LOCALLY PERIODIC HOMOGENIZATION OF REFLECTED DIFFUSION

REFLECTED DIFFUSION

ABOUBAKARY DIAKHABY AND YOUSSEF OUKNINE Received 27 July 2005; Accepted 15 February 2006

We study the homogenization of reflected SDEs with locally periodic coefficients and highly oscillating drift. Our method is entirely probabilistic, and builds upon earlier works of Tanaka, Bench´erif-Madani and Pardoux, and Bensoussan et al. We extend, to Tanaka’s theorem locally periodic case.

Copyright © 2006 A. Diakhaby and Y. Ouknine. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis- tribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

LetL^εbe a uniformly elliptic second-order partial differential operator of the form (2.11) indexed by a parameterε >0. The homogenization problem for an elliptic equation con- sists in computing the limit asε↓0, of the solutionu_ε(x) ofL^εu_ε= f in a domainDof R^d subject to an appropriate boundary condition under the assumption that the coeffi- cientsa^{i j}(x),bⁱ(x), andcⁱ(x) are periodic, almost periodic, or more generally, stationary random fields. In a probabilistic approach the problem becomes the following. What is the limit of the laws of the diffusion processesX_t^εwith generatorL^εasε→0? This kind of problem has been studied for diffusion processes in the whole ofR^dby Fre˘ıdlin [5,6] and Bensoussan et al. [3], and in the case of the presence of boundary conditions by Tanaka [13]; and in this paper, we will consider the case of locally periodic coefficients and gen- eralize [13, Theorem 2.2]. This result of Tanaka is used by Ouknine and Pardoux [9], the authors have combined the probabilistic approach of Pardoux [10] with backward stochastic differential equations, in order to derive homogenization results for semilinear parabolic PDEs with periodic highly oscillating drift and nonlinear term and nonlinear Neumann boundary conditions. We note that Bench´erif-Madani and Pardoux [1,2] deal in the locally periodic case with the same problem with a Cauchy boundary condition.

The paper is organized as follows. InSection 2, we give our assumptions, notation, and the problem formulation. InSection 3we deal with the main result and its proof.

Hindawi Publishing Corporation

Journal of Applied Mathematics and Stochastic Analysis Volume 2006, Article ID 37643, Pages1–17

DOI10.1155/JAMSA/2006/37643

2. Reflected diffusion with rapidly oscillating and locally periodic coefficients

LetD= {(x¹,. . .,x^d)∈R^d,x¹>0}, the functionsσ:R^d⊗R^d→R^d⊗R^d,b:R^d⊗R^d→ R^d,c:R^d⊗R^d→R^d, and γ:∂D⊗∂D(^∼=R^d−1⊗R^d−1)→R^d are locally periodic (i.e., periodic with respect to the second variable; of period 1 in each direction inD), and γ¹(x,y)=1. We notea=σ(x,y)^tσ(x,y)/2 and we define some family of operators in- dexed byxand acting ony. By convention∂imeans∂yi:

L_x= d i,j=1

a^{i j}

x,y ε

∂_i∂_j+1 ε

d i=1

bⁱ

x,y ε

∂_i+ d i=1

cⁱ

x,y ε

∂_i,

L_x,y= d i,j=1

a^{i j}(x,y)∂_i∂_j+ d i=1

bⁱ(x,y)∂_i,

Γ_x= d i=1

γⁱ

x,y ε

∂i,

Γ^εx,y= d i=1

γⁱ(x,y)∂_i.

(2.1)

2.1. Assumptions on the coefficients. Our standing assumptions are the following.

(H.1) Global Lipschitz condition: there exists a constantcsuch that for anyζ=a,b,c, andγ,

ζ(x,y)−ζ(x,y)≤c x−x + y−y ∀x,x∈R^d; y,y∈T^d. (2.2) (H.2) The partial derivatives∂_xζ(x,y) as well as the mixed derivatives∂²_xyζ(x,y) exist and are continuous,ζ=a,b,c, andγ,x∈R^d,y∈T^d.

(H.3) The coefficients are bounded, that is, there exists a constantcsuch that for any ζ=a,b,c, andγ,

ζ(x,y)≤c, x∈R^d, y∈T^d. (2.3) The system

L_x insideD,

Γxu=0 on∂D (2.4)

determines a unique diffusion process onD, which is called (L_x,Γx)-diffusion.

By requirement there exist aLx,y-diffusion onR^dwith generatorLx,yand byY-perio- dicity assumption on the coefficients this process induces diffusion processU^x on the d-dimensional torus T^d, moreover this diffusion process is ergodic. We denote bym(x,·) its unique invariant measure. In order for the process with generatorL_xto have a limit in law asε→0, we need the following condition to be in force.

(H.4) Centering condition: for allx,

T^db(x,u)m(x,du)=0. (2.5)

2.2. Notations. We use the following notation for any functionsζ(x,y) orξ(x):

ζ

X_s,X_t

≡ζ(s,t), Δs,rξ(_·)≡ξ(r)₋ξ(s),

∂iζ(x,y)≡∂yiζ(x,y).

(2.6)

First we notice that under (H.4), there exists a unique periodic solutionb^kofLb^k= −b^k for eachk=1,. . .,d, with zero integral against the measurem(x,·). That solution is given byb^k(x,u)=_∞

0 Eu{b^k(x,U_t^x)}dtwhere underPu,U^xstarts fromu.

We set

b=

⎡

⎢⎢

⎢⎢

⎣ b¹

... b^d

⎤

⎥⎥

⎥⎥

⎦, ∇yb=

⎡

⎢⎢

⎢⎢

⎣

∂1b¹···∂db¹ ...···...

∂1b^d···∂db^d

⎤

⎥⎥

⎥⎥

⎦,

a0(x)₌

T^d

I+∇yba^tI+∇yb(x,u)m(x,du),

c0(x)=

T^d

I+∇ybc(x,u)m(x,du),

L⁰(x)= d i,j=1

a^{i j}0(x)∂i∂j+ d i=1

c0ⁱ(x)∂i.

(2.7)

We writeu=H^xϕfor the solutionuof

Lxu=0 inD

u=ϕ on∂D. (2.8)

Then H^x sends functions defined on∂D to functions defined on D, while Γ^xH sends functions on∂D to functions on ∂D, where Γ^x=_d

i=1γⁱ(x,y)∂_i. There exist a unique Markov process on∂Dwith generatorΓ^xH^x. By the periodicity assumption this Markov process induces a Markov process on the torus T^d−1; letm(x, ·) be the invariant measure of the induced Markov process. We set

γ0(x)=

T^d−1

I+∇ybγ(x,u)m(x,du),

Γ⁰(x)= d i=1

γⁱ₀(x)∂_i.

(2.9)

Given ad-dimensional Brownian motion{Bt;t≥0}defined on a probability space (Ω,Ᏺ, P), (X^ε,φ^ε) is the unique solution with values inD×R+of the following reflected SDE:

dX_t^ε=σ

X_t^ε,X_t^ε ε

dBt+1

εb

X_t^ε,X_t^ε ε

dt+c

X_t^ε,X_t^ε

ε

dt+γ

X_t^ε,X_t^ε ε

dφ^ε_t, t≥0, X_t^ε,1≥0, φ^εis continuous and increasing,

_t

0X_s^1,εdφ^ε_s=0, t≥0, X₀^ε=x,

(2.10)

whereX^ε,1denotes the first component of the processX^ε. We recall thatD=R^d+, so that X^εlives inD, that is,X^ε,1remains nonnegative, andφ^εincreases when and only whenX^ε,1 is zero, just to keep it nonnegative.

Let

L= d i,j=1

a^{i j}

x,x ε

∂_xi∂_xj+1 ε

d i=1

bⁱ

x,x ε

∂_xi+ d i=1

cⁱ

x,x ε

∂_xi,

Γ= d i=1

γⁱ

x,x ε

∂_xi

(2.11)

be the operators acting onx, so the diffusion processX_t^εis an (L,Γ)-diffusion.

We define

C0(x,y)=

∂_xbb+I+∂_ybc+1

2Tr∂²_xybσσ^∗

(x,y), S0(x,y)=

I+∂ybσ(x,y), A0(x,y)=S0S^∗₀(x,y), C0(x)=

T^dC0(x,u)m(x,du), A0(x)=

T^dA0(x,u)m(x,du), γ0(x)=

T^d−1

I+∂ybγ(x,y)m(x,d y),

L⁰= d i,j=1

A^{i j}₀(x)∂_xi∂_xj+ d i=1

Cⁱ₀(x)∂_xi, Γ⁰= d i=1

γ₀ⁱ(x)∂_xi, dXt=A^1/2₀ Xt

dBt+C0

Xt dt+γ0

Xt

dφt, t≥0, X_t¹≥0, φis continuous and increasing,

_t

0X_s¹dφs=0, t≥0, X0=x,

(2.12)

The operatorsL⁰andΓ⁰are acting onx.

3. Main result

We can now state our main results, which are a generalization of Tanaka [13, Theorem 2.2].

Theorem 3.1. Under the assumptions (H.1), (H.2), (H.3), and (H.4), the (L^ε,Γ^ε)-diffusion processX^εconverges in law to an (L⁰,Γ⁰)-diffusionXasε↓0. Moreover,

X^ε,M^Xε,φ^ε=⇒

X,M^X,φ, (3.1)

whereM^X (resp.,M^Xε) is the martingale part ofX(resp.,X^ε), andφ(resp.,φ^ε) is the local time ofX¹(resp.,X^ε,1) at 0.

Remark 3.2 (Skorohod equation). We haveX_s^ε,1=U_s^,1+φ^ε_s, where U_t=x+

_t

0

σ

X_s^ε,X_s^ε

ε

dBs+1 εb

X_s^ε,X_s^ε

ε

ds+c

X_s^ε,X_s^ε ε

ds

, (3.2)

so by [13, Proposition 3.2], φ^ε_t−φ_s^ε≤ max

s≤t1≤t2≤t

U_t1^,1−U_t2^,1≤ sup

s≤t1≤t2≤t

U_t2−U_t1, 0≤s≤t, (3.3) and using the boundedness of the coefficients, with probability one, we have

φ^ε_t−φ^ε_s≤c(t−s) + (t−s)^1/2+⁻¹(t−s), 0≤s≤t, (3.4) and fort−s≤²,

φ^ε_t−φ^ε_s≤c(2 +) ∀t,s, 0≤s≤t≤s+²≤T. (3.5) LetU_·be the unique diffusion inR^d, solution in law to the stochastic differential equa- tion for 0≤t≤T,

Ut= _t

0C0

Us

ds+ _t

0A^1/2₀ Us

dBs. (3.6)

Then we have from [1], thatU converge in law sense toU (i.e.,U⇒U). We have, by the Skorohod equation, that (X^ε,1,φ^ε) is associated toU^ε,1:

X^ε,1=U^ε,1+φ^ε ∀>0. (3.7)

According to the above result, the above remark, and Słomi ´nski [11, Corollary A.3], we have the following.

Lemma 3.3. (X^ε,1,U^ε,1,φ^ε)⇒(X¹,U¹,φ) where (X¹,φ) is associated toU¹. By [1, Lemmas 21 and 22] and the above remark, we have also the following.

Lemma 3.4. For anyp >0, there exists a constantcpsuch that for allε >0, E

φ_T^εp< c_p. (3.8)

Proof. We have from the remark thatX_s^ε,1=U_s^,1+φ^ε_sand sinceU_t=U_t+R^εtby [1, Lem- mas 21 and 22], for any 0≤t≤T,∃csuch thatE U_t ^p< candE Rt p< c. From (3.3), we haveE(φ^ε_T)^p≤E(max0_≤t1≤t2≤t|U_t1^,1−U_t2^,1|)^p≤E(sup_{0≤t1≤t2≤t}|U_t2−U_t1|)^p<∞.

From [1, Lemma 1] it is easy to reach the following result.

Lemma 3.5. Let h(x,y) be a continuous bounded function onR^d×T^d such that for all x∈R^d,_Tdh(x,u)m(x,du)=0. Then

⁻¹ _t

sh(r,r)dr=⁻¹ _t

s

Δs,rh(·,r) +Δs,rL_·,rh(s,r)dr

+ _t

s∂yh(s,r)c(r,r)dr+γ(r,r)dφ_r +

_t

s∂yh(s,r)σ(r,r)dBr+Δt,sh(s,·).

(3.9)

Proof. Use the It ˆo-Krylov formula to computeΔs,th(s,·), wherehis the solution of the Poisson equation, and the fact thatL_s,rh(s,r) +h(s,r)=0.

Let us take a fine enough equidistant subdivision, ultimately depending on, of the interval [0,T] by means of the pointsti,i=0,. . ., [T/Δt]=N, wheret0=0,Δti=ti+1−ti. We denote byt_∗the largestti belowt, byt^∗the leastti abovet, and byNt the integer [t/Δt] fort≤T. Applying the preceding lemma tob(x,y) on eachΔtiwe can derive a representation ofX_t^εin which the singularity is removed by introducing a multiplicative small corrector term.

Let us first define, for 0≤s≤T, F^0,s_∗,s=

I+∂_ybs_∗,sc(s,s), G^0,s_∗,s=

I+∂ybs_∗,sσ(s,s), γ^0,s_∗,s=

I+∂ybs_∗,sγ(s,s), R^0,

s_∗,s=Δs∗,sb(·,s) +Δs∗,sL_·,sbs_∗,s,

(3.10)

and state the following as in [1].

Corollary 3.6. With the notation above, for 0≤t≤T,

X_t^ε_∗=x+ _t∗

0 F^0,s_∗,sds+ _t∗

0 G^0,s_∗,sdBs+ε⁻¹ _t∗

0 R^0, s_∗,sds +ε

Nt−1 i=0

Δti+1,tibt_i,· +

_t∗

0 γ^0,s_∗,sdφ_s.

(3.11)

Proof. Write downX_t^ε_∗and useLemma 3.5to change⁻¹_t∗

0 b(s,s)dsby^N_i=^t₀⁻¹⁻¹_ti+1

ti b(s,

s)ds.

We need the following.

Lemma 3.7 [1, Lemma 2]. Under the conditions above, there exists a constantc >0 such that for allx∈R^dandyin T^d,

b(x,y)+∂_xb(x,y)+∂_yb(x,y)+∂²_yb(x,y)+∂²_xyb(x,y)≤c (3.12)

and these derivatives are continuous.

Now we can give a result about tightness.

Lemma 3.8. There exists a constantcsuch that for all>0 and 0≤s < t≤T,

E

sup

s≤v≤t

X_v^ε−X_s^ε4

≤c(t−s)²+⁴+E

φ_t −φ_r⁴. (3.13)

Proof. Lett_ibe as inCorollary 3.6and let 0≤s < v≤t≤T, we can write

X_v^ε−X_s^ε≤X_v^ε−X_v^ε_∗+X_v^ε_∗−X_s^ε_∗+X_s^ε−X_s^ε_∗. (3.14)

By Lemmas3.5and3.7, for 0≤r_∗≤v≤r≤T,

X_v^ε−X_v^ε_∗≤c

⎛

⎝ ⁻¹_v

r∗X_u^ε−X_r^ε_∗du+v−r_∗ +_r^v_∗G^0,r_∗,udBu++φ_v−φ_r∗

⎞

⎠. (3.15)

Therefore by H¨older and convexity,

X_v^ε−X_v^ε_∗⁴≤c

⎛

⎜⎝⁻⁴

v−r_∗³_r^v_∗X_u^ε−X_r^ε_∗⁴du+v−r_∗⁴ +_r^v_∗G^0,r_∗,udB_u⁴+⁴+φ_v−φ_r∗⁴

⎞

⎟⎠. (3.16)

Hence

E

sup

r∗≤v≤r

X_v^ε−X_r^ε_∗⁴

≤c

⎛

⎜⎜

⎝ ⁻⁴

r−r_∗³_r^r_∗E

sup

r∗≤v≤u

X_v^ε−X_r^ε_∗⁴

du

+r−r_∗⁴+r−r_∗²+⁴+E

φ_r−φ_r∗⁴

⎞

⎟⎟

⎠. (3.17)

By the Gronwall-Bellman lemma, E

sup

r∗≤v≤r

X_v^ε−X_r^ε_∗⁴

≤c

⎛

⎝

r−r_∗⁴+r−r_∗²+⁴ +E

φ_r−φ_r∗⁴

⎞

⎠e^{c−4(r−r∗)4}. (3.18)

We now chooseΔti=², by this and (3.5) E

sup

r∗≤v≤r

X_v^ε−X_r^ε_∗⁴

≤c⁴+E

φ_r−φ_r∗⁴≤c⁴. (3.19) Since, fromLemma 3.7, the functionb(·,y) is Lipschitz onR^duniformly iny∈T^d, we have by convexity fors≤v≤t,

E sup

s≤v≤t

Nt−1 i=Ns+1

Δti+1,tib·,ti

4!

≤cE

" _Nt−1

i=Ns+1

X_t^ε_i−X_t^ε_i−1

#4

≤c t−s

Δti

4 c⁴

. (3.20) Hence

E sup

s≤v≤t

Nt−1 i=Ns+1

Δti+1,tib·,ti

4!

≤c(t−s)⁴. (3.21) So byCorollary 3.6, we have

E

sup

s≤v≤t

X_v^ε_∗−X_s^ε_∗⁴

≤c⁴+E

φ_t −φ_s⁴+ (t−s)²+ (t−s)⁴

≤c^$4+ (t−s)²+E

φ_t−φ_s^4%

(3.22)

which implies the result.

We can now state the following.

Theorem 3.9. Under the assumptions on the coefficients, the family of processes {X^ε, 0<≤1}is tight inC[0,T].

Proof. By Billingsley [4, Theorem 8.3], it suffices to check that for anyαandδ >0, there exist 0<0≤1 and 0< θ≤Tsuch that

θ⁻¹P

sup

s≤v≤s+θ

X_v^ε−X_s^ε> δ

< α (3.23)

for alls≤T−θand≤0. We have by the Markov-Chebychev inequality P

sup

s≤v≤s+θ

X_v^ε−X_s^ε> δ

≤ 1 δ^4E

sup

s≤v≤s+θ

X_v^ε−X_s^ε 4

≤ c δ⁴

θ²+⁴+E

φ_s+θ−φ_s⁴. (3.24)

By continuity ofφ_t, there existθ0such thatE(φ_s+θ−φ_s)≤(by (3.5), we haveE(φ^ε_t− φ^ε_s)≤c, 0≤s≤t≤s+²) for all 0< θ≤θ0≤T. For anyαandδ >0, there exist 0<0≤ 1 and 0< θ≤θ0≤Tsuch thatcθ[θ²+ 2⁴]< αδ⁴for 0<<0, this ends the proof.

We can recover our processes as a main term which converges in law, plus an asymp- totically small term. ByCorollary 3.6, we have

X_t^ε_∗=x+ _t∗

0

I+∂ybs_∗,sc(s,s)ds+ _t∗

0

I+∂ybs_∗,sσ(s,s)dBs

+ _t∗

0

I+∂ybs_∗,sγ(s,s)dφ_s+ε

Nt−1 i=0

Δti+1,tibti,·

+

Nt−1 i=0

ε⁻¹

Δti

Δti,sb(·,s) +Δti,sL_·,sbti,sds.

(3.25) Define

F₂^1,(s)I+∂_ybs_∗,sc(s,s), G^1,(s)I+∂ybs_∗,sσ(s,s), γ^1,(s)I+∂ybs_∗,sγ(s,s),

(3.26)

R^1,_Nt ^Nt− 1 i=0

ε⁻¹

Δti

Δti,sb(·,s) +Δti,sL_·,sbt_i,sds, (3.27)

S^1,_Nt ε

Nt−1 i=0

Δti+1,tibti,·

=ε

Nt−1 i=1

Δti−1,tib·,ti

+ b(0, 0)−bt_Nt−1,t_Nt

S^2,_Nt +R^2,_Nt.

(3.28)

So we have now

X_t^ε_∗ =x+ _t∗

0 F₂^1,(s)ds+ _t∗

0 G^1,(s)dB_s+ _t∗

0 γ^1,(s)dφ_s+S^1,_Nt +R^1,_Nt, S^1,_Nt S^2,_Nt +R^2,_Nt.

(3.29)