Diffusions in random environment and ballistic behavior

Diffusions in random environment and ballistic behavior

Tom Schmitz

Department of Mathematics, ETH Zurich, CH-8092 Zurich, Switzerland Received 11 October 2004; received in revised form 26 June 2005; accepted 30 August 2005

Available online 29 March 2006

Abstract

In this article we investigate the ballistic behavior of diffusions in random environment. We introduce conditions in the spirit of (T )and(T)of the discrete setting, cf. [A.-S. Sznitman, On a class of transient random walks in random environment, Ann. Probab.

29 (2) (2001) 723–764; A.-S. Sznitman, An effective criterion for ballistic behavior of random walks in random environment, Probab. Theory Related Fields 122 (4) (2002) 509–544], that imply, whend2, a law of large numbers with non-vanishing limiting velocity (which we refer to as ‘ballistic behavior’) and a central limit theorem with non-degenerate covariance matrix.

As an application of our results, we consider the class of diffusions where the diffusion matrix is the identity, and give a concrete criterion on the drift term under which the diffusion in random environment exhibits ballistic behavior. This criterion provides examples of diffusions in random environment with ballistic behavior, beyond what was previously known.

©2006 Elsevier Masson SAS. All rights reserved.

Résumé

On étudie dans cet article le comportement balistique de diffusions en milieu aléatoire. On montre que certaines conditions(T ) et(T), d’abord introduites dans le cadre discret, cf. [A.-S. Sznitman, On a class of transient random walks in random environment, Ann. Probab. 29 (2) (2001) 723–764 ; A.-S. Sznitman, An effective criterion for ballistic behavior of random walks in random environment, Probab. Theory Related Fields 122 (4) (2002) 509–544], entraînent en dimension supérieure une loi des grands nombres avec une vitesse limite non nulle (ce qu’on appelle « comportement balistique »), et un théorème limite central avec une matrice de covariance non dégénérée. Pour illustrer ces résultats, on considère la classe de diffusions où la matrice de diffusion est l’identité, et on donne un critère concret sur la dérive qui entraîne le comportement balistique de la diffusion en milieu aléatoire.

Ce critère fournit de nouveaux exemples de diffusions en milieu aléatoire avec comportement balistique.

©2006 Elsevier Masson SAS. All rights reserved.

Keywords:Diffusions in random environment; Ballistic behavior; Condition (T)

1. Introduction

The method of “the environment viewed from the particle” has played a prominent role in the investigation of random motions in random environment, see for instance [12,18,21,23–26]. In the continuous space–time setting, it applies successfully when one can construct, most often explicitly, an invariant measure for the process of the environment viewed from the particle, which is absolutely continuous with respect to the static measure of the random

E-mail address:schmitz@math.ethz.ch (T. Schmitz).

0246-0203/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpb.2005.08.003

medium, see [7,13–17,19,22–25]. However, the existence of such invariant measures is hard to prove in the general setting. The case of Brownian motion with a random drift which is either incompressible or the gradient of a stationary function, is tractable, see [23,24]. But many examples fall outside this framework, and only recent developments go beyond it, for they require new techniques, see [13–15,17].

Progress has recently been made in the discrete setting for random walks in random environment in higher dimen- sions, in particular with the help of the renewal-type arguments introduced in Sznitman and Zerner [36], see [3–6, 31–35,37]. It is natural, but not straightforward, to try to transpose these results to the continuous space–time setting, and thus propose a new approach to multidimensional diffusions in random environment, when no invariant measure is a priori known. The first step in this direction was taken up in Shen [28], where, in the spirit of Sznitman and Zerner [36], certain regeneration times providing a renewal structure are introduced. Then a sufficient condition for a ’bal- listic’ strong law of large numbers (‘ballistic’ means that the limiting velocity does not vanish, which we refer to as ballistic behavior) and a central limit theorem governing corrections to the law of large numbers, with non-degenerate covariance matrix, is given in terms of these regeneration times.

In this article we show that under condition (T), see (1.12) for the definition, whend2, the diffusion in ran- dom environment satisfies the aforementioned sufficient condition of Shen [28]. We formulate the rather geometric condition(T)and are able to restate it equivalently in terms of the renewal structure of Shen [28], see Theorem 3.1.

With(T)we are then able to derive tail estimates on the first regeneration time which in particular imply the above mentioned sufficient condition of Shen [28], see Theorem 4.5. In the discrete i.i.d. setting, condition (T) was intro- duced in the work of Sznitman, see [32] and [33], and some of our arguments are inspired by [32] and [33]. As an application of our methods, we give concrete examples. In particular, we recover and extend results of Komorowski and Krupa [15].

Before describing our results in more details, let us recall the setting.

The random environment is described by a probability space(Ω,A,P). We assume that there exists a group {t_x: x∈R^d}of transformations onΩ, jointly measurable inx,ω, which preserve the probabilityP:

t_xP=P. (1.1)

On(Ω,A,P)we consider bounded measurable functionsb(·):Ω→R^dandσ (·):Ω→R^d×d, as well as two finite constantsb,¯ σ >¯ 0 such that for allω∈Ω

b(ω)b,¯ σ (ω)σ ,¯ (1.2) where| · |denotes the Euclidean norm for vectors resp. for square matrices. We write

b(x, ω)=b tx(ω)

, σ (x, ω)=σ tx(ω)

.

We further assume thatb(·, ω)andσ (·, ω)are Lipschitz continuous, i.e. there is a constantK >0 such that for all ω∈Ω,x, y∈R^d,

b(x, ω)−b(y, ω)+σ (x, ω)−σ (y, ω)K|x−y|. (1.3)

σ σ^t(x, ω)is uniformly elliptic, i.e. there is a constantν >0 such that for allω∈Ω,x, y∈R^d, 1

ν|y|²σ^t(x, ω)y²ν|y|², (1.4)

where σ^t denotes the transposed matrix of σ. For a Borel subset F ⊂R^d, we define the σ-field generated by b(x, ω), σ (x, ω), forx∈F by

HF def=σ

b(x,·), σ (x,·): x∈F

, (1.5)

and assume finite range dependence: there is anR >0 such that for all Borel subsetsF, F⊂R^d withd(F, F)^def= inf{|x−x|: x∈F, x∈F}> R,

HF andHF areP-independent. (1.6)

We denote by(C(R₊,R^d),F, W )the canonical Wiener space, and with(B_t)_t0thed-dimensional Brownian motion (which is independent from(Ω,A,P)). The diffusion process in the random environmentωis described by the family

of laws(Px,ω)_x∈Rd (we call them thequenchedlaws) on(C(R₊,R^d),F)of the solution of the stochastic differential equation

dXt=σ (X_t, ω)dBt+b(X_t, ω)dt,

X₀=x, x∈R^d, ω∈Ω. (1.7)

The second order linear differential operator associated to the stochastic differential equation (1.7) is given by:

Lω def= 1

2 d i,j=1

a_ij(x, ω) ∂²

∂x_i∂x_j + d j=1

b_j(x, ω) ∂

∂x_j. (1.8)

To restore some stationarity to the problem, it is convenient to introduce theannealedlawsP_x, which are defined as the semi-direct products:

P_x^def=P×P_x,ω, forx∈R^d. (1.9)

Of course the Markov property is typically lost under the annealed laws.

Let us now explain the purpose of this work. The main object is to introduce sufficient conditions for ballistic behavior of the diffusion in random environment whend 2. These conditions are expressed in terms of another condition(T )_γ which is defined as follows. Consider, for|l| =1 a unit vector ofR^d,b, L >0, the slabs

U_l,b,L^def=

x∈R^d: −bL < x·l < L .

We say thatcondition(T )_γ holds relative tol∈S^d−1, in shorthand notation(T )_γ|l, if for alll∈S^d−1in a neighbor- hood ofl, and for allb >0,

lim sup

L→∞ L^−γlogP0[XT_U

l,b,L·l<0]<0, (1.10)

whereT_U

l,b,Ldenotes the exit time ofX_·out of the slabU_l,b,L, see (2.1) for the definition.

The aforementioned sufficient conditions for ballistic behavior are then condition(T )relative to the directionl, in shorthand notation(T )|l, which refers to the case where

(1.10)holds forγ=1, (1.11)

or the weaker condition(T)relative to the directionl, in shorthand notation(T)|l, which refers to the case where

(1.10)holds for allγ∈(0,1). (1.12)

Clearly(T )implies(T)which itself implies(T )_γ for allγ∈(0,1). We expect these conditions all to be equivalent, cf. Sznitman [33,35], however this remains an open question. The conditions(T )and(T)are not effective conditions which can be checked by direct inspection of the environment restricted to a bounded domain ofR^d. In the discrete i.i.d. setting, Sznitman [33] proved the equivalence between a certain effective criterion and condition(T). With the help of the effective criterion he also proved that(T )_γand(T)are equivalent for¹₂< γ <1. We believe that a similar effective criterion holds in the continuous setting, and it is in the spirit of this belief that we formulate all our results in Section 3 and 4 in terms of condition(T)resp.(T )_γ. Later, in Section 5, we verify the stronger condition (T) for a large class of examples.

In Theorem 3.1 we show that the definition of condition(T )_γ|l, see (1.10), which is of a rather geometric nature, has an equivalent formulation in terms of transience of the diffusion in directionland a stretched exponential control of the size of the trajectory up to the first regeneration timeτ₁(see Subsection 2.3 for the precise definition):

P₀-a.s. lim

t→∞X_t·l= ∞, (1.13)

and for someμ >0, E0

exp

μ sup

0tτ1

|Xt|^γ

<∞. (1.14)

Following Shen [28], the successive regeneration timesτ_k,k1, are defined on an enlarged probability space which is obtained by adding some suitable auxiliary i.i.d. Bernoulli variables, cf. Subsection 2.3. The quenched measure on the enlarged space, which couples the trajectories to the Bernoulli variables, is denoted byP_x,ω, andP_xrefers to the

annealed measureP×Px,ω, cf. Subsection 2.2. Loosely speaking, the first regeneration timeτ1is the first integer time where the diffusion process in random environment reaches a local maximum in a given directionl∈S^d−1, some auxiliary Bernoulli variable takes value one, and from then on the diffusion process never backtracks.

The strategy of the proof of the above mentioned equivalence statement is similar to that of the analogue statement in the discrete i.i.d. setting, see Sznitman [33]. Nevertheless, changes appear in several places, due among others to the fact that the regeneration timeτ₁is more complicated than in the discrete setting.

Theorem 3.1 is very useful because conditions (1.10) and (1.13), (1.14) have different flavors. Condition (1.13), (1.14) is especially useful when studying asymptotic properties of the diffusion process, whereas (1.10) is more adequate to construct examples.

Together with the crucial renewal property (see Theorem 2.2) induced by the regeneration timesτk,k1, the formulation (1.14) is instrumental in showing that under(T), and whend2,

lim sup

u→∞ (logu)^−αlogP0[τ1> u]<0, forα <1+d−1

d+1, (1.15)

see Theorem 4.5. The proof again uses a strategy close to the proof in the discrete case, see Sznitman [32]. We prove a seed estimate, see Lemma 4.4, which is then propagated to the right scale by performing a renormalisation step, see Lemma 4.3. Interestingly enough, we do not require condition(T)to prove the renormalisation lemma.

Under the assumption of (1.13) and the finiteness of the first and the second moment ofτ1, the Theorems 3.2 and 3.3 in Shen [28] imply that:

P₀-a.s., X_t

t →v, v=0, deterministic, withv·l >0, (1.16)

and underP0,B_·^s=(Xs·−s·v)/√

sconverges in law onC(R+,R^d), ass→ ∞, to a

Brownian motionB_·with non-degenerate deterministic covariance matrix. (1.17) Hence, when condition(T)holds, andd2, Theorem 4.5, see also (1.15), yields a ballistic law of large numbers and a central limit theorem governing corrections to the law of large numbers. Incidentally let us mention that as in the discrete setting, cf. Sznitman [33,35], condition(T)is a natural contender for the characterisation of ballistic diffusions in random environment whend2. However at present there are no rigorous results in that direction.

As an application of our methods, we provide a rich class of examples exhibiting ballistic behavior. We first consider the case where, for somel∈S^d−1and all ω∈Ω, allx∈R^d,b(x, ω)·l remains uniformly positive, and show in Proposition 5.1 that condition(T )|lholds. Hence we recover and extend the main result of Komorowski and Krupa [15] (which only asserts (1.16) whenσ=Id).

Then we consider the case whereσ in (1.7) is the identity. We prove in Theorem 5.2 that, whend1, there is a constantc_e(b, K, d, R) >¯ 0 such that, forl∈S^d−1, the inequality

E

b(0, ω)·l

+

> c_eE

b(0, ω)·l

−

(1.18)

implies condition(T )|l (and hence condition(T)|l). Clearly, whenσ =Id, the result of Proposition 5.1 is included in Theorem 5.2. Note that Theorem 5.2 covers additional situations where b(0, ω)·l changes sign in every unit directionl. This provides new examples of ballistic diffusions in random environment. More details are included in Remark 5.7 at the end of Section 5.

To prove Theorem 5.2, we verify the geometric formulation (1.11) of condition (T). However it is a difficult task to compute the exit distribution of the diffusion out of large slabs underP0, since the Markov property is lost underP0. In the spirit of Kalikow [10], we restore a Markovian character to the exit problem by virtue of Proposition 5.4. With the help of Proposition 5.4, we show that condition(T )is implied by a certain condition (K), see (5.23), which has a similar flavor as Kalikow’s condition in the discrete i.i.d. setting, see Sznitman and Zerner [36]. The proof of Theorem 5.2 is then carried out by checking condition (K). These steps are similar in spirit to the strategy used in the discrete setting, cf. lecture 5 of [4]. However, difficulties arise in the continuous space–time framework.

Let us now describe the organisation of this article.

In Section 2, we recall the coupling construction which leads to the measuresP_x,ωresp.P_x, cf. Proposition 2.1. On this new probability space one constructs the regeneration timesτ_k,k1, which provide the crucial renewal structure, cf. Theorem 2.2. These results have been obtained in Shen [28]; we recall them for the convenience of the reader.

In Section 3, we prove the equivalence of (1.10) and (1.13) (1.14), see Theorem 3.1.

In Section 4, we show that, whend2, condition(T)implies (1.15), see Theorem 4.5. Proposition 4.2 highlights the importance of large deviation controls of the exit probability of large slabs. The renormalisation step is carried out in Lemma 4.3, and a seed estimate is provided in Lemma 4.4.

In Section 5, we show that condition(T )(in the geometric formulation (1.10)) holds either under the assumption of the uniform positivity ofb(x, ω)·l for some unit vectorl and allω∈Ω, allx∈R^d, or under the assumption of σ=Id and (1.18).

In Appendix A, we provide some results on continuous local martingales and Green functions, that we use through- out this article.

Convention on constants. Unless otherwise stated, constants only depend on the quantitiesν,b,¯ σ , K, R, d, γ¯ . In particular they are independent of the environmentω. Generic positive constants are denoted byc. Dependence on additional parameters appears in the notation. For example,c(p, L)means that the constantcdepends onpandL and onν,b,¯ σ , K, R, d, γ¯ . When constants or positive numbers are not numerated, their value may change from line to line.

2. The regeneration times and the renewal structure

In this section, we recall the definition of the coupled measuresP_x,ω(resp.P_x) and of the regeneration timesτ_k, k1, given in Shen [28]. We then cite the resulting renewal structure, see Theorem 2.2. For the proofs or further details, we refer the reader to Shen [28].

2.1. Notation

We introduce some additional notation. Forx∈R^d,d1, we letB_r(x)denote thed-dimensional open Euclidean ball with radiusrcentered inx. ForU⊆R^d, we denote withUits closure, with diam(U )^def=sup{|x−y|: x, y∈U} its diameter, and, for measurableU, with|U|its Lebesgue measure. A domain stands for a connected open subset ofR^d. For two subsetsU,V ofR^d, we define their mutual distance byd(U, V )^def=inf{|x−y|:x∈U, y∈V}. For x∈R, we definex^def=sup{k∈Z: kx}andx^def=inf{k∈Z: kx}. For a discrete setA, we denote with #Aits cardinality. For an open setUinR^dandu∈Rwe define the(Ft)t0-stopping times ((Ft)t0denotes the canonical right-continuous filtration on(C(R+,R^d),F)): the exit time fromU,

T_U^def=inf{t0:X_t∈/U}, (2.1)

and the entrance times into the half-spaces{x·lu}resp.{x·lu},

T_u^ldef=inf{t0:X_t·lu}, T_u^ldef=inf{t0:X_t·lu}. (2.2) We define as well the maximal value of the process(X_s·l)_s0till timet,

M(t )^def=sup{X_s·l: 0st}, (2.3)

and the first return time of the process(X_s·l)_s0to the level−Rrelative to the starting point, as well as its rounded value,

J^def=inf

t0:(X_t−X₀)·l−R

, D^def= J. (2.4)

2.2. The coupled measures

We need further notations. We letlbe a fixed unit vector, and

U^xdef=B_6R(x+5Rl), B^xdef=B_R(x+9Rl). (2.5)

We denote byλjthe canonical coordinates on{0,1}^N. Further, letSm

def=σ{λ0, . . . , λm},m∈N, denote the canonical filtration on{0,1}^Ngenerated by(λ_m)_m∈NandS^def=σ{

mSm}be the canonicalσ-algebra. We also write fort0:

Zt

def=Ft⊗St, Z^def=F⊗S=σ

m∈N

Zm

. (2.6)

We also consider the shift operators{θ_m: m∈N}, withθ_m:(C(R₊,R^d)× {0,1}^N,Z)→(C(R₊,R^d)× {0,1}^N,Z), such that

θ_m(X_·, λ_·)=(X_m+·, λ_m+·). (2.7)

Then from Theorem 2.1 in Shen [28], one has the following measures, coupling the diffusion in random environment with a sequence of Bernoulli variables:

Proposition 2.1.There existsp >0, such that for everyω∈Ω andx∈R^d, there exists a probability measurePx,ω

on(C(R₊,R^d)× {0,1}^N,Z)depending measurably onωandx, such that

(1) UnderP_x,ω,(X_t)_t0isP_x,ω-distributed, and theλ_m,m0, are i.i.d. Bernoulli variables with success probabil- ityp.

(2) Form1,λ_mis independent ofFm⊗Sm−1underP_x,ω. Conditioned onZm,X_·◦θ_mhas the same law asX_· underP_X^λm

m,ω, where fory∈R^d,λ∈ {0,1},P_y,ω^λ denotes the lawPy,ω[ · |λ0=λ]. (3) P_x,ω¹ almost surely,X_s∈U^xfors∈ [0,1](recall(2.5)).

(4) UnderP_x,ω¹ ,X₁is uniformly distributed onB^x(recall(2.5)).

We then introduce the new annealed measures on(Ω×C(R₊,R^d)× {0,1}^N,A⊗Z):

P_x^def=P×P_x,ω and E_x^def=E×E_x,ω. (2.8)

2.3. The regeneration timesτ_kand the renewal structure

To define the first regeneration timeτ₁, we introduce a sequence of integer-valued(Zt)_t0-stopping times N_k, k1, such that, at these times, the Bernoulli variable takes the value one, and the process (X_t ·l)_t0 in essence reaches a new maximum. Proposition 2.1 now shows thatfor every environmentω∈Ω, conditionally onZN_k the position of the diffusion at timeN_k+1is uniformly distributed on the ballB^XNk underP_0,ω. We defineτ₁as the first N_k+1 such that, after time N_k+1, the process(X_t·l)_t0never goes below the levelX_Nk+1·l−R. In essence, the distance between the positionsX_τ1−1 andX_τ1 is large enough to obtain, in view of finite range dependence, independence of the parts of the trajectory(Xt −X0)tτ₁−1and(Xτ₁+t−Xτ₁)t0underP0, so that the diffusion regenerates at timeτ1underP0. We define the regeneration timesτk,k2, in an iterative fashion, and we provide the renewal structure in Theorem 2.2.

In fact, the precise definition ofτ1 relies on several sequences of stopping times. First, fora >0, introduce the (Ft)_t0-stopping timesV_k(a),k0 (recallM(t )in (2.3) andT_uin (2.2)):

V₀(a)^def=T_M(0)+a, V_k+1(a)^def=T_M(Vk(a))+R. (2.9) In view of the Markov property, see point (2) of Proposition 2.1, we want the stopping timesN_k(a),k1, to be integer-valued. Therefore we introduce in an intermediate step the (integer-valued) stopping timesN_k(a)where the processX_t·lessentially reaches a maximum:

⎧⎪

⎨

⎪⎩

N₁(a)^def=inf V_k(a)

: k0, sup

s∈[Vk,Vk]

l·(X_s−X_Vk)<R 2

, Nk+1(a)^def=N1(3R)◦θ_N

k(a)+Nk(a), k1

(2.10)

(by convention we setNk+1= ∞ifNk= ∞). In the spirit of the comment at the beginning of this subsection, we define the(Zt)t0-stopping timeN1as

N1(a)^def=infNk(a): k1, λ_Nk(a)=1

, N1def

=N1(3R), (2.11)

as well as the(Zt)_t0-stopping times

S₁^def=N₁+1,

R₁^def=S₁+D◦θ_S1. (2.12)

The(Zt)_t0-stopping timesN_k+1,S_k+1andR_k+1are defined in an iterative fashion fork1:

⎧⎪

⎨

⎪⎩

N_k+1^def=R_k+N₁(a_k)◦θ_Rk witha_k^def=M(R_k)−X_Rk·l+RR, S_k+1^def=N_k+1+1,

R_k+1^def=S_k+1+D◦θ_Sk+1

(2.13)

(the shiftθ_Rk isnotapplied toa_kin the above definition).

It follows from the above definitions thatP0-a.s.,X_Sk X_Nk·l+10R,k1. Notice that for allk1, the(Zt)_t0- stopping timesN_k,S_k andR_k are integer-valued, possibly equal to infinity, and we have 1N₁S₁R₁N₂ S₂R₂· · ·∞.

The firstregeneration timeτ₁is defined, as in [36], by τ1def

=inf

S_k: S_k<∞, R_k= ∞

∞. (2.14)

We define the sequence of random variablesτ_k,k1, iteratively on the event{τ₁<∞}, by viewingτ_kas a function of(X_·, λ_·):

τ_k+1

(X_·, λ_·)def

=τ₁

(X_·, λ_·) +τ_k

(X_τ1+·−X_τ1, λ_τ1+·)

, k1, (2.15)

and set by conventionτ_k+1= ∞on{τ_k= ∞}. Observe that for eachk1,τ_kis either infinite or a positive integer.

By convention, we setτ0=0. The random variablesτk,k0, provide a renewal structure, see also Theorem 2.5 in Shen [28], which will be crucial in the proof of Theorem 3.1.

Theorem 2.2((Renewal Structure)). Assume thatP0-a.s.,τ1<∞. Then under the measureP0, the random variables Z_k^def=(X_{(τk+·)∧(τk+1−1)}−X_τk;X_τk+1−X_τk;τ_k+1−τ_k),k0, are independent. Furthermore,Z_k,k1, underP0, have the distribution ofZ₀=(X_{·∧(τ1−1)}−X₀;X_τ1−X₀;τ₁)underP₀[ · |D= ∞].

The following proposition is also established in [28] (see Lemma 2.3 and Proposition 2.7 therein):

Proposition 2.3. P₀-a.s.τ₁<∞ if and only ifP₀-a.s. limt→∞X_t ·l= ∞. Furthermore P₀-a.s. τ₁<∞ implies P₀[D= ∞]>0 (recall the definition ofDin(2.4)).

3. Equivalent formulations of condition(T )γ

In this section, we provide an equivalent formulation of the condition(T )γ|l, cf. (1.10), in terms of a stretched exponential estimate on the size of the trajectoryXt, 0tτ1.

Theorem 3.1.Letl∈S^d−1,0< γ 1. The following two conditions are equivalent

(T )_γ|l, (3.1)

P0-a.s. lim

t→∞Xt·l= ∞, and for someμ >0, E0

exp

μ sup

0tτ1

|Xt|^γ

<∞. (3.2)

3.1. The proof of (3.1)⇒(3.2) Let us first show that

P₀-a.s. lim

t→∞X_t·l= ∞. (3.3)

We choose an orthonormal basis(f_i)_1idofR^dwithf₁=l. By definition of condition(T )_γ|l, there are unit vectors l_i,+,l_i,−inRf₁+Rf_i, 2id, such that:

li,±·f1>0, li,+·fi >0, li,−·fi<0, and, forl=l,li,+,li,−, 2id,b >0,

lim sup

L→∞ L^−γlogP₀[X_T

Ul,b,L·l<0]<0. (3.4)

Consider the open setD^def= {x∈R^d, |x·l|<1, x·li,±>−1, 2id}.Dis a bounded set, hence we can find numbersai,±>0, 2id, such that

D⊆

x∈R^d: x·l_i,±< a_i,±, 2id .

Since(T )_γ holds relative tolandl_i,±, 2id, writing P₀

T_LD< T_L^l

P₀T₋^l_L< T_L^l +

d i=2

P₀T₋^li,+_L < T_La^li,+

i,+

+ d i=2

P₀T₋^li,−_L < T_La^li,−

i,−

,

we find by (3.4) that lim sup

L→∞ L^−γlogP0

T_LD< T_L^l

<0. (3.5)

SinceP₀[T_L^l= ∞]P₀[T_LD< T_L^l], and the left-hand side increases withL, (3.5) implies that P₀-a.s. lim sup

t→∞ X_t·l= ∞. As a next step we observe that

lim sup

L→∞ L^−γlogP₀T_L^l

2

◦θ_Tl L< T_4L^l

3

◦θ_Tl L

<0. (3.6)

Indeed:

P₀T_L^l

2

◦θ_Tl L< T_4L^l

3

◦θ_Tl L

P₀

T_LD< T_L^l

+P₀T_L^l

2

◦θ_Tl L< T_4L^l

3

◦θ_Tl

L, T_LD=T_L^l

, (3.7)

and by (3.5) we only need to estimate the second term on the right-hand side of (3.7). We define

∂₊D^def=

x∈∂D: x·l=1 ,

and let(B1(xi))i∈I,xi∈∂₊LD,I a finite set with cardinality growing polynomially inL, be a cover of∂₊LDby unit balls, see above (2.2) for the notation. It follows from the strong Markov property and the stationarity of the measure Pthat

P0T_L^l

2 ◦θ_Tl L< T_4L^l

3 ◦θ_Tl

L, T_LD=T_L^l

i∈I

E E0,ω

PX

T lL

,ωT_L^l

2

< T_4L^l

3

, X_Tl

L∈B1(xi)

i∈I

E sup

x∈B₁(x_i)

P_x,ωT_L^l

2

< T_4L^l

3

=

i∈I

E sup

x∈B₁(0)

P_x,ωT^l

−^L₂ < T_L^l

3

. (3.8)

For large enoughL, it follows from the strong Markov property that for allω∈Ω, the functionx→Px,ωT₋^lL

2

< TL^l 3

isLω-harmonic onB3(0), (3.9)

see for instance [11] p. 364f. Harnack’s inequality (see [8] p. 250) states that there is a constantcH>1 such that for allLω-harmonic functionsuonB₃(x),x∈R^d,

sup

y∈B₁(x)

u(y)c_H inf

y∈B1(x)u(y), (3.10)

which shows that E

sup

x∈B₁(0)

P_x,ωT^l

−^L₂ < T_L^l

3

c_H P₀T^l

−^L₂ < T_L^l

3

. (3.11)

Inserting (3.11) in (3.8), we see that (3.6) follows from (3.1). From an application of Borel–Cantelli’s lemma we obtain thatP₀-a.s. for large integerL,

T_4L^l

3

<T_L^l

2 ◦θ_Tl L+T_L^l.

So on a set of full P0-measure we can construct an integer-valued sequence L_k ∞, with L_k+1= ⁴₃L_k and T_L^l

k+1<T_Lk^l

2

◦θ_Tl Lk +T_L^l

k,k0. This shows (3.3).

We now show that for someμ >0 E₀

exp

μ sup

0tτ1

|X_t|^γ

<∞. (3.12)

The proof is divided into several propositions. In a first step, we study the integrability properties of the random variable (recall (2.4))

M^def=sup

(Xt−X0)·l: 0tJ

, (3.13)

i.e.Mis the maximal relative displacement ofX_.in the directionlbefore it goes an amount ofR below its starting point. By virtue of Proposition 2.3 and (3.3), we know thatP₀[D= ∞] =P₀[J = ∞]>0 (recall (2.4)). Hence we cannot expectMto be finite. Nevertheless we have the following proposition:

Proposition 3.2.There isμ1>0such that E₀

exp μ₁M^γ

, J <∞

1−P₀[J= ∞]

2 .

Proof. LetL_k=(⁴₃)^k. By our previous result (3.6), we see that there isμ >0 such that for large integersk:

P₀[L_kM < L_k+1, J <∞]P₀T_L^l

k/2◦θ_Tl Lk < T_4L^l

k/3◦θ_Tl Lk

exp

−μL^γ_k

. (3.14)

Letk₀be large enough such that

kk₀exp{−^μ₂L^γ_k}^P0[J₄^=∞]. Further, letμ₁>0 such that 0< (⁴₃)^γμ₁<^μ₂. Then (3.14) shows that fork₀large enough,

E0

exp μ1M^γ

, J <∞

exp

μ1L^γ_k

0

P0[J <∞] +

kk0

exp

μ1L^γ_k+1

P0[L_kM < L_k+1, J <∞]

exp

μ₁L^γ_k

0

1−P₀[J= ∞]

+

kk₀

exp

−μ 2L^γ_k

exp

μ₁L^γ_k

0

1−P₀[J= ∞]

+P₀[J= ∞]

4 1−P₀[J= ∞]

2 ,

providedμ1>0 is chosen small enough in the last inequality. 2

As a next step, we shall prove the integrability of exp{μ (X_τ1·l)^γ}under the extended annealed measureP₀. Recall the(Zt)_t0-stopping times(V_k(a))_k0,(N_k(a))_k0andN₁(a)defined in Subsection 2.3. As we will see in the proof of Proposition 3.4, exp{μ ((X_N1(a)−X₀)·l)^γ}will play a key role in studying the integrability of exp{μ (X_τ1·l)^γ} underP₀. Let us therefore start with the following proposition, which only assumes that,P₀-a.s., limt→∞X_t·l= ∞, which we have established in (3.3).

Proposition 3.3.Assume thatlimt→∞Xt·l= ∞P0-a.s. Then, for eachμ2>0there isμ3>0, such that forP-a.e.

ω∈Ω: sup

x,aR

E_x,ω exp

μ₃

(X_N1(a)−X₀)·lγ

−a^γ

1+μ₂. (3.15)

Proof. DefineA_l^def= {limt→∞X_t·l= ∞}. Observe that

forP-a.e.ωand for everyx∈R^d, P_x,ω[A_l] =1. (3.16) Indeed, by the stationarity of the measureP,P_y[A_l] =1 for ally ∈R^d. Hence

dy Py[A^c_l] =0, and by applying Fubini’s Theorem it follows that there is aP-null setΓ ⊂Ω, such that for allω /∈Γ andy outside a Lebesgue null setN(ω)⊂R^d,P_y,ω[A^c_l] =0. Observe that for allx∈R^d, andω∈Ω,P_x,ω[A_l] =P_x,ω[A_l◦θ₁]. It follows from the Markov property that for allx∈R^d, andω /∈Γ,P_x,ω[A_l◦θ₁] =

R^dP_y,ω[A_l]p_ω(1, x, y)dy=1, wherep_ω(s, x, y)is the transition density function underP_x,ω(that is, for every open subsetUofR^d,P_x,ω[X_s∈U] =

Up_ω(s, x, y)dy).

The claim (3.16) now follows. WhenP_x,ω[A_l] =1, Proposition 4.8 in Shen [28] shows that (3.15) holds for allω∈Ω, whenγ=1. By the same proof as given there, Proposition 3.3 follows from (3.16) whenγ=1. When 0< γ <1, usingβ^γ−α^γ β−αforβ1∨α, and (3.15) withγ=1, we findμ₃∈(0,1)such that

sup

x,aR

Ex,ω

exp μ3

(XN₁(a)−X0)·lγ

−a^γ

sup

x,aR

E_x,ω exp

μ₃

(X_N1(a)−X₀)·lγ

−a^γ

, (X_N1(a)−X₀)·l1∨a

+e4.

By Jensen’s inequality, ifn1 is large enough, we find sup

x,aR

E_x,ω

exp μ₃

n

(X_N1(a)−X₀)·lγ

−a^γ

4^1/n1+μ₂, which shows (3.15). 2

Proposition 3.4.There existsμ₄>0such that E₀

exp

μ₄(X_τ1·l)^γ

<∞. (3.17)

Proof. Using that,P0-a.s.,XS_k·lXN_k·l+10R,k1 (see the remark following (2.13)) we observe that E₀

exp

μ₄(X_τ1·l)^γ

=

k1

E₀ exp

μ₄(X_Sk·l)^γ

, S_k<∞, D◦θ_Sk= ∞

exp

μ₄(10R)^γ

k1

E₀ exp

μ₄(X_Nk·l)^γ

, N_k<∞def

=exp

μ₄(10R)^γ

k1

h_k. (3.18)

Observe that, fork1, see (2.13),

l·X_Nk+1=l·X_Rk+l·(X_N1(ak)−X₀)◦θ_Rk,

witha_k =M(R_k)−l·X_Rk +R ∈ZRk (in fact for any m1, a_k ·1_{Rk=m} is Fm⊗Sm−1-measurable, and λ_m is independent ofFm⊗Sm−1). We recall that the shift θ_Rk isnotapplied toa_k. Therefore, by the strong Markov property, cf. Proposition 2.1, and, by applying Proposition 3.3 (notice thata_kR,k1, see (2.13)), we see that for allμ₂>0, there isμ₄∈(0, μ3)such that:

hk+1E E0,ω

exp

μ4(l·XR_k)^γ

, Rk<∞,EX_Rk,ω

exp μ4

l·(XN₁(a_k)−X0)γ E E_0,ω

exp

μ₄(l·X_Rk)^γ

, R_k<∞, (1+μ₂)e^μ4akγ

. (3.19)

Observe that withMfrom (3.13) andZ₁as in Lemma A.1 of Appendix A, the following inequalities hold, whenR_k is finite: