Homogenization of periodic semilinear hypoelliptic PDEs

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

ALASSANEDIÉDHIOU, ÉTIENNEPARDOUX

Homogenization of periodic semilinear hypoelliptic PDEs

Tome XVI, n^o2 (2007), p. 253-283.

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Annales de la Facult´e des Sciences de Toulouse Vol. XVI, n^◦2, 2007 pp. 253–283

Homogenization of periodic semilinear hypoelliptic PDEs

Alassane Di´edhiou¹, Etienne Pardoux´ ²

ABSTRACT.— We establish homogenization results for both linear and semilinear partial differential equations of parabolic type, when the linear second order PDE operator satisfies a hypoellipticity asumption, rather than the usual ellipticity condition. Our method of proof is essentially probabilistic.

R^´ESUM´E.— Nous ´etablissons des r´esultats d’homog´en´eisation d’´equa- tions aux d´eriv´ees partielles paraboliques lin´eaires et semi–lin´eaires, sous une hypoth`ese d’hypoellipticit´e de l’op´erateur aux d´eriv´ees partielles du second ordre, au lieu de l’hypoth`ese usuelle d’ellipticit´e. Notre m´ethode de d´emonstration est essentiellement probabiliste.

1. Introduction

Our aim is to homogenize two classes of periodic semilinear parabolic PDEs, namely

∂u^ε

∂t (t, x) =Lεu^ε(t, x) +1 εe(x

ε, u^ε(t, x)) +f(^x_ε, u^ε(t, x)), u^ε(0, x) =g(x),

(1.1)

and ∂u^ε

∂t (t, x) =Lεu^ε(t, x) +f(^x_ε, u^ε(t, x),∇u^ε(t, x)σ(x ε)) u^ε(0, x) =g(x), x∈IR^d,

(1.2) where Lε is a second order PDE operator (see (1.5)). The novelty of our result lies mainly in the fact that the matrix of second order coefficients of

(∗) Re¸cu le 8 avril 2005, accept´e le 31 octobre 2006

(1) D´epartement de Math´ematiques-Informatique, Facult´e des Sciences et Technique, Universit´e Cheikh Anta Diop, B.P. 5005 Dakar-Fann, S´en´egal.

asana@ucad.sn

(2) L.A.T.P, Universit´e de Provence, 39 rue F. Joliot Curie, 13453 Marseille cedex 13, France.

pardoux@cmi.univ-mrs.fr

L_ε is not assumed to be elliptic, but instead we formulate a hypoellipticity condition of H¨ormander type, see the end of this section.

There is by now quite a vast literature concerning the homogenization of second order elliptic and parabolic PDEs with a possibly degenerating matrix of second order coefficientsa, see among others [1], [2], [3], [6], [15].

But, as far as we know, in these works, either the coefficientais allowed to degenerate on sets of measure zero only, or else the equation is linear.

Our method of proof will be mainly probabilistic. We consider the SDE, forε0, x∈IR^d,

X_t^ε=x+ t

0

c(X_s^ε ε )ds+1

ε t

0

b(X_s^ε ε )ds+

d j=1

t 0

σj(X_s^ε

ε )dW_s^j (1.3)

where {W_t^j, j= 1, . . . , d;t0} is a standard d-dimensional Brownian mo- tion. The functionsc, bandσ_j, j= 1, . . . , dbelong toC^∞(IR^d,IR^d) and are periodic with period 1in each direction.

Under the above conditions, there exists a unique solution{X_t^ε, t0}of (1.3). Setting ˜X_t^ε=¹_εX_ε^ε2t, then we get with a newd–dimensional standard Brownian motion{Wt, t0}, which in fact depends on ε:

X˜_t^ε=x ε +ε

_t

0

c( ˜X_s^ε)ds+ _t

0

b( ˜X_s^ε)ds+ d j=1

_t

0

σ_j( ˜X_s^ε)dW_s^j. (1.4)

We shall assume that the matrixσ(x) of columns vectorsσ_j(x) satisfies the strong H¨ormander condition, given by the

Definition 1.1. — LetH(n, x)be the set of Lie brackets of(σ_j(x))_1jd of order lower than nat the point x∈IR^d.

We say that the matrixσsatisfies the strong H¨ormander condition (called SHC) if for all x∈ IR^d, there exists nx∈IN such that H(nx, x) generates IR^d.

Now we are going to study the ergodic properties of the processes{X˜_t^ε, t0}like in ´E. Pardoux [13] under the above strong H¨ormander condition.

Let us consider the infinitesimal generator of{X˜_t^ε, t0}: L_ε = 1

2 d i,j=1

a_ij(x) ∂²

∂xixj

+ d i=1

(b_i(x) +εc_i(x)) ∂

∂xi

= 1

2 d i,j=1

∂

∂xj

(aij(x) ∂

∂xi

) + d i=1

(˜bi(x) +εci(x)) ∂

∂xi

(1.5)

where ˜b_i(x) = b_i(x)− ¹₂ d j=1

∂

∂x_ja_ij(x). We will denote by L the operator L0.

In all the rest of the paper, we assume that the condition

The matrix σsatisfies the SHC (A)

is satisfied.

The paper is organized as follows. Section 2 contains several preliminary results, which are our tools for the homogenization, and which we extend from the classical elliptic case to our hypoelliptic setting. Section 3 applies the above preliminaries to the homogenization of a linear parabolic equation.

Finally section 4 studies the homogenization of equation (1.1), and section 5 that of equation (1.2).

2. Preliminaries 2.1. Invariant measure

We rewrite the equation (1.4) in the form:

X˜_t^ε=x ε +

t 0

b^ε( ˜X_s^ε)ds+ t

0

σ( ˜X_s^ε)dW_s^ε (2.1) whereb^ε( ˜X_s^ε) =εc( ˜X_s^ε) +b( ˜X_s^ε), or in Stratonovich form

dX˜_t^ε=σ₀( ˜X_t^ε)dt+ d j=1

_t

0

σ_j( ˜X_t^ε)◦dW_t^j,

where σ₀ⁱ = b^ε_i −¹₂ m k=1

d j=1

∂_kσⁱ_jσ_j^k. From now on, we consider the process {X˜_t^ε}as taking its values in thed–dimensional torusT^d:= IR^d/Z^d.

DefineH =L²([0, T],IR^d), leth∈H and Φ(h) be the solution of:

Φ(h)_t= x ε +

_t

0

σ₀(Φ(h)_s)ds+ d j=1

_t

0

σ_j(Φ(h)_s)h^j_sds

Proposition 2.1. — We have

support(X˜_t^ε) = {Φ(h)_t(x);h∈H}

= T^d.

Proof. — The first equality follows from the Stroock-Varadhan support theorem see [16], and the second is a well known consequence of condition (A), see e.g. V. Jurdjevic [9].

Letµ^ε_t(x, dy) denote the law of ˜X_t^εandp^ε_t(x, y) its density.

Theorem 2.2. — The densityp^ε_t(x, y)is strictly positive for all(t, x, y)∈ IR^∗₊×T^d×T^d.

Proof. — This is again a consequence of condition (A), see Michel, Par- doux [11] theorem 3.3.6.1.

We have the

Lemma 2.3. — For all ε 0, the T^d-valued diffusion process {X˜_t^ε, t0} of generatorLε, has a unique invariant probability µε.

Proof. — Since{X˜_t^ε, t0}is a homogeneous Feller process with values in a compact set, µ_ε exists. The proof of the uniqueness is the same as in E. Pardoux [13] by using the fact that, since the transition density´ p^ε_t(x, y) is strictly positive, any invariant measure has a strictly positive density.

Lemma 2.4. — For any fixedt >0 the function [0,1]×T^d×T^d −→ IR+

(ε, x, y) −→ p^ε_t(x, y) is continuous.

Proof. — In order to prove this Lemma it suffices to prove that the func- tion (ε, x)−→p^ε_t(x, .) from [0,1]×T^d intoC(T^d) is continuous. Consider the map

(ε, x) −→ µ^ε_t(x, dy) [0,1]×T^d −→ E,

where E denotes the set of probability measures on T^d equiped with the topology of weak convergence. This map is continuous, since the map (ε, x)−→X˜_t^εwith values inL²(Ω) is continuous. It now suffices to show that the densities are equicontinuous, in order to deduce from Ascoli’s theorem the wished continuity. We know that

||p^ε_·(x,·)||L¹([t−α,t+α]×T^d)= 2α, so for some large enoughn(whose value depends ond),

||p^ε_·(x,·)||H⁻ⁿ([t−α,t+α]×T^d)C(α),

then by the hypoellipticity of _∂t^∂ −L^∗_ε(cf. e.g. Lemma 5.2 p.122 of [17]) for allm >0, there existsC(m)>0 such that

||p^ε_.(x, .)||H^m([t−α,t+α]×T^d)C(m), and from the Sobolev embedding we deduce that

sup

ε∈[0,1],x∈T^d||p^ε_t(x, .)||C¹(T^d)C, (2.2) which establishes the wished equicontinuity. .

Since p^ε_t(x, y) > 0, ∀(ε, x, y) ∈ [0,1]×T^d×T^d and p_t is a continu- ous function of (ε, x, y) on this compact set, for each t > 0, there exists (ε₀, x₀, y₀)∈[0,1]×T^d×T^dsuch that

ct:= inf

ε,x,yp^ε_t(x, y) =p^ε_t⁰(x0, y0)>0.

We now prove the

Lemma 2.5. — For any t >0, andx, x∈T^d, we have

||p^ε_t(x, .)−p^ε_t(x, .)||L¹(T^d)2(1−c₁)^[t].

Proof. — For any coupling ofX_t^xandX_t^x we have

||p^εt(x, .)−p^ε_t(x, .)||L¹(T^d)2IP(X_t^x=X_t^x).

We first define a coupling of (X_n^x, X_n^x, n= 0,1,2, . . . , [t]). Let us con- sider a map F_ε : T^d×[0,1] −→ T^d, such that if the random variable η has the uniform distribution on [0,1], the random variableFε(x, η) has the probability density ^pε1(x,y)_1−c^−c1

1 .

Let (U₁, ξ₁, η₁, η₁, . . . , U_n, ξ_n, η_n, η_n, . . .) be independent random vari- ables such that the random variables ξ_n, η_n, η_n are uniformly distributed over [0,1], and

U_n=

1with probabilityc₁ 0 with probability 1−c₁.

We now define recursively the sequences X_n^x, X_n^x, n 1. For each n0, ifX_n^x=X_n^x, then we set

X_n+1^x =X_n+1^x =U_n+1ξ_n+1+ (1−U_n+1)F(X_n^x, η_n+1), if not then we set

X_n+1^x = U_n+1ξ_n+1+ (1−U_n+1)F(X_n^x, η_n+1) X_n+1^x = Un+1ξn+1+ (1−Un+1)F(X_n^x, η_n+1).

So we have

IP(X_n^x=X_n^x) = IP(U1= 0, U2= 0, . . . , Un= 0) = (1−c1)ⁿ. Similarly we defineFt−[t](x, .) such thatFt−[t](x, η) possesses the prob- ability densityp^ε_t−[t](x, y) and ifX_[t]^x =X_[t]^x, then we set

X_t^x=X_t^x =F_t−[t](X_[t]^x, η_[t]+1),else

X_t^x = F_t−[t](X_[t]^x, η_[t]+1) X_t^x = F_t−[t](X_[t]^x, η_[t]+1).

Hence we get IP(X_t^x = X_t^x) = IP(X_[t]^x =X_[t]^x) = (1−c1)^[t]. Since the densities ofX_t^x, X_t^x are respectivelyp^ε_t(x, y) andp^ε_t(x, y),we have that:

||µ^ε_t(x, .)−µ_ε||T V =

|p^ε_t(x, y)−p^ε(y)|dy

=

|p^ε_t(x, y)−

µε(dx)p^ε_t(x, y)|dy

=

|

µε(dx)(p^ε_t(x, y)−p^ε_t(x, y))|dy µ_ε(dx)|p^ε_t(x, y)−p^ε_t(x, y)|dy 2(1−c1)^[t].

Hence we have the

Lemma 2.6. — There exists a constant ρ >0 such that for any ε 0 andf ∈L^∞(T^d),

|IE(f( ˜X_t^ε))−

f(x)µ_ε(dx)|||f||L^∞(T^d)e^−ρ[t]

Iff is centered with respect toµ_ε, i.e.

T^df(x)µ_ε(dx) = 0, then we get

|IE(f( ˜X_t^ε))|||f||L^∞(T^d)e^−ρ[t], t >0

We shall need the following result Lemma 2.7. —

µ_ε⇒µ

(in the sense of weak convergence of probability measures), as ε→0.

Proof. — The collection{µε, ε >0}is tight, since these are measures on the compact setT^d. For eachε >0,t >0,f ∈C(T^d),

T^d

f(x)µε(dx) =

T^d

IExf( ˜X_t^ε)µε(dx). (2.3) But as ε → 0, clearly ˜X_t^ε → X˜_t in L²(Ω), uniformly with respect to the starting point x. Hence taking the limit in (2.3) along a subsequence {ε_k} along whichµ_εk converges weakly toν, we deduce that

T^d

f(x)ν(dx) =

T^d

IExf( ˜Xt)ν(dx).

This is true for all t >0 and allf ∈C(T^d). Hence all accumulation points of the collection{µε}, asε→0, equal the invariant measureµ, andµε⇒µ, as ε→0.

2.2. Ergodic theorem

From Lemma 2.6 and Lemma 2.7, we deduce the

Proposition 2.8. — Iff ∈L^∞(T^d),then for any t >0, _t

0

f X_s^ε

ε

ds−→t

T^d

f(x)µ(dx) in probability, asε−→0.

Proof. — Set ˜f_ε(x) =f(x)−

T^df(x)µ_ε(dx). It follows from Lemma 2.7 that

εlim−→0

T^d

f(x)µε(dx) =

T^d

f(x)µ(dx),

at least for f ∈ C(T^d). However, an argument very similar to that used to prove (2.2) above yields that the densityp_εof the invariant measure µ_ε satisfies

pεL^∞(T^d)C, ∀ε0.

This allows us to extend the above convergence to f ∈L^∞(T^d). Hence it suffices to show thatt

0f˜ε(^X_ε^sε)ds→0, asε→0.We haveX_s^ε=εX˜^εs

ε2,from which we deduce that_t

0f˜ε(^X_ε^sε)ds=ε²_ε^t₂

0 f˜ε( ˜X_u^ε)du.

From the Markov property of the process ˜X^ε, and Lemma 2.6, we get:

IE[(

_t

0

f˜_ε( ˜X_u^ε)du)²] = 2IE[

_t

0

_s

0

f˜_ε( ˜X_s^ε) ˜f_ε( ˜X_u^ε)]dsdu 2C||f˜ε||²_L∞(T^d)

t 0

s 0

e^−ρ[s−u]dsdu 2Ce^ρ||f˜ε||²_L∞(T^d)

t 0

s 0

e^−ρ[s−u]dsdu

= 2Ce^ρρ⁻²(−1 +ρt+e^−ρt)||f˜ε||²_L∞(T^d), hence

IE





ε^{2 t}

ε2

0

f˜ε( ˜X_u^ε)du 2

2Ce^ρρ⁻²

−ε⁴+ρε²t+ε⁴e^−ρεt2

||f˜ε||²_L∞(T^d),

from which the Proposition follows.

2.3. The Poisson equation We have the

Theorem 2.9. — Iff ∈C^∞(T^d)is such that

T^df(x)µ(dx) = 0then the PDE

Lfˆ(x) +f(x) = 0, x∈T^d

has a solution fˆ ∈ C^∞(T^d), which is given by the probabilistic formula fˆ(x) =_+∞

0 IEx(f( ˜Xt))dt.

Proof. — Let us consider the parabolic PDE:





∂u(t, x)

∂t −Lu(t, x) = 0, t >0, x∈T^d u(0, x) =f(x), x∈T^d

This equation has a solution inC^∞(IR₊×T^d), by the hypoellipticity of the operator _∂t^∂ −L, which is given by the Feynman-Fac formula: u(t, x) = IEx[f( ˜Xt)].

By Lemma 2.6 withε= 0,u(t, .)−→0 inL^∞(T^d) at exponential speed as t→ ∞, and if we setv(t, x) =t

0u(s, x)ds,we have

||v(t, .)||_∞= sup

x∈T^d

|v(t, x)|C, and

v(t, x)−→v(x) = _+∞

0

IE_x[f( ˜X_t)]dt, as t−→ ∞.

By Alaoglu’s theorem (see e.g. A. Friedman[7] p.169) there exists a sequence t_n −→ ∞, such that v(t_n, .)−→v in L^∞(T^d) for the weak star topology.

Sinceu(t, x) =Lv(t, x) +f(x), we have

∀ϕ∈C^∞(T^d),(u(t_n), ϕ) = (v(t_n), L^∗ϕ) + (f, ϕ) and and letting ntend to infinity, we get

(v, L^∗ϕ) + (f, ϕ) = 0,∀ϕ∈C^∞(T^d), i.e.v solves the PDE

Lv+f = 0

in the sense of distributions. Then by the hypoellipticity ofL we have that v∈C^∞(T^d).

3. Homogenization of a linear parabolic equation

The functionsa, b, csatisfy the conditions of section 1. Let us consider the functions e belonging to C^∞(IR^d,IR), and f : IR^d −→ IR measurable and bounded, and both are periodic with period 1in each direction, and g ∈ C(IR^d) with at most polynomial growth at infinity. For ε > 0, we consider the linear PDE:





∂u^ε(t, x)

∂t =Lεu^ε(t, x) + (¹_εe(^x_ε) +f(^x_ε))u^ε(t, x), t >0, x∈IR^d, u^ε(0, x) =g(x), x∈IR^d,

(3.1)

where

L_ε= 1 2

d i,j=1

a_ij(x ε) ∂²

∂x_i∂x_j + d i=1

(1 εb_i(x

ε) +c_i(x ε)) ∂

∂x_i. We assume that

T^d

bi(x)µ(dx) = 0, i= 1, . . . , d,

T^d

e(x)µ(dx) = 0,

whereµis the invariant probability of the process ˜Xt= ˜X_t⁰, t0. If we set Y_t^ε=

_t

0

(1 εe(X_s^ε

ε ) +f(X_s^ε

ε ))ds, t0, then the solution of (3.1) is given by

u^ε(t, x) = IE_x[g(X_t^ε) exp(Y_t^ε)]

whereX_t^εis the solution of (1.1).

Let ˆe(x) =_∞

0 IEx[e( ˜Xt)]dt,and ˆbi(x) =

_∞

0

IEx[bi( ˜Xt)]dt, i= 1, . . . , d, x∈T^d be solutions of the Poisson equations

Lˆe(x) +e(x) = 0, Lˆb_i(x) +b_i(x) = 0, i= 1, . . . , d.

Let us define

A =

T^d

(I+∇ˆb)a(I+∇ˆb)^∗(x)µ(dx);

C =

T^d

(I+∇ˆb)(c+a∇e)(x)µ(dx);ˆ

D =

T^d

(1

2∇eˆ^∗.a∇ˆe+f+∇ec)(x)µ(dx).ˆ Thenu(t, x) = IE[g(x+Ct+A¹²Wt)]e^Dtis the solution of







∂u(t, x)

∂t = 1 2

d i,j=1

A_ij∂²u(t, x)

∂x_i∂x_j + d i=1

C_i∂u(t, x)

∂x_i +Du(t, x) u(0, x) =g(x), x∈IR^d,

(3.2)

Theorem 3.1. — For anyt0, x∈IR^d we have u^ε(t, x)−→u(t, x) whenε−→0.

Proof. — We know from Theorem 2.9 that the functions ˆb_iand ˆebelong toC^∞(T^d). We then can copy the proof of theorem 3.1in ´E. Pardoux [13].

4. Homogenization of a semilinear parabolic equation 1 Let us consider the semilinear parabolic equation





∂u^ε

∂t (t, x) =Lεu^ε(t, x) +¹_εe(^x_ε, u^ε(t, x)) +f(^x_ε, u^ε(t, x)), u^ε(0, x) =g(x).

(4.1) where

L_ε= 1 2

d i,j=1

a_ij(x ε) ∂²

∂xi∂xj

+ d i=1

(1 εb_i(x

ε) +c_i(x ε) ∂

∂xi

.

The assumptions ona, b, care the same as in the previous section.The func- tion g belongs to C(IR^d), with at most a polynomial growth at infinity.

Again the bi’s verify the condition

T^d

bi(x)µ(dx) = 0, i= 1, . . . , d.

We assume that

e, f : IR^d×IR−→IR

are measurable, periodic with respect to their first variable, with period one in each direction, the functione is C^∞ with respect tox, continuous iny uniformly with respect tox, twice continuously differentiable iny, uniformly with respect to x, and moreover for ally∈IR,

T^d

e(x, y)µ(dx) = 0,

and verifiese(x, y) =e0(x, y) +e1(x)y.We assume that there exists a con- stantK such that

|e1(x)|+|e0(x, y)|+|∂e₀

∂y(x, y)|+|∂²e₀

∂y² (x, y)|K, x∈T^d, y∈IR.

We assume also that for somek∈IR, allx∈IR,y, y∈IR, (f(x, y)−f(x, y))(y−y)k|y−y|²,

and

|f(x, y)|C(1+y²).

From the above assumptions oneand similarly as in Theorem 2.9, for each y∈IR, there exists a solution of the Poisson equation

Lˆe(x, y) +e(x, y) = 0, x∈T^d, y∈IR, which is given given by

ˆ e(x, y) =

_∞

0

IEx[e( ˜Xt, y)]dt.

The function y −→ IE_x[e( ˜X_t, y)] is twice differentiable with respect to y according to the assumptions one,and we get

|IEx[∂e

∂y( ˜Xt, y)]|Ke^−ρ[t]; |IEx[∂²e

∂y²( ˜Xt, y)]|Ke^−ρ[t].

We now prove that ˆebelongs toC²(T^d×IR) and the derivatives of order one and two with respecty verify the Poisson equations

L∂ˆe

∂y(x, y) +∂e

∂y(x, y) = 0; L∂²eˆ

∂y²(x, y) +∂²e

∂y²(x, y) = 0.

Forδ >0,we have

|e(x, yˆ +δ)−e(x, y)ˆ | _T

0

|IE_x[e( ˜X_t, y+δ)−e( ˜X_t, y)]|dt +

_∞

T

|IEx[e( ˜Xt, y+δ)−e( ˜Xt, y)]|dt

_T

0

|IE_x[e( ˜X_t, y+δ)−e( ˜X_t, y)]|dt+Ce^−ρT. Let us chooseT large enough such thatCe^−ρT < ^ε₂, and using the Lebesgue dominated convergence theorem we have, fo any ε >0, there exists η >0 such that

δ < η=⇒ |ˆe(x, y+δ)−ˆe(x, y)|ε.

By the same argment we show that the functions y −→ ^∂_∂y^eˆ(x, y) and y−→ _∂y^∂2eˆ2(x, y) are continuous.

Let us consider the map :

y−→(ˆe(., y),∂eˆ

∂y(., y),∂²eˆ y² (., y))

from IR into (C²(T^d))³. For any sequencey_n converging to y, we have by the hypoellipticity ofL, and the smoothness ofe, _∂y^∂e, _∂y^∂2e2 (see e.g. [17] )

||ˆe(., yn)||C³(T^d)+||∂ˆe

∂y(., yn)||C³(T^d)+||∂²ˆe

∂y²(., yn)||C³(T^d)C, so the functions ˆe(., y_n),^∂_∂y^eˆ(., y_n) and_∂y^∂2ˆe2(., y_n) are equicontinuous, together with their derivatives in x of order one and two. Then from Ascoli’s the- orem ˆe(., y_n), ^∂_∂y^eˆ(., y_n), ^∂_∂y^2ˆe2(., y_n) have subsequences converging uniformly in C²(T^d).Since the sequences ˆe(x, y_n),_∂y^∂ˆe(x, y_n),and ^∂_∂y^2ˆe2(x, y_n) converge respectively to ˆe(x, y), ^∂_∂y^eˆ(x, y), _∂y^∂2ˆe2(x, y) then

ˆ

e(., yn) −→ e(., y)ˆ

∂eˆ

∂y(., y_n) −→ ∂ˆe

∂y(., y),

∂²eˆ

∂y²(., y_n) −→ ∂²eˆ

∂y²(., y),

in C²(T^d). Hence ˆe, _∂y^∂ˆe and _∂y^∂2ˆe2 are continuous in (x, y) and their partial derivatives with respect toxof order one and two are also continuous. The limiting equation (4.1), can be formulated as











∂u

∂t(t, x) =1 2

d i,j=1

Aij

∂²u

∂xi∂xj

(t, x) + d i=1

Ci(u(t, x))∂u

∂xi

(t, x) +D(u(t, x)),

u(0, x) =g(x),

(4.2)

where

A =

T^d

(I+∇ˆb)a(I+∇ˆb)^∗(x)µ(dx), C(y) =

T^d

(I+∇ˆb)(c+a ∂²eˆ

∂x∂y(., y))(x)µ(dx), D(y) =

T^d

[∂ˆe

∂x(., y), c −∂ˆe

∂y(., y)e(., y) + ∂²eˆ^∗

∂x∂y(., y)a∂ˆe

∂x(., y) +f(., y)](x)µ(dx).

Then we have the following

Theorem 4.1. — For allt0, x∈IR^d,

u^ε(t, x)−→u(t, x), whenε−→0,

whereu^ε is the solution of the equation (4.1) anduthe solution of (4.2).

Proof. — The functions ˆb_i, i = 1, . . . , d, ˆe are smooth and with our assumptions on a, b, c, g, e and f we can follow the proof of Theorem 4.1 in ´E. Pardoux [13], which establishes the convergence of BSDEs. In fact considering the progressively measurable process {(Y_s^ε, Z_s^ε); 0 s t} in IR×IR^dsolution of the BSDE :

Y_s^ε=g(X_t^ε) +1 ε

t s

e(X_r^ε

ε , Y_r^ε)dr+ t

s

f(X_r^ε

ε , Y_r^ε)dr− t

s

Z_r^εdW_r, with

IE( sup

0st|Y_s^ε|²+ _t

0

||Z_s^ε||²ds)<∞, by ´E. Pardoux [14] the solution of (4.1) is given by

u^ε(t, x) =Y₀^ε.

In order to prove the above Theorem 4.1, it suffices to prove that Y₀^ε−→Y₀,

whereY0 is the value att= 0 of the solution of the FBSDE









Xs=x+ _s

0

C(Yr)dr+A¹²Bs, 0st Ys=g(Xt) +

t s

D(Yr)dr− t

s

ZrdBr, 0st.

5. Homogenization of a semilinear parabolic equation 2 We consider the semilinear parabolic equation





∂u^ε

∂t (t, x) =Lεu^ε(t, x) +f(x

ε, u^ε(t, x),∇u^ε(t, x)σ(x ε)) u^ε(0, x) =g(x), x∈IR^d,

(5.1)

where

L_ε= 1 2

d i,j=1

a_ij(x ε) ∂²

∂x_i∂x_j + d i=1

(1 εb_i(x

ε) +c_i(x ε)) ∂

∂x_i.

The functionsa, b, cverify the assumptions of the previous section, and we assume that

g∈W^2,p(IR^d),

for somep > d+ 1 ,peven, and

f : IR^d×IR×IR^d−→IR,

is continuous, periodic of period one in each direction with respect to its first argument, of classC¹ with respect to its second and third arguments, uniformly with respect to the first argument, withf_y bounded from above, and∇zf bounded. We assume that

|f(x, y, z)| K(1+|y|+|z|),

|f(t, y, z)−f(t, y, z)| K(|y−y|+|z−z|),

and moreover that for allx∈IR^d,f(x,·,·)∈C²(IR×IR^d), all the derivatives being bounded, uniformly with respect to x. Let us consider the progres- sively measurable process {(Y_s^ε, Z_s^ε); 0 s t} with values in IR×IR^d, solution of the BSDE:

Y_s^ε=g(X_t^ε) + t

s

f(X_r^ε

ε , Y_r^ε, Z_r^ε)dr− t

s

Z_r^εdBr,

with IE[ sup

0st|Y_s^ε|²+ _t

0

||Z_r^ε||²dr]<∞.

The solution of (5.1) is given by:u^ε(t, x) =Y₀^ε. We are going to study the limit of u^ε whenεtends to zero.

We first state and prove the (for the notion ofS–tightness, see [8], [10]) Proposition 5.1. — There exists a constantC >0 such that

|Ys^ε(ω)|C, ∀ε >0, 0st, ω∈Ω.

Moreover, the collection of continuous processes {Y_s^ε, 0 st}0<ε<ε0 is S–tight.

Proof. — Since g is bounded, it follows from Itˆo’s formula that for any α∈IR,

e^αs|Y_s^ε|²+ t

s

e^αr(α|Y_r^ε|²+|Zr|²)drc+ 2 t

s

e^αrY_r^εf(X^ε_r, Y_r^ε, Z_r^ε)dr

+2 _t

s

e^αrY_r^εZ_r^εdB_r Now from our assumption onf,

yf(x, y, z) K(|y|+y²+|y| × |z|) K+ (K+^K₂²)y²+|z|²,

hence, combining this with the previous inequality where we choose α = 2(K + ^K₂²), and taking the conditional expectation given Fs, we deduce that

|Y_s^ε|² 2K

α (e^α(t−s)−1 ) +ce^−αs,

from which the first result follows. It now follows easily from the above that sup

ε>0

IE

sup

0st|Y_s^ε|²+ t

0

|Z_s^ε|²ds

<∞, from which theS–tightness follows, since

|f(x, y, z)|K(1+|y|+|z|).

The limiting PDE can be formulated as:









∂u

∂t(t, x) =1 2

d i,j=1

Aij

∂²u

∂xi∂xj

(t, x) + d

i=1

Ci

∂u

∂xi

+f(u(t, x),∇u(t, x)) u(0, x) =g(x), x∈IR^d,

(5.2) where

A =

T^d

(I+∇ˆb)a(I+∇ˆb)^∗(x)µ(dx)

C =

T^d

(I+∇ˆb)c(x)µ(dx) f(y, z) =

T^d

f(x, y, z(I+∇ˆb)σ(x))µ(dx).

It follows from the above assumptions on f that f ∈ C²(IR×IR^d), with bounded derivatives. We shall assume w. l. o. g. that the orthonormal basis of IR^d has been choosen in such a way that the matrixAis of the form

A=

A 0 0 0

,

where A is a d ×d positive definite matrix, with d d. We set IR^d = Ed⊕Ed−d, whereEd is the subspace of IR^dof dimension d generated by the vectors ei, i= 1,2,· · ·, d after a new arrangement of the basis vectors of IR^dso we can obtain the wished form of A.

Note that from Jensen’s inequality

|f(y, z)|

|f(x, y, z(I+∇ˆb)σ(x))|µ(dx) K(1+|y|+

|z(I+∇ˆb)σ(x)|µ(dx)

K(1+|y|+

< Az, z >),

(5.3)

and

|f(y, z)−f(y, z)|

T^d

|f(x, y, z(I+∇ˆb)σ(x))

−f(x, y, z(I+∇ˆb)σ(x))|µ(dx)

K

|y−y|+

T^d

|(z−z)(I+∇ˆb)σ(x)|µ(dx)

K

|y−y|+

A(z−z), z−z

(5.4) We define HA(IR^d) ={u∈ L²(IR^d);√

A∇u∈ (L²(IR^d))^d}, and we define the following norm onHA(IR^d)

||v||HA(IR^d)=

||v||²_L²_(IR^d₎+||√

A∇v||²_(L²_(IR^d₎₎^d1₂ . We have by (5.3),||f(v,∇v)||L²(IR^d)C(1+||v||HA(IR^d)).

We can show the

Theorem 5.2. — Equation (5.2) has a unique solutionu in L²((0, T);

H¹(IR^d)),such that for all 1kd,

< A∇uk,∇uk >∈L¹((0, T)×IR^d), where

∂u

∂xk

=u_k ∈L²((0, T)×IR^d).

Moreover

u∈C(IR₊;L²(IR^d)).

Proof. —Step 1:

We first assume that the matrix A is elliptic, and we look for a solution u ∈L²(0, T;H¹(IR^d))∩C([0, T], L²(IR^d)). Let us prove the existence and uniqueness of the solution of the PDE. We setF =L²((0, T);H¹(IR^d)), and consider the map

Φ :F −→F,