Homogenization of a singular random one-dimensional PDE

www.imstat.org/aihp 2008, Vol. 44, No. 3, 519–543

DOI: 10.1214/07-AIHP134

© Association des Publications de l’Institut Henri Poincaré, 2008

Homogenization of a singular random one-dimensional PDE

Bogdan Iftimie

, Étienne Pardoux

and Andrey Piatnitski

aDepartment of Mathematics, Academy of Economical Studies, 70167 Bucharest, Romania. E-mail: iftimieb@csie.ase.ro bLATP, CMI, Universite de Provence, 39 rue Joliot-Curie, 13453 Marseille, Cedex 13013, France. E-mail: Etienne.Pardoux@cmi.univ-mrs.fr cNarvik University College, Postbox 385, 8505 Narvik, Norway and P.N. Lebedev Physical Institute RAS, Leninski prospect 53, 119991 Moscow,

Russia. E-mail: andrey@sci.lebedev.ru Received 21 February 2006; accepted 30 March 2007

Abstract. This paper deals with the homogenization problem for a one-dimensional parabolic PDE with random stationary mixing coefficients in the presence of a large zero order term. We show that under a proper choice of the scaling factor for the said zero order terms, the family of solutions of the studied problem converges in law, and describe the limit process. It should be noted that the limit dynamics remain random.

Résumé. Cet article traite de l’homogénéisation d’une équation aux dérivées partielles en dimension un d’espace, avec des co- efficients aléatoires stationnaires et mélangeants, en présence d’u terme d’ordre zéro fortement oscillant. Nous montrons qu’avec un choix convenable du facteur d’échelle de ce terme d’ordre zéro, les solutions du problème étudié convergent en loi, et nous décrivons le processus limite. On peut noter que la dynamique limite est elle aussi aléatoire.

MSC:74Q10

Keywords:Stochastic homogenization; Random operators

1. Introduction

Our goal is to study the limit, asε→0, of a linear parabolic PDE of the form

∂u^ε

∂t (t, x)=1 2

∂

∂x

a ·

ε ∂u^ε

∂x

(t, x)+ 1

√εc x

ε

u^ε(t, x), t≥0, x∈R;

(1.1) u^ε(0, x)=g(x), x∈R,

whereaandcare stationary random fields, andcis centered.

Let us recall (see [1]) that in the periodic case the equation

∂

∂tu=1 2

∂

∂x

a x

ε ∂

∂xu

+1 εc

x ε

u

admits homogenization under the natural conditionc =0 (·stands for the mean value) and that the homogenized operator takes the form

∂

∂tu=1 2aˆ ∂²

∂x²u+ ˆcu, with constantaˆandc.ˆ

In contrast with symmetric divergence form parabolic problems, in the presence of the lower order terms the asymptotic behaviour of operators with random coefficients might differ a lot from that of periodic operators.

Homogenization problem for parabolic operators whose coefficients are periodic in spatial variables and random stationary in time, were studied in [4,5,14]. It was shown that, under natural mixing assumptions on the coefficients, the critical rate of the potential growth is of order 1/ε. In this case the limit equation is a stochastic PDE.

If the oscillating potential is random stationary (statistically homogeneous) in spatial variables then the range of the oscillations (the power ofε⁻¹in front of potentialc) should depend on the spatial dimension.

In this work we deal with a one-dimensional spatial variable and show that the range of oscillation should be of order√¹

ε. This means that for larger powers of ¹_ε the family of solutions is not tight asε→0, while for smaller powers of ¹_ε the contribution of the potential is asymptotically negligible.

It turns out that the Dirichlet forms technique which is usually quite efficient in homogenization problems, does not apply to problem (1.1) because one cannot prove any lower bound for the quadratic form corresponding to the operator (1.1). This is due to the fact that the problem is stated on the whole lineR, and not on a compact interval, and that the coefficients of the operator are a.s. unbounded, see the discussion in Section 6. Instead we use the direct approach combining the Feynman–Kac formula with several correctors, Itô calculus and martingale convergence arguments.

The main result of the paper (see Theorem 2.2) states that under proper mixing conditions the solutionu^εof eq.

(1.1) converges in law to a random field u(x, t )=E

g(x+√

˜ aB_t)exp

c

˜ a

RL^y_t^−xW (dy)

,

whereB_·andW_·are independent Brownian motions, EandL^y_t are respectively the expectation and the local time related to√

˜

aB_·, anda,˜ c¯are constants.

The interpretation of this expression is given in the last section of the paper. It is shown that the effective equation is not a standard SPDE but rather a parabolic PDE with random coefficients.

Let us give an intuitive explanation of our result. The Feynman–Kac formula for the solution of eq. (1.1) yields u^ε(t, x)=E

g

X^ε,x_t exp

1

√ε _t

0

c Xs^ε,x

ε

ds

,

where E means expectation with respect to the law of the diffusion X^ε,x_· , the random field c being frozen (or

“quenched”). Under the assumptions which we shall make below, one can apply a version of the functional central limit theorem, which tells us that

W_ε(x)= 1

√ε _x

0

c y

ε

dy

converges weakly towardscW (x), whereW is a standard Wiener process. Now the exponent in the above Feynman–

Kac formula reads _t

0

W_ε X^ε,x_s

ds=

RW_ε(y)L^y,ε_t dy,

whereL^y,ε_t denotes the local time at timet and pointx of the diffusion process{X^ε,x}. One might expect that the last integral converges towards the integral of the limiting local time, with respect to the limiting Wiener process. This is one of the results which will be established in this paper.

Our paper is organized as follows. In Section 2, we formulate our assumptions, and the results. In Section 3, we prove some weak convergence results. Section 4 is devoted to the proof of the pointwise convergence of the sequence u^ε(t, x), while Section 5 is concerned with convergence in the space of continuous functions. Finally in Section 6 we discuss the limiting PDE.

2. Set up and statement of the main result We make the following assumptions:

(A.1) The initial conditiongbelongs toL²(R)∩Cb(R).

(A.2) The coefficients{a(x), x∈R}and{c(x), x∈R}are stationary random fields defined on a probability space (Σ,A, P ), and we assume that

0< c≤a(x)≤C, x∈R, P a.s., (2.1)

Ec(0)=0, c(0)L^∞(Σ )<∞, (2.2)

_∞

−∞

Ec(0)c(x) dx <∞, (2.3)

whereEdenotes expectation with respect to the probability measureP. (A.3) Let

Fx:=σ

a(y), c(y);y≤x

, F^x:=σ

a(y), c(y);y≥x .

We assume that the random fieldsa andcareφ-mixing in the following sense. Define, forh >0,φ (h) the mixing coefficient with respect to theσ-algebras from above, as

φ (h):= sup

{A∈Fx,B∈F^x+h,P (A)>0}

P (B|A)−P (B) . We suppose that

_∞

0

φ^1/2(h)dh <∞. (2.4)

Consider now the family of Dirichlet forms{E^ε,σ, ε >0, σ∈Σ}onL²(R)defined by E^ε,σ(u, v)=1

2

Ra x

ε, σ

u(x)v(x)dx,

with domainH¹(R). For eachε >0, σ∈Σ there exists a unique self-adjoint operatorL^ε,σ with domainD(L^ε,σ), such that

Eε^σ(u, v)= −

L^ε,σu, v

L²(R), foru∈D(L^ε,σ), v∈L²(R).

For each initial pointx∈R,σ∈Σandε >0, there exists a continuous Markov process{X_t^ε,σ,x,t≥0}defined on some probability space(Ω,F,P^ε,x,σ)whose generator isL^ε,σ and which starts at timet=0 fromx. The probability may depend on the three parametersε,x,σ.x will be fixed throughout this paper, so we drop it from now on. Note that the process{X^ε,x_t , t ≥0}is in fact defined on the probability space(Σ×Ω,A⊗F, Q^ε)(as such, it is not a Markov process), where the probabilityQ^εon the product spaceΣ×Ωis defined as

Q^ε(A)=

A

P (dσ )P^ε,σ(dω).

The Feynman–Kac formula allows us to write down an explicit formula for the solution of eq. (1.1):

u^ε(t, x)=E^ε,·

g X_t^ε,x

exp 1

√ε _t

0

c X^ε,x_s

ε

ds

, (2.5)

whereE^ε,σ denotes expectation with respect toP^ε,σ.

Considering assumption (A.2), we define the finite quantities c²=

_∞

−∞E

c(0)c(x)

dx, a˜=

E 1

a(x) ₋1

. (2.6)

In view of Theorem 5.1 and Lemma 5.1 from [11], we may state the following theorem.

Theorem 2.1. We have the following convergence,P a.s.:

X^ε,x_· ⇒X^x_· :=x+X_·,

inC([0,∞)),whereXis a one-dimensional Brownian motion defined on(Ω,F,P)such thatE(X²_t)= ˜at,for any t≥0.

The main result of this paper is the following theorem.

Theorem 2.2. Let u(t, x):=E

g(x+Xt)exp c

˜ a

RL^y_t^−xW (dy)

,

whereW denotes a one-dimensional standard Brownian motion defined on the probability space(Σ,A, P )andL^y_t is the local time at timet and pointyof the process{X_t, t≥0}defined on(Ω,F,P).

Thenu^ε⇒uin law inC(R₊×R),asε→0.

We introduce the notation Y_t^ε,x:= 1

√ε _t

0

c X_s^ε,x

ε

ds.

The first step in the proof of Theorem 2.2 is to establish the weak convergence of the pair(X_t^ε,x, Y_t^ε,x), which is done in the next section.

3. Weak convergence

The main result of this section is the following theorem.

Theorem 3.1. For eacht >0, X^ε,x_t , Y_t^ε,x

⇒ X_t^x, Y_t^x weakly,asε→0,with

Y_t^x:= c

˜ a

RL^y_t^−xW (dy),

where,as above,L^y_t is the local time at pointyand timet of the Brownian motion{X_t, t≥0}defined on(Ω,F,P), and{Wy, y∈R}is a Wiener process defined on(Σ,A, P ),so that(X, L)andWare independent.

Theorem 3.1 will follow easily from Propositions 3.7 and 3.10, as we shall see at the end of this section. Note that all we shall need in the next section is both Propositions 3.7 and 3.10.

Let us first state a consequence of Aronson’s estimate, see Lemma II.1.2 in [16]:

Lemma 3.2. There existsκ >0,which depends only oncandCin(2.1),such that for allε >0,r >0, P

sup

0≤s≤t

X_s^ε,x−x > r

≤κexp −r²

κt

.

We next prove the easiest part of the above result, i.e. we give a proof of Theorem 2.1, since we shall need some of its details later.

Proof of Theorem 2.1. Let{χ (x), x∈R}be the zero mean random process given by the formula χ (x)= ˜a

_x

0

dy a(y)−x.

We note that from Birkhof’s ergodic theorem (see e.g. Theorem 24.1 in [3]), χ (x)

x →0, P a.s., as|x| → ∞. (3.1)

Moreover, this random process satisfies the two relations:

a

1+χ

(x)=0, x∈R, (3.2)

and a(x)

1+χ(x)

= ˜a, x∈R. We now define

Z^ε_t =X^ε,x_t +εχ X_t^ε,x

ε

.

It follows from the Itô–Fukushima decomposition (see [7] or Theorem 0.10 in [11]) and (3.2) that (here and further below,M^Xε,xdenotes the martingale part of the processX^ε,x)

Z^ε_t =Z^ε₀+ _t

0

1+χ

X^ε,x_s ε

dM_s^Xε,x, t≥0,

hence{Z^ε}isP a.s. aP-martingale. Moreover, its quadratic variation is given by Z^ε

t= ˜a² _t

0

ds a(X_s^ε,x/ε).

It will be proved below in Lemma 3.9 that Z^ε

t→ ˜at

inQ^ε probability. It now follows from well-known results thatP a.s., Z^ε⇒x+√

˜ aB,

where{B_t, t≥0}is a standard Brownian motion defined on the probability space(Ω,F,P).

Moreover, for allT >0, sup

0≤t≤T

X^ε,x_t −Z_t^ε →0, Q^εa.s.,

consequentlyP a.s., X^ε,x⇒x+√

˜ aB

inPlaw.

LetΦ denote the solution of the ordinary differential equation:

aΦ

(x)=c(x), which is defined as follows:

Φ(x)= 1 a(x)

_x

0

c(y)dy,

(3.3) Φ(x)=

_x

0

1 a(z)

_z

0

c(y)dy

dz.

We let

cW_ε(x):=√ εa

x ε

Φ

x ε

= 1

√ε _x

0

c y

ε

dy (3.4)

and

F_ε(x):=ε^3/2Φ x

ε

= 1

√ε _x

0

1 a(z/ε)

_z

0

c y

ε

dydz

= c _x

0

1

a(z/ε)W_ε(z)dz. (3.5)

We first prove the following proposition.

Proposition 3.3. The sequence of random processes {Wε} converges weakly, as ε→0, in the space C(R), to a standard Wiener process{W}defined on(Σ,A, P ).

Proof. Denote, forx≥0,W_ε¹(x)=W_ε(x)andW_ε²(x)=W_ε(−x). According to assumptions (A.1), (A.2), (A.3) and the functional central limit theorem (see e.g. [2], pages 178, 179), it follows that

W_ε¹, W_ε²_D

→

W¹, W² ,

where{W¹(x), x≥0}and{W²(x), x≥0}are mutually independent standard Brownian motions. Finally we denote by{W (x), x∈R}the process defined by

W (x):=W¹(x), forx≥0, W (x):=W²(−x), forx <0.

It remains to show why Theorem 3.1 follows from Theorem 2.1 and Proposition 3.3.

First we define, forx∈R, kε(x)=

_x

0

1

a(z/ε)dz, k(x)=x

˜ a. We have the following lemma.

Lemma 3.4. Asε→0, k_ε→k inC(R), P a.s.

Proof. Sincea(·)is bounded away from zero,a⁻¹(·)is bounded. Hence the collection of random functionsk_εis tight inC(R). It then suffices to show that the finite dimensional marginals converge in law to those of the deterministic functionk. But from Birkhoff’s ergodic theorem, for anyx1, x2, . . . , x_n∈R,

kε(x1), kε(x2), . . . , kε(xn)

→ x1

˜ a ,x2

˜

a , . . . ,x_n

˜ a

inP a.s., asε→0.

Denote byC₊(R)the space of continuous and increasing functions onR, and byS the map fromC(R)×C₊(R) intoC(R)defined by

S (f, h)

(x):=

_x

0

f (z)dh(z).

We have the following lemma.

Lemma 3.5. The mappingSis continuous,fromE=C(R)×C₊(R),equipped with the product of the locally uniform topology ofC(R)×C(R),intoC(R),equipped with the locally uniform topology.

Proof. It suffices to show that for eachN >0, if{(f_n, h_n)} ⊂C([−N, N])×C₊([−N, N]), and

|xsup|≤N

f_n(x)−f (x) + h_n(x)−h(x) →0, asn→ ∞,

then sup

|x|≤N

S(fn, hn)(x)−S(f, h)(x) →0, asn→ ∞.

But this follows from Lemma 5.8 in [8].

We now have the following lemma.

Lemma 3.6. Asε→0, (Wε, Fε)→(W, F )

inC(R)×C(R)in law,whereF (x)=_a^c_˜x

0 W (z)dzandW_εandF_εare defined in(3.4)and(3.5)respectively.

Proof. Sincekε converges to a deterministic limit, it follows from a well-known theorem (see e.g. Theorem 4.4 in [2]) that the pair(W_ε, k_ε)converges. Hence from Lemma 3.5, the pair(W_ε, F_ε)=(W_ε, S(W_ε, k_ε))converges.

The next step is to show that the triple (Xε, Wε, Fε)converges. This is essentially a consequence of the three following facts:X_ε converges,(W_ε, F_ε)converges, and the two limitsXand(W, F )are defined on(Ω,F,P)and (Σ,A, P )respectively. We now prove that fact rigorously.

Proposition 3.7. For anyx∈R,asε→0, X^ε,x, W_ε, F_ε

→

X^x, W, F in law,inC(R+)×C(R)×C(R).

Proof:. We first choose two arbitrary functionalsΨ ∈Cb(C(R)×C(R))andΘ∈Cb(C(R₊)). We have

Σ×Ω

Ψ

W_ε(σ ), F_ε(σ ) Θ

X^ε,x(ω, σ )

P (dσ )P^ε,σ(dω)

=

Σ

Ψ

Wε(σ ), Fε(σ )

Ω

Θ

X^ε,x(ω, σ )

P^ε,σ(dω)

P (dσ )

=

Σ

Ψ

W_ε(σ ), F_ε(σ )

Ω

Θ

X^ε,x(ω, σ )

−Θ

X^x(ω)

P^ε,σ(dω)

P (dσ ) +

Ω

Θ X^x(ω)

P^ε,σ(dω)×

Σ

Ψ

W_ε(σ ), F_ε(σ ) P (dσ ).

Theorem 2.1 tells us that

Ω

Θ

X^ε,x(ω, σ )

P^ε,σ(dω)→

Ω

Θ X^x(ω)

P(dω), P a.s., and from Lemma 3.6,

Σ

Ψ

Wε(σ ), Fε(σ )

P (dσ )→

Σ

Ψ

W (σ ), F (σ ) P (dσ ),

asε→0. Hence, from the Bounded Convergence Theorem, we conclude that

Σ×Ω

Ψ

W_ε(σ ), F_ε(σ ) Θ

X^ε,x(ω, σ )

P (dσ )P^ε,σ(dω)

→

Ω

Θ X^x(ω)

P (dω)×

Σ

Ψ

W (σ ), F (σ ) P (dσ ).

It now suffices to note thatA:= {Ψ ⊗Θ, Ψ ∈Cb(C(R)×C(R)), Θ∈Cb(C(R+))}is a determining class onC(R)×

C(R)×C(R+).

It follows from Lemma 3.8 that ε^3/2Φ

X^ε,x_t ε

=ε^3/2Φ x

ε

+ 1 2√

ε _t

0

c X^ε,x_s

ε

ds+M_t^ε,x, (3.6)

in other words Fε

X^ε,x_t

=Fε(x)+1

2Y_t^ε,x+M_t^ε,x, where{M_t^ε,x, t≥0}is the continuous martingale

M_t^ε,x=√ ε

_t

0

Φ X_s^ε,x

ε

dM_s^Xε,x,

andM^Xε,xdenotes again the martingale part of the processX^ε,x. In particular the quadratic variation ofM^ε,xis given by the quantity

M^ε,x

t =ε _t

0

a X_s^ε,x

ε

Φ X_s^ε,x

ε 2

ds

= _t

0

1 a(X^ε,x_s /ε)

√ εΦ

Xs^ε,x

ε

a X^ε,xs

ε 2

ds

=c² _t

0

1 a(X^ε,xs /ε)

Wε

X_s^ε,x2

ds, and the joint quadratic variation ofM^ε,xandZ^εis

M^ε,x, Z^ε

t=√ ε

_t

0

Φ X^ε,x_s

ε

1+χ X^ε,x_s

ε

a X^ε,x_s

ε

ds

= ˜ac _t

0

W_ε(X_s^ε,x) a(Xs^ε,x/ε)ds.

Lemma 3.8. The identity(3.6)holdsQ^εa.s.

Proof. Let{X^ε,x,M_t , t≥0}denote the process{X^ε,x_t , t≥0}, killed when exiting the interval[−ε(M+1), ε(M+1)], andL^ε,σ,M its generator. For anyM >0, the random function

Φ_M(x)=

⎧⎨

⎩ _x

0 1 a(z)

_z

0cM(y)dydz, ifx≥0;

0 x

1 a(z)

0

z c_M(y)dydz, ifx <0,

with the random field{cM(x), −M−1≤x≤M+1}defined by

c_M(x)=

⎧⎨

⎩

β_M, if−M−1< x <−M; c(x), if−M≤x≤M; α_M, ifM < x < M+1, where

αM= − _M+1

0 (1/a(z))_z∧M

0 c(y)dydz _M+1

M (z−M)/a(z)dz ; βM= −

₀

−M−1(1/a(z))₀

z∨−Mc(y)dydz ₋M

−M−1(z+M)/a(z)dz satisfies

L^ε,σΦ_Mx

ε

=ε⁻²c_Mx

ε

, Φ_M(−M−1)=Φ_M(M+1)=0.

Consequently for eachε >0, ΦM

· ε

∈D L^ε,·,M

,

and by virtue of the Itô–Fukushima decomposition (see [7] or Theorem 0.10 in [11]), we get for|x| ≤M+1 ε^3/2Φ_M

X^ε,x

t∧τ_M^ε,x

ε

=ε^3/2Φ_M x

ε

+ 1 2√

ε _t∧τ^ε,x

M

0

c_M X_s^ε,x

ε

ds+M_t∧τ^ε,xε,x M

, where

τ_M^ε,x=inf

t≥0, X^ε,x_t =ε(M+1) .

The result follows, by lettingM→ ∞, with the help of Lemma 3.2.

Define h_ε(t ):=

_t

0

1 a(X_s^ε,x/ε)ds.

Lemma 3.9. Asε→0, h_ε(·)→ ·

˜ a inC(R+)in probability.

Proof. Denoteθ (x)=_a(x)¹ −¹_a˜. Thenθ (x)is a bounded stationary field with zero mean. Letting Θ(x)=

_x

0

1 a(z)

_z

0

θ (y)dy

and repeating the argument in Lemma 3.8, we get ε^3/2Θ

X_t^ε,x ε

=ε^3/2Θ x

ε

+ 1

√ε t

0

θ X_s^ε,x

ε

ds+M^εt, (3.7)

where M^ε

t=ε _t

0

a⁻¹ X_s^ε,x

ε

_X^ε,x_{s /ε}

0

θ (y)dy 2

ds.

In the same way as in the proof of Proposition 3.7, one can show that the families{ε^3/2Θ(^Xε,x·_ε )−ε^3/2Θ(^x_ε)}and {M^ε_·}are tight inC(R). Indeed,Θis constructed fromθexactly asΦfromc. Moreover,θis, exactly asc, a stationary mixing bounded and zero mean random field. Now, multiplying the relation (3.7) by√

ε, we conclude that _·

0

θ X^ε,x_s

ε

ds→0

inC(R₊)in probability. This implies the desired convergence.

For anyf ∈Cb(R)andε >0, let us define the process{N_t^f,ε;t≥0}by N_t^f,ε=

_t

0

f (X_s^ε,x)

a(Xs^ε,x/ε)dM_s^Xε,x

= 1

˜ a

_t

0

f X^ε,x_s

dZ_s^ε. We now prove the following proposition.

Proposition 3.10. P a.s., N^f,ε, X^ε,x

⇒

N^f, X^x

inC [0, t]

×C [0, t]

, where

N_s^f =1

˜ a

_s

0

f X_r^x

dX^x_r.

Proof. Since(X^ε,x−Z^ε)converges to zero in probability, (N_t^f,ε, X^ε,x_t )behaves as ε→0 exactly as(N_t^f,ε, Z_t^ε), hence we consider the two-dimensional martingale(N_t^f,ε, Z_t^ε), and compute its associated bracket process, which takes values in the set of 2×2 symmetric matrices. We have

N^f,ε Z^ε

t

=

⎛

⎝ _t

0

f²(X^ε,x_s )

a(X^ε,xs /ε)ds a˜_t

0 f (X^ε,x_s ) a(X^ε,xs /ε)ds

˜ a_t

0 f (X^ε,x_s )

a(X^ε,x_s /ε)ds a˜²_t

0 ds a(X^ε,x_s /ε)

⎞

⎠.

Combining Theorem 2.1, Lemmas 3.5 and 3.9 we obtain that thisR⁴-valued process convergesP a.s. inPlaw towards ₁

˜ a

_t

0f²(X_s^x)ds _t

0f (X^x_s)ds _t

0f (X^x_s)ds at˜

. We then conclude thatP a.s.,

N^f,ε,, X^ε,x

⇒

N^f, X^x inPlaw, where

N_t^f =1

˜ a

_t

0

f X^x_s

dX_s^x, t≥0.

The statement below is a straightforward consequence of Birkhoff’s ergodic theorem.

Proposition 3.11. For anyN >0,x∈Randf∈C(R)the following convergence holdsP a.s.

|y−supx|≤N

y x

f (z)

a(z/ε)dz−1

˜ a

y x

f (z)dz −→0.

We now establish the version of (3.6) forε=0, which is an Itô type formula for the process{F (x+Xt), t≥0}, whereF (y):=_a^c_˜y

0 W (z)dz, y∈R. More precisely, we have the following lemma.

Lemma 3.12. For anyt≥0andx∈R,we have F

X_t^x

−F (x)=c

˜ a

t 0

W X^x_s

dXs+c 2

RL^y−x_t W (dy). (3.8)

Proof. We prove this formula by using smooth approximations of the process{W}, obtained by convolution. Letρ be aC²₀(R)function such thatρ≥0, supp(ρ)⊆ [−1,1]and

Rρ(y)dy=1. Define now ρ_n(y):=nρ(ny), W_n(y):=

Rρ_n(y−z)W (z)dz= ₁

−1

ρ(z)W

y−z n

dz. (3.9)

From the uniform continuity ofWon compacts,Wn−W_C(K)→0,P a.s., asn→ ∞, for any compact setKinR. Moreover, taking into account the fact thatW is a standard Brownian motion, we get

E W_n⁴(y)

≤C 1

−1

E

W⁴

y−z

n

dz=C 1

−1

y−z

n 2

dz≤C 1+y²

, (3.10)

fory∈R. Set Fn(·):=c

˜ a

_·

0

Wn(y)dy.

Itô’s formula applied to the process{F_n(x+X_t), t≥0}gives Fn(x+Xt)−Fn(x)=c

˜ a

_t

0

Wn(x+Xs)dXs+c 2

_t

0

W_n(x+Xs)ds. (3.11)

Recall that{x+X_t, t≥0}is a non-standard Brownian motion independent of{W}. It is easy to see that the left-hand side in the last formula tends toF (x+X_t)−F (x),P ×Pa.s., asn→ ∞. Moreover,

E×E _t

0

W_n(x+X_s)−W (x+X_s) dXs

2

= ˜aE

E _t

0

W_n(x+X_s)−W (x+X_s)2

ds

≤ ˜aE _t

0

E

W_n(x+X_s)−W (x+X_s)2 ds

→0,

sinceW_n(x+X_s)→W (x+X_s), ds×P×Pa.e., asn→ ∞, and moreover the sequence{(W_n(x+X_s))², n≥1}is ds×P ×Puniformly integrable on[0, t] ×Σ×Ω, thanks to (3.10).

Finally, from the occupation time formula for continuous semimartingales (see e.g. Corollary 1.6, page 209 in [15]), with{L^·_t}denoting the the local time of the process{X_s, 0≤s≤t},

_t

0

W_n(x+Xs)ds=

RL^y_t^−xW_n(y)dy→

RL^y_t^−xW (dy),

inL²(Σ ),P a.s., asn→ ∞(for more details see Section 5.7 in [10]). We used again the fact that the Brownian motions{X_t, t≥0}and{W (y), y∈R}are independent. Passing now to the limit in the formula (3.11) we get the

desired result.

We can finally proceed with the following proof.

Proof of Theorem 3.1. Since the mapping Λ:C(R₊)×C(R)→C(R₊) defined by

Λ(x, f )(t )=f x(t )

is continuous, if we equip the three spaces with the topology of uniform convergence on compact sets, we first con- clude from Proposition 3.7 that

Wε

X_·^ε,x

⇒W X_·^x

.

Hence from the formulas forM^ε,xt andM^ε,x, Z^εtabove, and Lemma 3.9, we deduce easily that M^ε,x

·⇒c²

˜ a

_·

0

W X^x_s

ds, M^ε,x, Z^ε

·⇒c

_·

0

W X^x_s

ds, and consequently

M_·^ε,x⇒ c

˜ a

_·

0

W X^x_s

dXs.

From Proposition 3.7, those convergences are joint with those of(X_·^ε,x, F_ε). Consequently F_ε

X_t^ε,x

−F_ε(x)−M_t^ε,x⇒F X_t^x

−F (x)−c

˜ a

_t

0

W X^x_s

dXs.

The convergenceY_t^ε,x⇒Y_t^x now follows from (3.6) and (3.8). The result finally follows from the fact that all the

above convergences are joint with that ofX_·^ε,x.

4. Pointwise convergence of the sequenceu^ε

The first part of this section is devoted to establishing uniform integrability estimates for the exponent in the Feynman–

Kac formula (Propositions 4.4 and 4.5) which are essential for the proof of the pointwise convergence part of Theo- rem 2.2, to which the second part of this section is devoted.

We first define the followingR+-valued random variables, for 0< γ <1/2,ε≥0:

ξ_{γ ,ε}=sup

x∈R

|W_ε(x)| (1+ |x|)^1−γ. We have the following lemma.

Lemma 4.1. For any0< γ <1/2andε0>0,the collection of random variables{ξγ ,ε,0< ε≤ε0}is tight.

Proof. Due to the symmetry it is sufficient to estimate|Wε(x)|forx >0. We have E W_ε(r) ²

=ε _r/ε

0

_r/ε

0

E

c(s)c(t ) dsdt

≤2ε _r/ε

0

_∞

0

Ec(0)c(s) dsdt

≤2rc0. Denote

ηt= _∞

0

E

c(s+t )|Gt

ds.

By Proposition 7.2.6. in [6] the processη_t is stationary and|η_t| ≤c1a.s. with a non-random constantc1. Moreover, _t

0

c(r)dr−ηt

is a square integrableGt martingale. Denote it byNt. Clearly, W_ε(t )=

√ε

¯ c

_{t /ε}

0

c(s)ds=

√ε

¯

c Nt /ε+

√ε

¯ c η_{t /ε}, and thus we deduce from Doob’s inequality

E

sup

0≤t≤r

Wε(t ) ²

≤ 2

¯ c²E

sup

0≤t≤r/ε

√εNt

2 +2c²₁ε

c²

≤ 4

¯ c²E√

εNr/ε

2 +2c₁²ε

c²

≤8E W_ε(r) ²

+10c₁²ε c²

≤C(ε+r),

providedC=(16c0)∨(10c²₁/c²). Now forj≥1,M >0, P

sup

2^j−1<r≤2^j

|W_ε(r)| (1+r)^1−γ ≥M

≤P

sup

0≤r≤2^j

W_ε(r) ≥

1+2^j−11−γM

≤ C(ε+2^j) M²(1+2^j−1)^2−2γ

≤(ε∨1)2C M²

1+2^j−12γ−1

. Summing up overj≥1, we deduce that

P (ξγ ,ε≥M)≤2P

sup

r>0

|W_ε(r)| (1+r)^1−γ ≥M

≤(ε∨1)4C M²

∞

j=0

1+2^j2γ−1

≤(ε∨1)C M².

The lemma is established.

Remark 4.2. We can in fact show that,asε→0, ξγ ,ε⇒ξγ:=sup

x∈R

|W (x)| (1+ |x|)^1−γ,

provided again0< γ <1/2,but we shall not use that result.

We next state a result, which is an immediate consequence of Lemma 3.2.

Lemma 4.3. There exists a continuous mapping ρ:R₊×R₊×(0,2)→R₊

such that for allc >0,t >0,ε >0, 0< p <2, E^ε,·exp

c X^ε,x_t ^p

≤ρ(c, t, p).

We next establish the following proposition.

Proposition 4.4. For any0< γ <1/2,there exists a continuous mappingΨ_γ:R₊→R₊such that E^ε,·

exp

1

√ε _t

0

c X_s^ε,x

ε

ds 2

≤Ψγ(ξγ ,ε). (4.1)

Proof. Since from (3.6) and (3.5)

√1 ε

t 0

c X^ε,xs

ε

ds=2

c X_t^ε,x

x

Wε(y)

a(y/ε)dy−M_t^ε

,

we obtain E^ε,·

exp 1

√ε _t

0

c X^ε,xs

ε

ds 2

=E^ε,·exp

4c _Xt^ε,x

x

Wε(y)

a(y/ε)dy−4M_t^ε

≤

E^ε,·exp

8c _X^ε,x_t

x

W_ε(y) a(y/ε)dy

1/2

E^ε,·exp

−8M_t^ε1/2

. (4.2)

Clearly, it suffices to estimate each factor on the r.h.s. of (4.2) separately.

E^ε,·exp

8c _X^ε,x

t

x

W_ε(y) a(y/ε)dy

≤E^ε,·exp

8c c

_X^ε,x

t

x

Wε(y) dy

≤E^ε,·exp

8c c

X^ε,x_t x

ξ_{γ ,ε}

1+ |y|1−γ

dy

≤E^ε,·exp

c ξγ ,ε

2−γ

1+ X^ε,x_t ^2−γ−

1+ |x|2−γ

≤Ψ_γ¹(ξγ ,ε),

where we have used Lemma 3.2 for the last inequality. The second factor on the r.h.s. of (4.2) can be estimated as follows

E^ε,·exp

−8Mt^ε

≤

E^ε,·exp

−16Mt^ε−128 M^ε

t

1/2

E^ε,·exp 128

M^ε

t

1/2

.

The first term on the r.h.s. does not exceed 1. For the second one we have by Jensen’s inequality E^ε,·exp

128 M^ε

t

≤E^ε,·exp

c _t

0

W_ε² X_s^ε,x

ds

≤t⁻¹ _t

0 E^ε,·exp ct W_ε²

X^ε,x_s ds

≤ sup

0≤s≤t

E^ε,·exp

ct[ξ_{γ ,ε}]²

1+ X^ε,x_s ^2−2γ

≤Ψ_γ²(ξ_{γ ,ε}),

where we have again used Lemma 3.2 for the last inequality. The result clearly follows.

Clearly, the same proof allows us to establish the slightly more general proposition.

Proposition 4.5. Let {τ^ε, ε≥0}be a collection of stopping times such that 0≤τ^ε≤t P×P a.s.Then for any 0< γ <1/2,

E^ε,·

exp 1

√ε _τ^ε

0

c X_s^ε,x

ε

ds 2

≤Ψγ(ξγ ,ε), whereΨγ:R+→R+is the mapping which appeared in(4.1).

We can now proceed with the following proof.

Proof of the pointwise convergence in Theorem 2.2. We will now show that for each(t, x)∈R₊×R,u^ε(t, x)⇒ u(t, x), asε→0. We delete the parameters t andx. It suffices to show that for anyϕ∈C(R; [0,1]),ϕ Lipschitz continuous, asε→0,

Eϕ E^ε,·

g X^ε

exp Y^ε

→Eϕ E^ε,·

g(X)exp(Y )

, (4.3)