula to get V ε ( X , εs , X , εs , Y εs ) = V ε ( x , x , Y ε ) + Z s (cid:2) f ( X , εr ε , X , εr , Y εr ) − ¯ f ( X , εr , X , εr , Y εr ) (cid:3) dr + Z s T race (cid:2) a (1) ( X , εr , X , εr ) D x V ε ( X , εr , X , εr , Y εr ) (cid:3) dr + Z s [ D x V ε ( X , εr , X , εr , Y εr ) b (1) ( X , εr , X , εr ) − D y V ε ( X ,εr , X ,εr , Y εr ) f ( X ,εr ε , X ,εr , Y εr )] dr + Z s [ D x V ε ( X , εr , X , εr , Y εr ) σ ( X , εr , X , εr ) + D y V ε ( X ,εr , X ,εr , Y εr ) Z εr σ ( X ,εr ε , X ,εr )] dB r + 12 Z s D y V ε ( X ,εr , X ,εr , Y εr ) Z εr σσ ∗ ( X ,εr ε , X ,εr )( Z εr ) ∗ dr + 12 Z s D x D y V ε ( X ,εr , X ,εr , Y εr ) σσ ∗ ( X ,εr ε , X ,εr )( Z εr ) ∗ dr (4.5)In view of Lemma 4.7 and Corollary 4.3, V ε ( x , x , Y ε ) tends to zero as ε → .Using the fact taht {| X , εs | < √ ε } + 1 {| X , εs |≥√ ε } and Lemma 4.7, we obtain (cid:12)(cid:12) V ε ( X , εs , X , εs , Y εs ) (cid:12)(cid:12) ≤ ε (cid:20) (1 + | X , εs | + | Y εs | ) | β ( X , εs ε , X , εs , Y εs ) | (cid:21) + 1 {| X , εs |≥√ ε } | X , εs | (cid:20) (1 + | X , εs | + | Y εs | ) | β ( X , εs ε , X , εs , Y εs ) | (cid:21) From Lemma 4.1 and Proposition 4.2, we deduce thatIE (cid:18) sup ≤ s ≤ t | V ε ( X , εs , X , εs , Y εs ) | (cid:19) ≤ K ε + sup | x |≥√ ε sup ( x , y ) | β ( x ε , x , y ) | ! Then, since β satisfy respectively (2.2), the right hand side of the previous inequality tendsomogenization of semilinear PDEs with discontinuous coefficients 12to zero as ε −→ . Similarly, one can show that Z s T race (cid:2) a (1) ( X , εr , X , εr ) D x V ε ( X , εr , X , εr , Y εr ) (cid:3) dr + Z s [ D x V ε ( X , εr , X , εr , Y εr ) b (1) ( X , εr , X , εr ) − D y V ε ( X ,εr , X ,εr , Y εr ) f ( X ,εr ε , X ,εr , Y εr )] dr + Z s [ D x V ε ( X , εr , X , εr , Y εr ) σ ( X , εr , X , εr ) + D y V ε ( X ,εr , X ,εr , Y εr ) Z εr σ ( X ,εr ε , X ,εr )] dB r + 12 Z s D y V ε ( X ,εr , X ,εr , Y εr ) Z εr σσ ∗ ( X ,εr ε , X ,εr )( Z εr ) ∗ dr + 12 Z s D x D y V ε ( X ,εr , X ,εr , Y εr ) σσ ∗ ( X ,εr ε , X ,εr )( Z εr ) ∗ dr converges to zero in probability. Let us give an explanation concerning the one but last term,which is the most delicate one. (cid:12)(cid:12)(cid:12)(cid:12)Z s D y V ε ( X ,εr , X ,εr , Y εr ) Z εr σσ ∗ ( X ,εr ε , X ,εr )( Z εr ) ∗ dr (cid:12)(cid:12)(cid:12)(cid:12) ≤ C sup ≤ r ≤ s (cid:12)(cid:12) D y V ε ( X ,εr , X ,εr , Y εr ) (cid:12)(cid:12) Trace Z s Z εr σσ ∗ ( X ,εr ε , X ,εr )( Z εr ) ∗ dr Since { Trace R s Z εr σσ ∗ ( X ,εr ε , X ,εr )( Z εr ) ∗ dr, ≤ s ≤ t } is the increasing process associatedto a martingale which is uniformly L p ( IP ) − integrable for each p ∈ IN, its L p ( IP ) norm isbounded, for all p ≥ . Finally the same argument as above shows that sup ≤ r ≤ s (cid:12)(cid:12) D y V ε ( X ,εr , X ,εr , Y εr ) (cid:12)(cid:12) → in probability, as ε → . Lemma 4.9. Z . ¯ f ( X , εr , X , εr , Y εr ) dr law = ⇒ Z . ¯ f ( X r , X r , Y r ) dr on C ([0 , t ] , IR ) as ε −→ . For the proof of this Lemma, we need the following two results.
Lemma 4.10.
Let X s := x + Z s ¯ ϕ ( X r , X r ) dW r , ≤ s ≤ t , and, assume (A2-i), (B1).For ε > , let D εn := (cid:26) s : s ∈ [0 , t ] / | X ,εs | ≤ n (cid:27) .Define also D n := (cid:26) s : s ∈ [0 , t ] / | X s | ≤ n (cid:27) .Then, there exists a constant c > such that for each n ≥ , ε > ,IE | D εn | ≤ cn and IE | D n | ≤ cn , where | . | denotes the Lebesgue measure. Proof.
Consider the sequence (Ψ n ) of functions defined as follows, Ψ n ( x ) = − xn − n if x ≤ − nx if − n ≤ x ≤ nxn − n if x ≥ /n omogenization of semilinear PDEs with discontinuous coefficients 13We put, ¯ ϕ := ¯ a := ρ ( x , x ) − .Using Itô’s formula, we get Ψ n ( X s ) = Ψ n ( X ) + Z s Ψ ′ n ( X s ) ¯ ϕ ( X s , X s ) dW s + 12 Z s Ψ ” n ( X s ) ¯ ϕ ( X s , X s ) ds, s ∈ [0 , t ] Since ¯ ϕ is lower bounded by C , taking the expectation, we get C IE Z t [ − n , n ] ( X s ) ds ≤ IE Z t Ψ ” n ( X s ) ¯ ϕ ( X s , X s ) ds = 2 IE (cid:2) Ψ n ( X t ) − Ψ n ( x ) (cid:3) It follows that IE ( | D n | ) ≤ C − IE [Ψ n ( X t ) − Ψ n ( x )] ≤ c/n . The same argument, applies to D εn , allows us to show the first estimate. Lemma 4.11.
Consider a collection { Z ε , ε > } of real valued random variables, and a realvalued random variable Z . Assume that for each n ≥ , we have the decompositions Z ε = Z ,εn + Z ,εn Z = Z n + Z n , such that for each fixed n ≥ , Z ,εn ⇒ Z n IE | Z ,εn | ≤ c √ n IE | Z n | ≤ c √ n . Then Z ε ⇒ Z , as ε → . Proof.
The above assumptions imply that the collection of random variables { Z ε , ε > } is tight. Hence the result will follow from the fact thatIE Φ( Z ε ) → IE Φ( Z ) , as ε → for all Φ ∈ C b ( IR ) which is uniformly Lipschitz. Let Φ be such a function, and denote by K its Lipschitz constant. Then | IE Φ( Z ε ) − IE Φ( Z ) | ≤ IE | Φ( Z ε ) − Φ( Z ,εn ) | + + | IE Φ( Z ,εn ) − IE Φ( Z n ) | + IE | Φ( Z n ) − Φ( Z ) |≤ | IE Φ( Z ,εn ) − IE Φ( Z n ) | + 2 K c √ n . Hence lim sup ε → | IE Φ( Z ε ) − IE Φ( Z ) | ≤ K c √ n , for all n ≥ . The result follows.omogenization of semilinear PDEs with discontinuous coefficients 14 Proof of Lemma 4.9.
For each n ≥ , define a function θ n ∈ C ( IR , [0 , such that θ n ( x ) = 0 for | x | ≤ n , and θ n ( x ) = 1 for | x | ≥ n . We have Z t ¯ f ( X ,εs , X ,εs , Y εs ) ds = Z t ¯ f ( X ,εs , X ,εs , Y εs ) θ n ( X ,εs ) ds + Z t ¯ f ( X ,εs , X ,εs , Y εs )[1 − θ n ( X ,εs )] ds = Z ,εn + Z ,εn Z t ¯ f ( X s , X s , Y s ) ds = Z t ¯ f ( X s , X s , Y s ) θ n ( X s ) ds + Z t ¯ f ( X s , X s , Y s )[1 − θ n ( X s )] ds = Z n + Z n Note that the mapping ( x , x , y ) Z t ¯ f ( x s , x s , y s ) θ n ( x s ) ds is continuous from C ([0 , t ]) × D ([0 , t ]) equipped with the product of the sup–norm and the S topologies into IR. Hence from Proposition 4.5, Z ,εn = ⇒ Z n as ε → , for each fixed n ≥ .Moreover, from Lemma 4.10, the linear growth property of ¯ f , Lemma 4.1 and Proposition4.2, we deduce that E | Z ,εn | ≤ c √ n , E | Z n | ≤ c √ n . Lemma 4.9 now follows from Lemma 4.11. (cid:4)
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]. 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 ( ¯ Y s , ¯ Z s , ≤ s ≤ t ) be the unique solution to BSDE (1.5). Then,for every s ∈ [0 , t ] ,IE | Y s − ¯ Y s | + IE (cid:18) [ M − Z . ¯ Z r dM Xr ] t − [ M − Z . ¯ Z r dM Xr ] s (cid:19) = 0 . Proof.
For every s ∈ [0 , t ] \ D , we have Y s = H ( X t ) + R ts ¯ f ( X r , Y r ) dr − ( M t − M s )¯ Y s = H ( X t ) + R ts ¯ f ( X r , ¯ Y r ) dr − R ts ¯ Z r dM Xr Arguing as in [24], we show that ¯ M := R .s ¯ Z r dM Xr is a F s -martingale.Since ¯ f satisfies condition ( C , we get by Itô’s formula, thatIE | Y s − ¯ Y s | + IE (cid:18) [ M − Z . ¯ Z r dM Xr ] t − [ M − Z . ¯ Z r dM Xr ] s (cid:19) ≤ C IE Z ts | Y r − ¯ Y r | dr. Therefore, Gronwall’s lemma yields that IE | Y s − ¯ Y s | = 0 , ∀ s ∈ [0 , t ] − D .Since ¯ Y is continuous, Y is càd-lag and D is countable, then Y s = ¯ Y s , IP- a.s, ∀ s ∈ [0 , t ] .Moreover, we deduce that, IE (cid:18) [ M − Z . ¯ Z r dM Xr ] t − [ M − Z . ¯ Z r dM Xr ] s (cid:19) = 0 . omogenization of semilinear PDEs with discontinuous coefficients 15As a consequence of Proposition 4.12, we have Corollary 4.13. (cid:18) Y ε , Z . Z εr dM X ε r (cid:19) law = ⇒ (cid:18) ¯ Y , Z · ¯ Z r dM Xr (cid:19) . Theorem 3.3 is proved.
Since the SDE (3.1) is weakly unique ([18]), the martingale problem associated to X =( X , X ) is well posed. We then have the following: Proposition 5.1. ( i ) For any t > , x ∈ IR d , the BSDE Y t, xs = H ( X xt ) + Z ts ¯ f ( X xr , Y t, xr ) dr − Z ts Z t, xr dM X x r , ≤ s ≤ t. admits a unique solution ( Y t, xs , Z t, xs ) ≤ s ≤ t such that the component ( Y t, xs ) ≤ s ≤ t is boundedand Y t, x is deterministic. ( ii ) If moreover, the deterministic function, ( t, x ) ∈ [0 , T ] × IR d +1 v ( t, x ) := Y t, x belongs to C (cid:0) [0 , T ] × IR d +1 , IR (cid:1) , then it is a L p -viscosity solution of the PDE (3.2). Remark.
The continuity of the map ( t, x ) v ( t, x ) := Y t, x , 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 existenceand uniqueness for the BSDE Y t, xs = H ( X xt ) + Z ts ¯ f ( X xr , Y t, xr ) dr − Z ts Z t, xr dB r , ≤ s ≤ t. Since f satisfies (C) and ρ is bounded, one can easily verify that ¯ f is uniformly Lipschitzin y uniformly with respect to ( x , x ) and satisfies ( C1 ) - ( ii ) . Existence and uniqueness ofsolution follow then from standard results for BSDEs, see e. g. [22]. Moreover, since H isuniformly bounded and ¯ f satisfies the linear growth condition ( C1 ) - ( ii ) , one can prove thatthe solution Y t, x is bounded, see e. g. [1]. Finally, since ( Y t, xs ) is F Xs − adapted then Y t, x ismeasurable with respect to a trivial σ − algebra and hence it is deterministic. ( ii ) Assume that the function v ( t, x ) := Y t,x belongs to C (cid:0) [0 , T ] × IR d +1 , IR (cid:1) . We onlyprove that v is a L p –viscosity sub–solution. The proof of the super–solution property canbe 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. ∂u∂t ( t, x ) = ( ¯ Lu )( t, x ) + ¯ f ( x, u ( t, x )) t ∈ [0 , T ] ,u ( T, x ) = H ( x ) . (5.1)Working with this backward PDE will simplify the details of the proofs below.Let X t,xs be the unique weak solution to SDE (3.1). We will establish that the solution Y of the Markovian BSDE Y t, xs = H ( X t,xT ) + Z Ts ¯ f ( X t,xr , Y t, xr ) dr − Z Ts Z t, xr dM X t,x r , ≤ t ≤ s ≤ T. (5.2)omogenization of semilinear PDEs with discontinuous coefficients 16define a L p − viscosity sub–solution to the problem (5.1) by puting u ( t, x ) := Y t,xt .Let ϕ ∈ W , p, loc (cid:0) [0 , T ] × IR d +1 , IR (cid:1) , let ( b t, b x ) ∈ [0 , T ] × IR d +1 be a point which is a localmaximum of u − ϕ . Since p > d + 2 , then ϕ has a continuous version which we consider fromnow on. We assume without loss of generality that v ( b t, b x ) = ϕ ( b t, b x ) (5.3)We will argue by contradiction. Assume that there exists ε, α > such that ∂ϕ∂s ( s, x ) + ¯ Lϕ ( s, x ) + ¯ f ( x, u ( s, x )) < − ε, λ – a.e. in B α ( b t, b x ) . (5.4)where λ denote the Lebesgue measure.Since ( b t, b x ) is a local maximum of u − ϕ , we can find a positive number α ′ (which we cansuppose equal to α ) such that u ( t, x ) ≤ ϕ ( t, x ) in B α ( b t, b x ) (5.5)Define τ = inf n s ≥ b t, ; | X b t, b xs − b x | > α o ∧ ( b t + α ) Since X is a Markov diffusion and ¯ f is uniformly Lipschitz in y and satisfies condition ( C1 ) - ( ii ) , then arguing as in [10], one can show that for every r ∈ [ b t, b t + α ] , Y b t, b xr = u ( r, X b t, b xr ) .Hence, the process ( ¯ Y s , ¯ Z s ) := (( Y b t, b xs ∧ τ ) , [0 , τ ] ( s )( Z b t, b xs )) s ∈ [ b t, b t + α ] solves the BSDE ¯ Y s = u ( τ, X b t, b xτ ) + Z b t + αs [0 , τ ] ¯ f ( r, X b t, b xr , u ( r, X b t, b xr )) dr − Z b t + αs ¯ Z r dM X b t, b x r , s ∈ [ b t, b t + α ] . On other hand, by Itô-Krylov formula, the process ( b Y s , b Z s ) s ∈ [ b t, b t + α ] , defined by ( b Y s , b Z s ) := (cid:16) ϕ ( s ∧ τ, X b t, b xs ∧ τ ) , [0 , τ ] ( s ) ∇ ϕ ( s, X b t, b xs ) (cid:17) solves the BSDE b Y s = ϕ ( τ, X b t, b xτ ) − Z b t + αs [0 , τ ] [( ∂ϕ∂r + ¯ Lϕ )( r, X b t, b xr )] dr − Z b t + αs b Z r dM X b t, b x r . From the choice of τ , ( τ, X b t, b xτ ) ∈ B α ( b t, b x ) . Therefore, u ( τ, X b t, b xτ ) ≤ ϕ ( τ, X b t, b xτ ) .Let A := { ( t, x ) ∈ B α ( b t, b x ) , [ ∂ϕ∂s + ¯ Lϕ + ¯ f ( ., u ( . ))]( t, x ) < − ε } and ¯ A := B α ( b t, b x ) \ A thecomplement of A . By (5.4), λ ( ¯ A ) = .Since the diffusion { X ˆ t, ˆ xs , s ≥ t } is nondegenerate, Krylov’s inequality ([17], Ch. 2, Sec. 2 &
3) implies that ¯ A ( r, X b t, b xr ) = 0 dr × d IP − a.e. It follows thatIE Z b t + α b t − [0 , τ ] [( ∂ϕ∂r + ¯ Lϕ )( r, X b t, b xr ) + ¯ f ( r, X b t, b xr , u ( r, X b t, b xr ))]) dr ≥ IE ( τ − b t ) ε > (5.6)omogenization of semilinear PDEs with discontinuous coefficients 17This implies that [ − [0 , τ ] [( ∂ϕ∂r + ¯ Lϕ )( r, X b t, b xr ) + ¯ f ( r, X b t, b xr , u ( r, X b t, b xr ))])] > on a set of dt × d IP positive measure. Therefore, the strict comparison theorem (Remark 2.5 in [23])shows that ¯ Y b t < b Y b t , that is u ( b t, b x ) < ϕ ( b t, b x ) , which contradicts our assumption (5.3).Under assumptions (A), (B) , the SDE (3 . has a unique weak solution, see [18]. Wethen have the following continuity property. Proposition 5.2. (Continuity in law of the map x X x. )Assume (A), (B) . Let X xs be the unique weak solution of the SDE (3 . , and X ns := x n + Z s ¯ b ( X nr ) dr + Z s ¯ σ ( X nr ) dB r , ≤ s ≤ t Assume that x n → x = ( x , x ) ∈ IR d as n → ∞ . Then X n law = ⇒ X x . Proof.
Since ¯ b and ¯ σ satisfy (A), (B) , one can easily check that the sequence X n is tightin C ([0 , t ] × IR d +1 ) . By Prokhorov’s theorem, there exists a subsequence (denoted also by X n ) which converges weakly to a process b X . We shall show that b X is a weak solution ofSDE (3 . . • Step 1:
For every ϕ ∈ C ∞ c ( IR d ) , ∀ u ∈ [0 , t ] , ϕ ( b X r ) − Z u ¯ Lϕ ( b X v ) dv is a F b X -martingale.All we need to show is that for every ϕ ∈ C ∞ c ( IR d ) , every ≤ s ≤ u and every function Φ s of ( X x n r ) ≤ r
From step 1 , there exists a F b X -Brownian motion b B such that, b X s = x + Z s ¯ b ( b X r ) dr + Z s ¯ σ ( b X r ) d b B r , ≤ s ≤ t. Weak uniqueness of the SDE (3.1) allows us to deduce that b X = X x in law sense.omogenization of semilinear PDEs with discontinuous coefficients 18 Proposition 5.3.
Assume (A), (B), (C) . Then, ( i ) lim ε → Y ε = Y t,x . ( ii ) The map ( t, x ) Y t, x is continuous. ( iii ) For p > d + 2 , the function v ( t, x ) := Y t, x is a L p -viscosity solution to the PDE (1.6). Proof. ( i ) Let Y t,x be the limit process defined in Proposition 4.5. We have Y ε = H ( X εt ) + Z t f ( X ,εr ε , X , εr , Y εr ) dr − M εt Y t,x = H ( X xt ) + Z t ¯ f ( X xr , Y t,xr ) dr − M t From Jakubowski [14], the projection: y y t is continuous from D ([0 , t ]; IR ) into IR for the S –topology. We then deduce from the convergence of the above right–hand sides that Y ε converges towards Y in distribution. Since Y ε and Y are deterministic, this means exactlythat Y ε → Y ( ii ) Let ( t n , x n ) → ( t, x ) . We assume that t > t n > . We have, Y t n , x n s = H ( X x n t n ) + Z t n s ¯ f ( X x n r , Y t n , x n r ) dr − Z t n s Z t n , x n r dM X xn r , ≤ s ≤ t n , (5.7)where X x n law ⇒ X x .Since H is a bounded continuous function and ¯ f satisfies ( C , one can easily show that thesequence { ( Y t n , x n , R . [ s,t n ] ( u ) Z x n r dM X xn r ) } n ∈ IN ∗ is tight in D ([0 , t ]; IR ) .Let us rewrite the equation (5.7) as follows Y t n , x n s = H ( X x n t n ) + Z ts ¯ f ( X x n r , Y t n , x n r ) dr − Z ts [ s,t n ] ( u ) Z t n , x n r dM X xn r (5.8) − Z tt n ¯ f ( X x n r , Y t n , x n r ) dr, ≤ s ≤ t. = A n + A n • Convergence of A n Since ¯ f is bounded, IE (cid:12)(cid:12)(cid:12)(cid:12)Z tt n ¯ f ( X x n r , Y t n , x n r ) dr (cid:12)(cid:12)(cid:12)(cid:12) ≤ K | t − t n | . Hence A n tends to zero in proba-bility. • Convergence of A n Denote by ( Y ′ , M ′ ) the weak limit of { ( Y t n , x n , R . [ s,t n ] ( u ) Z x n r dM X xn r ) } n ∈ IN ∗ . The sameproof as that of Lemma 4.9 establishes that Z ts ¯ f ( X x n r , Y t n , x n r ) dr law = ⇒ Z ts ¯ f ( X xr , Y ′ r ) dr .Passing to the limit in (5.8), we obtain that Y ′ s = H ( X xt ) + Z ts ¯ f ( X xr , Y ′ r ) dr − ( M ′ t − M ′ s ) , s ∈ [0 , t ] ∩ D c . omogenization of semilinear PDEs with discontinuous coefficients 19The uniqueness of the considered BSDE ensures that ∀ s ∈ [0 , t ] , Y ′ s = Y t, xs IP-ps. Hence Y t n , x n law ⇒ Y t, x . As in ( i ) , one derive that Y t n , x n law = ⇒ Y t, x which yields to the continuity of Y t, x .Assertion ( iii ) follows from ( ii ) and the second statement of Proposition 5.1. A Appendix: S-topology
The S –topology has been introduced by Jakubowski ([14], 1997) as a topology defined onthe Skorohod space of càdlàg functions: D ([0 , T ]; IR ) . This topology is weaker than theSkorohod topology but tightness criteria are easier to establish. These criteria are the sameas the one used in Meyer-Zheng [20].Let N a, b ( z ) denotes the number of up-crossing of the function z ∈ D ([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 ε ) ε> is said to be S –tight if andonly if it is relatively compact for the S –topology.Let ( Y ε ) ε> be a family of stochastic processes in D ([0 , T ]; IR ) . Then this family is tight forthe S –topology if and only if ( k Y ε k ∞ ) ε> and ( N a, b ( Y ε )) ε> are tight for each a < b . Let (Ω , F , IP , ( F t ) t ≥ ) be a stochastic basis. If ( Y ) ≤ t ≤ T is a process in D ([0 , T ]; IR ) suchthat Y t is integrable for any t , the conditional variation of Y is defined by CV ( Y ) = sup n ≥ , ≤ t <...
Let ( Y ε ) ε> be a S -tight family of stochastic process whose trajectories belongto D ([0 , T ]; IR ) . Then there exists a sequence ( ε k ) k ∈ IN decreasing to zero, some process Y ∈ D ([0 , T ]; IR ) and a countable subset D ∈ [0 , T ] such that for any n ≥ and any ( t , ..., t n ) ∈ [0 , T ] \ D , ( Y ε k t , ..., Y ε k t n ) D ist −→ ( Y t , ..., Y t n ) Remark
A.3 . The projection π T : y ∈ ( D ([0 , T ]; IR ) , S ) y ( T ) is continuous (see Remark2.4, p.8 in Jakubowski [14]), but y y ( t ) is not continuous for each ≤ t ≤ T .omogenization of semilinear PDEs with discontinuous coefficients 20 Lemma A.4.
Let ( U ε , M ε ) be a multidimensional process in D ([0 , T ]; IR p ) ( p ∈ IN ∗ ) con-verging to ( U, M ) in the S-topology. Let ( F U ε t ) t ≥ (resp. ( F Ut ) t ≥ ) be the minimal completeadmissible filtration generated by U ε (resp. U ). We assume moreover that for every T > , sup ε> IE (cid:2) sup ≤ t ≤ T | M εt | (cid:3) < C T .If M ε is a F U ε -martingale and M is F U -adapted, then M is a F U -martingale. Lemma A.5.
Let ( Y ε ) ε> be a sequence of process converging weakly in D ([0 , T ]; IR p ) to Y .We assume that sup ε> IE (cid:2) sup ≤ t ≤ T | Y εt | (cid:3) < + ∞ . Then for any t ≥ , E (cid:2) sup ≤ t ≤ T | Y t | (cid:3) < + ∞ . Acknowlegement . The authors are sincerely grateful to the referee for many useful remarkswhich have leaded to an improvement of the paper.The second author thanks IMATH laboratorie of université du Sud Toulon-Var and theLATP laboratory of université de Provence, Marseille, France, for their kind hospitality.
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