Cut points and diffusive random walks in random environment

2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved 10.1016/S0246-0203(02)00019-5/FLA

CUT POINTS AND DIFFUSIVE RANDOM WALKS IN RANDOM ENVIRONMENT

POINTS DE COUPURE ET MARCHES ALÉATOIRES DIFFUSIVES EN MILIEU ALÉATOIRE

Erwin BOLTHAUSEN^a, Alain-Sol SZNITMAN^b, Ofer ZEITOUNI^c,∗

aInstitut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland bDepartement Mathematik, ETH-Zentrum, CH-8092 Zürich, Switzerland

cDepartments of Electrical Engineering and of Mathematics, Technion, Israel Institute of Technology, Haifa 32000, Israel

Received 1 February 2002, revised 1 July 2002

ABSTRACT. – We study in this work a special class of multidimensional random walks in random environment for which we are able to prove in a non-perturbative fashion both a law of large numbers and a functional central limit theorem. As an application we provide new examples of diffusive random walks in random environment. In particular we construct examples of diffusive walks which evolve in an environment for which the static expectation of the drift does not vanish.

2003 Éditions scientifiques et médicales Elsevier SAS

RÉSUMÉ. – On étudie dans cet article une classe de marches aléatoires en milieu aléatoire en dimension supérieure, pour lesquelles on prouve de manière non perturbative une loi des grands nombres et un théorème central limite fonctionnel. Comme application de ces résultats on construit de nouveaux exemples de marches aléatoires diffusives en milieu aléatoire. En particulier on présente des exemples de marches aléatoires diffusives qui évoluent dans un environnement aléatoire pour lequel l’espérance statique de la dérive n’est pas nulle.

2003 Éditions scientifiques et médicales Elsevier SAS

0. Introduction

Over the recent years there has been considerable interest in the study of random walks in random environment. The asymptotic behavior of this canonical model of

∗Corresponding author.

E-mail address: zeitouni@math.umn.edu (O. Zeitouni).

random motion in a random medium remains quite mysterious, especially in the multi- dimensional situation. Recent advances have mainly been concerned with the ballistic situation where the walk has a non-degenerate asymptotic velocity, see [15,12,13,16].

Diffusive behavior has remained largely unexplored, except for the the work of Lawler [7] when the walk has no local drift, and of Bricmont and Kupiainen [2], for small isotropic perturbations of the simple random walk in dimension d 3. The present article provides new examples of walks with diffusive behavior. It studies a special class of walks for which we are able to derive in a non-perturbative fashion the law of large numbers as well as central limit theorems. Interestingly, proofs are simple when compared to [2].

We now describe the setting. We consider two integersd15,d21, and writed= d1+d2. We viewZ^d1 andZ^d2as the respective subspaces ofZ^dof vectors with vanishing lastd2and vanishing firstd1components. Throughout this work we study random walks in random environment for which the Z^d1-projection evolves according to a standard random walk, and the random environment only affects theZ^d2-component. Specifically we consider a numberκ∈(0,_2d¹

1)(the ellipticity constant for theZ^d1-component) and a (2d1+1)-vector governing the jump-distribution of theZ^d1-components of the walk:

q(e)_{|e|1, e∈Z}d1, with q(e)=1, q(e)=q(−e) >0, for|e|1, e∈Z^d1,

andq(e)κ, fore=0, (0.1)

and introduce

Pq(·)the set of(2d)-vectorsp(e)_|e|=1, withp(e)∈ [0,1], for alle∈Z^d,|e| =1,

|e|=1p(e)=1, andp(e)=q(e), fore∈Z^d1,|e| =1. (0.2) The random environment is an element ω=(ω(x,·))_x∈Z^d of=Pq(^Zd·), endowed with the product σ-algebra and the product measure P=µ^⊗Zd, whereµis a probability on Pq(·)governing the distribution of the environment at a single site. The random walk in the random environmentωis the canonical Markov chain(Xn)n0on(Z^d)^Nwith state spaceZ^d, and “quenched” lawPx,ωstarting fromx∈Z^d, under which

Px,ω[Xn+1=Xn+e|X0, . . . , Xn]^Px,ω-a.s.

= ω(Xn, e), n0, |e| =1, Px,ω[X₀=x] =1.

(0.3)

The annealed laws are then defined as the semi-direct products on×(Z^d)^N:

Px=P×Px,ω, forx∈Z^d. (0.4)

Our very choice of environmentsωinforces theZ^d1-projection ofXnto evolve under Px,ω as a random walk with jump distribution q(·). We assume symmetry ofq(·) for otherwise we would be in a non-nestling situation where the law of large numbers and the central limit theorem have been proven in [15,12]. The assumption d15, enables to exploit the presence of cut times of the random walk, where loosely speaking past and future of the random walk have no intersection, (for the precise definition, see (1.4)).

These cut times play a somewhat similar role to the regeneration times employed in [15, 12], although they do not provide a renewal structure.

In the above setting we are able to derive a law of large numbers:

P0-a.s., Xn

n →v (with a deterministicv). (0.5) Further assuming that either the law of the environment is invariant under the antipodal transformation (cf. (2.1), in this case v = 0), and d1 7, or without symmetry assumption that d1 13, we obtain a functional central limit theorem under the quenched measure:

P-a.s., underP0,ω, the Skorohod-space valuedB.^n=√1n

X_[·n]− [·n]v

converges in law to a Brownian motion with deterministic covariance. (0.6) One can of course replace the quenched measure by the annealed measureP0in (0.6).

The above result in particular provides examples of diffusive behavior beyond current knowledge. It can also be applied to certain small perturbations of the standard random walk. Forε∈(0,1), following [14], we define

Sε=the set of(2d)-vectorsp(e)_|e|=1, withp(e)−_2d¹_4d^ε, for alle,

and_ep(e)=1, (0.7)

and writed(x, ω)for the local drift:

d(x, ω)=

e

ω(x, e)e. (0.8)

It is shown in [14], that forη >0, and smallεdepending ond andη, when the single site distribution is concentrated onSε, and the static expectation of the local driftE[d(0, ω)] has size bigger thanε^5/2−η, whend=3,ε^3−η, whend4, the walk has a non-vanishing limiting velocity (much more is known, see [14]). One can wonder whether the same remains true for arbitrarily small non-vanishing E[d(0, ω)]. We show here that this is not the case and provide examples whend7, of single site distributions concentrated on Sε, for arbitrarily smallε, with E[d(0, ω)] =0, but vanishing limiting velocity v, and even with diffusive behavior, whend 15. We also construct further examples of analogous behavior for walks which are not small perturbations of the simple random walk.

Let us now explain how this article is organized.

In Section 1, we provide an alternative representation of the law of the walk under the annealed measure which takes advantage of the cut times. We then derive the law of large numbers in Theorem 1.4.

In Section 2, we prove the functional central limit theorem under the annealed measure. The case with antipodal symmetry and d17 is covered by Theorem 2.1, the general case withd113, is treated in Theorem 2.2.

Section 3 explains how the functional central limit theorem under the annealed measure can be strengthened to a similar statement under the quenched measure.

Section 4 provides examples of walks which are small perturbations of the simple random walk, for whichE[d(0, ω)] =0, but the limiting velocity vanishes,(d7), and which behave diffusively,(d15).

Section 5 contains further examples of analogous behavior, which in a certain sense are small perturbations of a one-dimensional random walk in a random environment.

1. An alternative representation ofP₀and a law of large numbers

In this section we first introduce some further notations and provide a special representation of the walk under the measureP0, see Proposition 1.2. This representation will provide an easy comparison of the walk underP0with a process constructed as an additive functional over a probability space with an ergodic shift. This will lead to a law of large numbers, cf. Theorem 1.4.

We begin with some notations. We denote by(ei)1idthe canonical basis ofR^d, and by| · |the Euclidean distance onR^d. ForU a subset ofZ^d,|U|denotes the cardinality ofU and∂U the boundary ofU:∂U= {x∈Z^d\U,∃y∈U,|x−y| =1}. The drift will be theR^d-valued function onPq(·):

d(p)=

|e|=1

p(e)e=

i>d1

p(ei)−p(−ei)ei, forp(·)∈Pq(·). (1.1)

To represent the random walk governing the evolution of the Z^d1-projection of the RWRE, we consider the product space

W_∗=e∈Z^d1, |e|1^Z,

endowed with the product σ-algebra W∗ and the product measure P =q^⊗Z, (in the notation of (0.1)). We denote by (θn)n∈Z the canonical shift onW_∗ and by(In)n∈Z the canonical coordinates. We then define, forw∈W_∗,

X¹_n=X_n¹(w)=





I1+ · · · +In, n1,

0, n=0,

−(In+1+ · · · +I0), n−1.

(1.2) Observe thatX¹_n, n0, andX¹_n, n0, are two independent random walks onZ^d1 with jump-distribution q, and that

X_n¹◦θk=X_n¹_+k−X_k¹, n, k∈Z. (1.3) The set of cut times where “future” and “past” ofX¹. have no intersection will play an important role in this article. Specifically, forw∈W_∗, we consider

D(w)=n∈Z, X¹_{(−∞,n−1]}(w)∩X_[¹_n,∞)(w)= ∅, (1.4) as well as the stationary point process

N (w, dk )=

n∈Z

δn(dk)1n∈D(w). (1.5)

It will be convenient to restrict P to the shift-invariant set of full P-measure (cf.

Lemma 1.1 below)

W =w∈W_∗, Nw, (−∞,0]=Nw,[0,∞)= ∞. (1.6) We will write W for the restriction of W∗ to W. We collect some useful properties relative to the point processN in the following

LEMMA 1.1. –

P (0∈D) >0. (1.7)

P (W )=1, and onW, N (w, dk)=

m∈Z

δT^m(w)(dk), (1.8) where T^m(w), m∈Z are Z-valued variables on W, increasing with m and such that T⁰0< T¹.

P^def=P[· |0∈D]is invariant underθ^def=θ_T¹. (1.9)

T¹dP=P[0∈D]⁻¹. (1.10)

f dP=

T¹−1 0

f ◦θkdP T¹dP , (1.11)

forf bounded measurable onW.

PT¹> nc(logn)^1+d1−24n^−(d1−24), n1, (1.12) for a positive constantcdepending only ond1andq(·).

Proof. – The claim (1.7) follows from the fact that X_n¹, n0, and X₋¹_n, n0, are independent random walks on Z^d1, d15, with jump distribution q(·) using classical estimates on the decrease of the transition probability, cf. Spitzer [11], p. 75, and similar arguments as in Section 3.2 of Lawler [8] or Section 4 of Erdös and Taylor [5]. Using the ergodicity of θ and (1.7), P (W )=1 follows and (1.8) is straightforward. Up to a different normalizationPis the Palm measure attached to the stationary point processN, cf. Neveu [9], Chapter II, (see in particular (10), p. 317). The statements (1.9), (1.10) (Kac’s lemma), and (1.11) are then standard. We now turn to the proof of (1.12).

We consider an integerL1, and write:

kj=1+Lj, forj0. (1.13)

Then forJ 1:

PT¹> k2J

=PNw,[1, k2J]=0

0j <2J+1

PX₍¹_{−∞,kj−1]}∩X_[¹_kj+1,∞)= ∅ +Pfor all 0j <2J +1, X¹_(−∞,k

j−1]∩X¹_[k

j+1,∞)= ∅, andNw,[1, k2J]=0

def=a1+a2. (1.14) We first bound a2. To this end note that when N (w,[1, k2J])=0 and X₍¹_{−∞,kj−1]}∩ X¹_[k

j+1,∞)= ∅for 0j <2J +1, then for any 1j2J,

∅ =X₍¹_{−∞,kj−1]}∩X_[¹_kj,∞)=X_[¹_{kj−1,kj−1]}∩X_[kj,kj+1−1]. Hence using independence, we see that

a2PX_[−¹ _L,−1]∩X¹_[0,L−1]= ∅^J P[0∈/D]^J. (1.15) We now turn to the control ofa1. We observe that

a1(2J +1)PX₍¹_−∞,−1]∩X¹_[L,∞)= ∅

(2J +1)

i1,jL

PX¹_i+j=0(2J +1)

kL

kPX_k¹=0

(2J +1)constL^−(d1−24), (1.16)

using [11], p. 75, in the last step. Choosing a large enoughγ depending ond1,q(·), and settingJ = [γlogn],L= [_3Jⁿ], (1.12) now follows from (1.15), (1.16). ✷

We will now provide an alternative representation of the law of the walk under the annealed measureP0. We letW=(Z^d2)^Nstand for the space ofZ^d2-valued trajectories (w(k)) k0and

I(w)=k0, X_k¹(w)=X¹_k+1(w), forw∈W, (1.17) denote the non-negative idle times ofX¹. We specify a probability kernelK(w, dwdω) fromW toW×through:



























ωisP-distributed,

w(0)=0,

for anyk0, conditionally onω,w(0), . . . , w(k),

w(k+1)−w(k) equals 0, whenkT¹ork /∈I(w), e, with probability ^ω(X

1

k+^{w(k), e)}

q(0) , for any e= ±ei, i > d1, ifk < T¹andk∈I(w).

(1.18)

We can then consider the spaces

.0=W ×(W×)^N and .s=W×(W×)^Z, (1.19) endowed with their productσ-fields, (the subscript “0” refers toP0and the subscript “s” to stationary) and the probabilities

Q0=P ×M0, Qs=P ×Ms, (1.20)

where M0 and Ms stand for the respective kernels from W to (W×)^N and W to (W×)^Zdefined by

M0(w, dγ0)=K(w, dw0dω0)⊗

m1

K(θT^mw, dwmdωm), (1.21) (withγ₀=(w, γ0)=(w, (wm, ωm)m0)), and similarly

Ms(w, dγs)=

m∈Z

K(θT^mw, dwmdωm), (1.22) (with γ_s =(w, γs)=(w, (wm, ωm)m∈Z)). We will shortly see that (.0, Q0) is helpful in providing a representation of X. ^{under P}0, whereas (.s, Qs) possesses useful stationarity properties.

We now define on.0theZ^d2-valued processX_k²,k0, via X²₀=0, X²_k=w0(k), for 0kT¹,

X²_(Tm+k)∧T^m+1 =X²_Tm+wm

k∧T^m+1−T^m form1, k0, (1.23) in the notations of (1.21). In the sequel we will especially be interested in theZ^d-valued process defined on.0:

Zk=X¹_k+X_k², k0, (1.24)

and by thePq(·)-valued process (see (0.2)):

σk=ω0(Zk,·), when 0kT¹,

ωm(Zk−ZT^m,·), whenT^mk < T^m+1, m1. (1.25) The above processes will easily be compared with the processes defined on.s:

Z_k^s=X_k¹+X^2,s_k , k∈Z, (1.26)

σ_k^s=ωm

Z_k^s−Z_T^sm,·, forT^mk < T^m+1, (1.27) in the notations of (1.22), with

X^2,s₀ =0 and X^2,s_(Tm+k)∧T^m+1 =X_T^2,sm+wm

k∧T^m+1−T^m, form∈Z, k0.

(1.28) The next two propositions clarify the interest of the above objects.

PROPOSITION 1.2. – UnderQ0,(Zk, σk)k0has the same law as(Xk, ω(Xk,·))k0

underP0.

Proof. – For ω∈, the Z^d1-projection of X. ^{under P}0,ω has same law as (X¹_k)k0

underP. Further forω∈ifYk, k0, is aZ^d2-valued process such thatY0=0 and for k0, conditionally onX.^1,Y0, . . . , Yk, the incrementYk+1−Yk is









0, whenk /∈I(w),

takes the valueewith probability ω(X¹_k+Yk, e)

q(0) ,fore= ±ei, i > d₁, whenk∈I(w),

(1.29)

then

X¹_k+Yk, ωX_k¹+Yk,·_k0is distributed asXk, ω(Xk,·)_k0underP0,ω. (1.30) Letting (ω(x,·))_x∈Z^d be i.i.d. µ-distributed (see below (0.2)), and replacing P0,ω with P0 the above identity of laws holds true as well. But the subsets of Z^d1: X_[¹_0,T1−1], X¹_[T1,T²−1], . . . , X_[¹_Tm,T^m+1−1], . . .are disjoint. Hence if(ωm)m0is an i.i.d. sequence with common distribution P, and one replaces in (1.29), and in the first expression of (1.30) ωwithω0, if 0k < T¹andωm(· −(X_T¹m+YT^m),·), ifT^mk < T^m+1, the identity in law is still preserved. Our claim now follows straightforwardly. ✷

To take advantage of the stationarity property on (.s, Qs), we introduce on .s the flow(4k)k∈Zvia:

4k(γ)=θkw, (wn+m, ωn+m)m∈Z

, onTn(w)k < Tn+1(w), (1.31) withγ as below (1.22). This is the natural flow extending(θk)k∈Z, if one views(wm, ωm) as marks of theδT^m, form∈Z.

PROPOSITION 1.3. – Z^s_n=

n−1

k=0

Z₁^s◦4k, forn1, (1.32)

σ_n^s=σ₀^s◦4n, forn∈Z, (1.33)

41preservesQs and in fact(.s, 41, Qs)is ergodic. (1.34) Proof. – Both (1.32) and (1.33) follow by direct inspection using (1.26)–(1.28). The fact 41 preserves Qs is checked by a straightforward calculation. Let us show the ergodicity of(.s, 41, Qs). The Palm measure

Qs

def=Qs(· |0∈D)=P×Ms (1.35) attached to the stationary point process N preserves

4 =4_T¹ (1.36)

(see Neveu [9], p. 338), and the analogue of (1.11) with4,Qs,Qs in place ofθ, P ,P and f bounded measurable holds as well. Our claim is equivalent to the ergodicity of (.s∩ {0∈D},4, Qs). LetAbe measurable subset of .s ∩ {0∈D}invariant under4 and ε >0. We can find an integer mε 1 and a measurable subset Aε depending only onw,(wm, ωm)_|m|m_ε, such that:

E^Qs|1A−1A^ε|ε. (1.37)

Then forL0,

Qs(A)=E^Qs[1A1A◦4L] =E^Qs[1Aε1Aε◦4L] +cε, (1.38)

with |cε|2ε. On the other hand if L >2mε, conditioning on the w component and using the fact that the(wm, ωm)m∈Zare independent conditionally onw(see (1.22)), the above equals

E^P Qs(Aε|w)Qs(Aε|w)◦θL

+cε. As a result

2εlimN→∞

Qs(A)− 1 N

N−1 L=0

E^P Qs(Aε|w)Qs(Aε|w)◦θL

, (1.39)

but(W ∩ {0∈D},θ , P ) is ergodic as a consequence of the ergodicity of(W, θ, P )and

1 N

_N−1

0 Qs(Aε|w)◦θL L¹(P )

−→Qs(Aε). We thus find with (1.37) and the above that Qs(A)−Qs(A)² Qs(A)−Qs(Aε)²+2ε4ε.

Lettingεtend to 0, we see thatQs(A)=0 or 1, and our claim follows. ✷

We will now apply the above to the derivation of a law of large numbers. In particular this will prove the existence of a (possibly vanishing) asymptotic velocity for the walk under the annealed measure P0, when the single site distribution µ is concentrated on Pq(·), (see (0.1), (0.2), withd15,d21).

THEOREM 1.4. – Let7be a bounded measurable function onPq(·), then P0-a.s., 1

n

n−1

k=0

7ω(Xk,·)−→

n→∞E^Qs7σ₀^s, (1.40) and moreover in the notation of (1.1),

P0-a.s., Xn

n →v^def=E^Qsdσ₀^s=E^QsZ^s₁. (1.41) Proof. – In view of Proposition 1.2, it suffices to prove similar statements with(Zk)k0

and(σk)k0in place of(Xk)k0and(ω(Xk,·))k0.

In the notations of (1.19), we consider the kernelMfromW to(W×)×(W×)^Z: M(w, dγ )=K(w, dw₀dω₀)⊗

m∈Z

K(θT^mw, dwmdωm), (1.42) for γ = (w, γ )=(w, (w₀, ω₀), (wm, ωm)m∈Z), and the probability Q on the space .=W ×(W×)×(W×)^Zdefined as the semi-direct productQ=P ×M. Then the applications

γ ∈.−→⁸⁰ γ₀=w, (w₀, ω₀), (wm, ωm)m1

∈.0

γ ∈.−→^8s γ_s=w, (wm, ωm)m∈Z

∈.s,

respectively mapQontoQ0andQs. Moreover with a slight abuse of notations, we see that

Q-a.s., ZT¹+k−ZT¹=Z^s_T1+k−Z_T^s1, σk+T¹=σ_k^s_+T1, k0. (1.43) As a result we find that for7as in (1.40)

Q-a.s., 1 n

n−1

k=0

7(σk)−1 n

n−1

k=0

7σ_k^s→0. (1.44)

In view of Proposition 1.3 we can apply the ergodic theorem to the second expression in (1.44), and (1.40) follows. By (1.43), we also see that

Q-a.s., Zn−Z_n^s2T¹∧n, (1.45) and from Proposition 1.3 and the ergodic theorem we conclude that

P0-a.s., Xn

n →E^QsZ₁^s. (1.46)

Moreover by a martingale argument (underP0,ω), E0[Xn] =E0

_n−1

k=0

dω(Xk,·)

, (1.47)

and by (1.40) we now conclude that

E^QsZ₁^s=E^Qsdσ₀^s, finishing the proof of Theorem 1.4. ✷

2. Central limit theorem under the annealed measure

In the setting of the previous sections, we now present two central limit theorems for the walk under the measure P0. Theorem 2.1 requires a symmetry assumption on the law of the environment, cf. (2.1) below, and holds whend17, on the other hand Theorem 2.2 makes no symmetry assumption, but holds whend113. We will later use Theorem 2.2 when providing in Sections 4 and 5 examples of diffusive behavior of the walk in biased environments.

For the first theorem, we assume the following “antipodal symmetry” of the single site distribution (see below (0.2))

µis invariant underp(e)_|e|=1→p(−e)_|e|=1. (2.1) Note that when (2.1) holds, E0[Xn] = 0, for n0, and the limiting velocity v in (1.41) necessarily vanishes. In what follows we denote by D(R₊,R^d) the set ofR^d- valued functions on R+, which are right continuous with left limits, which is tacitly

endowed with the Skorohod topology and its Borel σ-algebra, (cf. Chapter 3 of Ethier and Kurtz [6]).

THEOREM 2.1 (d17, under (2.1)). – UnderP0, the D(R+,R^d)-valued sequence B.^{n= √1}nX_[·n] converges in law to a Brownian motion with covariance matrixA given in (2.14).

Proof. – In view of Proposition 1.2 and (1.45), it suffices to show that underQs, √¹

nZ^s_[·n]converges in law to a Brownian motion with

covariance matrixA. (2.2)

Define the non-decreasing sequence kn, n0, Qs-a.s. surely tending to infinity such thatT^knn < T^kn+1, and

:m=Z_T^sm−Z_T^s0, form0. (2.3) Note thatQs-a.s., for anyT >0:

sup

tT

1

√nZ^s_{[t n]}− 1

√n:k_[tn]

2 sup

0kk_[T n]

(T^k+1−T^k)

√n . (2.4)

From (1.12) andd17, we see that forγ <³₂,

E^PT^1γ<∞ (2.5)

and using (1.11) we conclude that forγ < ⁵₂,

E^PT^1γ=E^QsT^1γ<∞. (2.6) Using stationarity, we see that foru >0,

P

sup

0k[T n]

(T^k+1−T^k)

√n > u

(T n+1)PT¹>√ nu (T n+1)

n E^PT¹², T¹>√

nu−→^(2.6)

n→∞0.

On the other hand sup_{0k[T n]}^(Tk+√1−_n^Tk) is invariant underθ_T0, and by (1.11) the image ofP underθ_T⁰ isT¹P / T¹dP, so that the above calculation also proves that

sup

0k[T n]

(T^k+1−T^k)

√n _n−→_→∞0 inP (orQs)-probability. (2.7) SinceQs-a.s.,knnfor alln, we see from (2.4), (2.7) that our claim will follow if we show (2.2) with ^√1_{nk[t n]}in place of ^√1_nZ_[^s_{t n]}.

Observe then that conditionally onw, underQs, the variablesZ_T^sk+1−Z^s_Tk,k0, are independent, cf. (1.22), (1.26), (1.28), with zero mean thanks to (2.1). Further from the ergodic theorem:

Qs-a.s., 1 n

0k<n

Z_T^sk+1−Z_T^sk

Z^s_Tk+1 −Z_T^sk

t

→E^QsZ_T^s1

Z_T^s1

tdef

=A. (2.8) Using the martingale central limit theorem, see Durrett [4], p. 374, or Ethier and Kurtz [6], p. 340, it follows from (2.6), (2.8) that

forP-a.e.w, conditionally onwunderQs, ^√1_n:_·nconverges in law to a Brownian motion with covariance matrixA, provided :s, s0, stands for the linear interpolation of:m, m0.

(2.9)

Noting that ^√1_n:_·n is invariant under 4_T⁰ and the image of Qs under 4_T⁰ is T¹Qs/ T¹dP, it follows that

underQs,^√1_n:_·nconverges in law to a Brownian motion with

covariance matrixA. (2.10)

From the ergodic theorem, we know that T^m

m →E^PT¹ Qs-a.s., (2.11)

and by similar arguments as above the same holds true Qs-a.s. It then follows thatQs- a.s. ^k_nⁿ →1/T¹dP, and with the help of Dini’s theorem:

Qs-a.s., for allT >0, sup

0tT

k_[t n]

n − t

E^P[T¹]

=0. (2.12)

From (2.10) and (2.12), we then conclude that

underQs, ^√1_n:k_[·n]converges in law to a Brownian motion (2.13) with covariance matrix

A=E^QsZ_T^s1

Z_T^s1

t

/E^PT¹ =A/E ^PT¹, (2.14) which finishes the proof of our claim. ✷

We now turn to the second theorem which does not require the symmetry assump- tion (2.1), and covers situations with possibly non-vanishing limiting velocity v, see (1.41).

THEOREM 2.2 (d₁ 13). – Under P₀, the D(R+,R^d)-valued sequence B.^{n =}

√1

n(X_[·n]− [·n]v) converges in law to a Brownian motion with covariance matrix A given in (2.20).

Proof. – By Proposition 1.2 and (1.45), it suffices to prove a similar result for the sequence

√1 n

Z_[·^s_n]− [·n]v^(1.32)=^−(1.41) 1

√n

[·n]−1 k=0

Y·4k, (2.15)

with the notation

Y =Z₁^s−E^QsZ₁^s. (2.16) We now introduce on.s, see (1.19), the filtration

Gk=σZ_n^s₊₁−Z_n^s, n < k, fork0,

=σZ_n^s, nk, sinceZ^s₀=0. ✷ (2.17) The main step in proving Theorem 2.2 is provided by an adaptation of Gordin’s method:

LEMMA 2.3. – There is aG∈L²(.s,G0, Qs)such that Mn

def=G◦4n−G+Z_n^s−nv=G◦4n−G+

n−1

k=0

Y◦4kis a(Gn)-martingale. (2.18) Let us for the time being admit Lemma 2.3 and explain how we conclude the proof of Theorem 2.2. Observe that for anyε >0:

Qs

sup

1mn

|G·4m|> ε√

nnQs

|G|> ε√ n ε⁻²E^QsG²,|G|> ε√

n_n−→_→∞0, (2.19) so that it suffices to prove that ^√1_nM_[·n] converges in law to conclude that ^√1_n(Z^s_[·n]− [·n]v)converges in law to the same limit. However

Mn=

n−1

k=0

(G◦41−G+Y )◦4k

is a martingale with stationary increments and from the theorem of Billingsley and Ibragimov, see Durrett [4], p. 375, it follows that

underQs, ^√1_nM_[·n]converges in law to a Brownian motion with

covariance matrixA=E^Qs(G◦41−G+Y )(G·41−G+Y )^t, (2.20) which proves Theorem 2.2.

Proof of Lemma 2.3. – To simplify notations, we drop the superscriptQswhen writing expectations or conditional expectations. It follows from (1.12) that

T¹∈L⁴(Qs) orL⁴(P ). (2.21)