An effective criterion and a new example for ballistic diffusions in random environment
aa r X i v : . [ m a t h . P R ] J un The Annals of Probability (cid:13)
Institute of Mathematical Statistics, 2008
AN EFFECTIVE CRITERION AND A NEW EXAMPLE FORBALLISTIC DIFFUSIONS IN RANDOM ENVIRONMENT
By Laurent Goergen
ETH Zurich
In the setting of multidimensional diffusions in random environ-ment, we carry on the investigation of condition ( T ′ ), introduced bySznitman [ Ann. Probab. (2001) 723–764] and by Schmitz [ Ann.Inst. H. Poincar´e Probab. Statist. (2006) 683–714] respectivelyin the discrete and continuous setting, and which implies a law oflarge numbers with nonvanishing limiting velocity (ballistic behav-ior) as well as a central limit theorem. Specifically, we show thatwhen d ≥
2, ( T ′ ) is equivalent to an effective condition that can bechecked by local inspection of the environment. When d = 1, we provethat condition ( T ′ ) is merely equivalent to almost sure transience. Asan application of the effective criterion, we show that when d ≥ ε > ε − η , η >
0, satisfies condition ( T ′ ) when ε is small and hence exhibitsballistic behavior. This class of diffusions contains new examples ofballistic behavior which in particular do not fulfill the condition in[ Ann. Inst. H. Poincar´e Probab. Statist. (2006) 683–714], (5.4)therein, related to Kalikow’s condition.
1. Introduction.
Diffusions in random environment emerged about 25years ago from homogenization theory in the study of disordered media;see, for instance, [3]. Within the rich field of “random motions in randommedia,” they are closely related to the discrete model of “random walks inrandom environment”; see [9, 22].In the one-dimensional discrete setting a complete characterization of bal-listic behavior, which refers to the situation where the motion tends to in-finity in some direction with nonvanishing velocity, was established alreadyin 1975 by Solomon [15]; see also [6, 10]. In the multidimensional setting,however, such a characterization has not been found yet, but a great deal
Received August 2006; revised May 2007.
AMS 2000 subject classifications.
Key words and phrases.
Diffusions in random environment, ballistic behavior, effectivecriterion, condition ( T ′ ), perturbed Brownian motion. This is an electronic reprint of the original article published by theInstitute of Mathematical Statistics in
The Annals of Probability ,2008, Vol. 36, No. 3, 1093–1133. This reprint differs from the original inpagination and typographic detail. 1
L. GOERGEN of progress has been made over the last seven years: the so-called condi-tions ( T ) and ( T ′ ) introduced by Sznitman (see [18, 19]) for random walksin random environment seem to be promising candidates for an equivalentdescription of ballistic behavior when the space dimension d ≥
2. In essence,one possible formulation of condition ( T ) [see (1.12)] requires exponentialdecay of the probability that the trajectory exits a slab of growing widththrough one side rather than the other. These conditions have interestingconsequences such as a ballistic law of large numbers and a central limittheorem. Their analogues in the setting of diffusions have been developedby Schmitz (see [11, 12]), and he used a previous result of Shen [13, 14] toshow that they imply the same asymptotic behavior as mentioned beforein the discrete setup. The drawback of the definitions of conditions ( T ) or( T ′ ) as they were stated in [11] [see (1.10)] is their asymptotic nature whichmakes them difficult to check by local considerations. To remedy this prob-lem, we provide in the first part of this article an effective criterion, in thespirit of [19], which is equivalent to ( T ′ ) (see Theorem 2.6), and which canbe checked by inspection of the environment in a finite box.In the second part of this work, which is related to [20] in the discrete set-ting, we use the effective criterion to show that when d ≥
4, Brownian motionperturbed with a small random drift satisfying the assumption (1.16), fulfillscondition ( T ′ ); see Theorem 3.1. As we will see below, this class of diffusionscontains new examples for ballistic behavior beyond prior knowledge.Before we discuss our results any further, we first describe the model.The random environment is specified by a probability space (Ω , A , P ) onwhich acts a jointly measurable group { t x ; x ∈ R d } of P -preserving transfor-mations, with d ≥
1. The diffusion matrix and the drift of the diffusion inrandom environment are stationary functions a ( x, ω ) , b ( x, ω ) , x ∈ R d , ω ∈ Ω,with respective values in the space of nonnegative d × d matrices and in R d ,that is, a ( x + y, ω ) = a ( x, t y ω ) , (1.1) b ( x + y, ω ) = b ( x, t y ω ) for x, y ∈ R d , ω ∈ Ω . We assume that these functions are bounded and uniformly Lipschitz, thatis, there is a ¯
K >
1, such that for x, y ∈ R d , ω ∈ Ω, | b ( x, ω ) | + | a ( x, ω ) | ≤ ¯ K, (1.2) | b ( x, ω ) − b ( y, ω ) | + | a ( x, ω ) − a ( y, ω ) | ≤ ¯ K | x − y | , where | · | denotes the Euclidean norm for vectors and matrices. Further,we assume that the diffusion matrix is uniformly elliptic, that is, there is a ν > x, y ∈ R d , ω ∈ Ω:1 ν | y | ≤ y · a ( x, ω ) y ≤ ν | y | . (1.3) N EFFECTIVE CRITERION AND A NEW EXAMPLE The coefficients a, b satisfy a condition of finite range dependence: for A ⊂ R d , we define H A = σ ( a ( x, · ) , b ( x, · ); x ∈ A ) , (1.4)and assume that for some R > H A and H B are independent under P whenever d ( A, B ) ≥ R, (1.5)where d ( A, B ) is the mutual Euclidean distance between A and B . Withthe above regularity assumptions on a and b , for any ω ∈ Ω, x ∈ R d , themartingale problem attached to x and the operator L ω = d X i,j =1 a ij ( · , ω ) ∂ ij + d X i =1 b i ( · , ω ) ∂ i (1.6)is well posed; see [17] or [2], page 130. The corresponding law P x,ω on C ( R + , R d ), unique solution of the above martingale problem, describes thediffusion in the environment ω and starting from x . We write E x,ω for theexpectation under P x,ω and we denote the canonical process on C ( R + , R d )with ( X t ) t ≥ . Observe that P x,ω is the law of the solution of the stochasticdifferential equation dX t = σ ( X t , ω ) dβ t + b ( X t , ω ) dt, (1.7) X = x, P x,ω -a.s. , where, for instance, σ ( · , ω ) is the square root of a ( · , ω ) and β is some d -dimensional Brownian motion under P x,ω . The laws P x,ω are usually called“quenched laws” of the diffusion in random environment. To restore transla-tion invariance, we consider the so-called “annealed laws” P x , x ∈ R d , whichare defined as semi-direct products: P x def = P × P x,ω . (1.8)Of course the Markov property is typically lost under the annealed laws.We now come back to the object of this work. We start by recalling thedefinition of conditions ( T ) and ( T ′ ) as stated in [11]. These conditions areexpressed in terms of another condition ( T ) γ defined as follows. For a unitvector ℓ of R d , d ≥
1, and any u ∈ R , consider the stopping times T ℓu = inf { t ≥ X t · ℓ ≥ u } , ˜ T ℓu = inf { t ≥ X t · ℓ ≤ u } . (1.9)For γ ∈ (0 , T ) γ holds relative to ℓ , in shorthandnotation ( T ) γ | ℓ , if for all unit vectors ℓ ′ in some neighborhood of ℓ and forall b >
0, lim sup L →∞ L − γ log P [ ˜ T ℓ ′ − bL < T ℓ ′ L ] < . (1.10) L. GOERGEN
Condition ( T ′ ) relative to ℓ is then the requirement that(1.10) holds for all γ ∈ (0 , , (1.11)and condition ( T ) relative to ℓ refers to the case where(1.10) holds for γ = 1 . (1.12)It is clear that ( T ) implies ( T ′ ) and we show in Theorem 2.6 that ( T ′ )is equivalent to ( T ) γ when γ ∈ ( , T ) γ , γ ∈ (0 ,
1] are all equivalent.Let us also mention that for γ ∈ (0 , T ) γ relative to ℓ is inessence equivalent to almost sure transience in direction ℓ together withfiniteness of a stretched exponential moment of the size of the trajectory upto a certain regeneration time; see [11], Theorem 3.1 therein or [19] for asimilar result in the discrete setting. The latter formulation of condition ( T ′ )is especially appropriate to study the asymptotic properties of the diffusion.Indeed Schmitz showed in [11], Theorem 4.5 (see also [18]) that when d ≥ T ′ ), they are not “effective conditions” that can be checked by local inspec-tion of the environment. Concrete examples where ( T ′ ) holds, besides theeasy case where the projection of the drift on some unit vector is uniformlybounded away from 0 (see [11], Proposition 5.1), originate from a strongercondition going back to Kalikow; see [7, 21]. For instance, it is shown in[11], Theorem 5.2, and [12], Theorem 2.1 that there exists a constant c e > K, ν, R, d [see (1.2)–(1.5)], such that condition ( T ) holdswhen E [( b (0 , ω ) · ℓ ) + ] ≥ c e E [( b (0 , ω ) · ℓ ) − ] . (1.13)In the first part of this work we derive an effective criterion in the abovesense. We show (see Theorem 2.6) that when d ≥ ℓ , ( T ′ ) | ℓ is in essence equivalent toinf B,a ∈ (0 , { c ( d ) ˜ L d − L d − E [ ρ aB ] } < , (1.14)with ρ B = P ,ω [ X T B / ∈ ∂ + B ] P ,ω [ X T B ∈ ∂ + B ] , (1.15)provided in the above infimum, B runs over all large boxes transversal to ℓ consisting of the points x with x · ℓ ∈ ( − L + R + 2 , L + 2) and other co-ordinates in an orthonormal basis with first vector ℓ , smaller in absolute N EFFECTIVE CRITERION AND A NEW EXAMPLE value than ˜ L , for L ≥ c ′ ( d ) , R + 2 ≤ ˜ L < L . In the above formula for ρ B , T B denotes the exit time from B and ∂ + B is the part of the boundary of B where x · ℓ = L + 2. The proof of Theorem 2.6 follows the strategy ofSznitman [19] and the sufficiency of the effective criterion is obtained by aninduction argument along a growing sequence of boxes B k that tend to looklike infinite slabs and in which suitable moments of ρ B k are used to controlmoments of ρ B k +1 . This allows us to deduce the asymptotic exit behavior(1.10) from slabs. As a first application of the effective criterion, we showthe equivalence between ( T ′ ) and ( T ) γ when γ ∈ ( , ρ B in(1.15) reminds us of the decisive quantity appearing in the one-dimensionaltheorem of Solomon [15]. We will see in Section 2.1 that when d = 1, the box B is replaced with an interval ( − L, L ) and the existence of an a ∈ (0 ,
1] and a
L > R such that E [ ρ a ( − L,L ) ] < T ′ ) and ( T ) as well as to al-most sure convergence to + ∞ . Hence in opposition to the multidimensionalcase, condition ( T ) does not imply ballistic behavior when d = 1.In the second part of this article we use the effective criterion to constructa new class of ballistic diffusions. We show (see Theorem 3.1) that when d ≥
4, for any η >
0, Brownian motion perturbed with a random drift b ( · , ω )such that sup x ∈ R d ,ω ∈ Ω | b ( x, ω ) | ≤ ε and E [ b (0 , ω ) · e ] ≥ ε − η for ε > , (1.16)satisfies the effective criterion with ℓ = e if ε is small enough. The con-ditions (1.16) allow for laws P of the environment such that (1.13) doesnot apply. Indeed, since the constant c e is larger than 1, as one can seefrom an inspection of the proof of [11], Theorem 2.5, (1.13) requires that E [ b (0 , ω ) · e ] is larger than ( c e − E [( b (0 , ω ) · e ) − ] which can be chosento be of order ε under (1.16). Note that in the discrete setting, Sznitman(see [20]) obtained similar results under conditions significantly weaker than(1.16). Indeed, he showed that a discrete version of the effective criterionis satisfied by randomly perturbed simple random walk on Z d with a drift d (0 , ω ) def = E ,ω [ X − X ] of size ε such that E [ d (0 , ω ) · e ] is larger than ε / − η when d = 3, respectively larger than ε − η when d ≥
4. The strength of thisresult in contrast to ours is that it includes expected drifts of an order notlarger than ε , which enabled him to construct examples for condition ( T ′ )where Kalikow’s condition (see, e.g., [20], (5.3) therein) fails. Consideringcondition (5.23) of [11] as a continuous analogue of Kalikow’s condition,we believe that such examples also exist in our setting. Since, however, thecontinuous setup with the finite range dependence tends to complicate thearguments, we did not attempt to retrieve the full strength of Sznitman’sresult.Let us now briefly describe the proof leading to the new example. In orderto verify the effective criterion (1.14) under (1.16), we slice a large box B L. GOERGEN [as defined below (1.14)] into thinner slabs transversal to e and propagategood controls on the exit behavior out of these slabs to the box B using arefinement of the estimate (see Lemma 2.3 and Proposition 3.3) that wasinstrumental in the induction argument leading to the effective criterion. Theheart of the matter is then to prove these good controls for the thinner slabs.To this end, we express the probability that the trajectory exits through theright side of a slab with the help of the Green operator of the diffusionkilled when exiting the slab; see (3.23). This quantity is linked to the Greenoperator of killed Brownian motion via a certain perturbation equality; see(3.40). For Brownian motion, however, an explicit formula obtained by thewell-known “method of the images” from electrostatics [see (3.30)] allows usto compute all necessary estimates.Let us finally explain how this article is organized. In Section 2, we firstintroduce some notation and then we show the equivalence between the ef-fective criterion and condition ( T ′ ) when d ≥
2; see Theorem 2.6. The keyestimate for the induction step is given by Proposition 2.2. In Section 2.1, wediscuss the one-dimensional case. In Section 3, we use the effective criterionto show that a certain perturbed Brownian motion satisfies condition ( T ′ )when d ≥
4. In Section 3.1, we state the main Theorem 3.1 and a refine-ment of Lemma 2.3; see Proposition 3.3. In Section 3.2, we define the Greenoperators and Green’s functions for which we provide certain deterministicestimates in the case of Brownian motion; see Lemmas 3.7 and 3.9. We alsoprove a perturbation equality; see Proposition 3.8. In Section 3.3, we use theresults from the previous sections to prove the main Theorem 3.1. In Ap-pendix A.1, we give the proof of Lemma 2.3 which is similar to that of [19],Proposition 1.2. In Appendix A.2, we prove Lemma 3.9 using a techniquesimilar to [20], Lemma 2.1.
Convention on constants.
Unless otherwise stated, constants only dependon the quantities d, ¯ K, ν, R . We denote with c positive constants with valueschanging from place to place and with c , c , . . . positive constants withvalues fixed at their first appearance. Dependence on additional parametersappears in the notation.
2. An effective criterion when d ≥ . In this section we show that con-dition ( T ′ ) [see (1.11)] is equivalent to the effective criterion [see (2.53)]which controls the exit probability from some finite box. By an inductionargument we propagate this control to larger boxes that tend to look likeinfinite slabs. Then one can infer the fast decay of exit probabilities fromslabs through “the left” side as required by condition ( T ′ ).We first need some notation. For A, B ⊂ R d an open and a closed set,we denote with T A = inf { t ≥ X t / ∈ A } the exit time from A and with H B = inf { t ≥ X t ∈ B } the entrance time into B . For any stopping time S , we call S = 0, S k +1 = S ◦ θ S k + S k , k ≥
0, the iterates of S . Here, θ t N EFFECTIVE CRITERION AND A NEW EXAMPLE denotes the canonical time shift. We consider a direction ℓ ∈ S d − and arotation R of R d such that R ( e ) = ℓ . The vectors e i , i = 1 , . . . , d , constitutethe canonical basis. As a shortcut notation for the stopping times in (1.9),we write T u = T ℓu and ˜ T u = ˜ T ℓu , u ∈ R . Moreover, we introduce | z | ⊥ = max j ≥ | z · R ( e j ) | for x ∈ R d . (2.1)For positive numbers L, L ′ , ˜ L , we introduce the box B = B ( R , L, L ′ , ˜ L ) def = R (( − L, L ′ ) × ( − ˜ L, ˜ L ) d − ) , (2.2)and the positive, respectively negative, part of its boundary ∂ + B = ∂B ∩ { x ∈ R d : ℓ · x = L ′ } , ∂ − B = ∂B \ ∂ + B. (2.3)We also define the following random variables: for ω ∈ Ω, p B ( ω ) = P ,ω [ X T B ∈ ∂ + B ] = 1 − q B ( ω ) , (2.4) ρ B ( ω ) = q B ( ω ) p B ( ω ) ∈ [0 , ∞ ] . (2.5)In the sequel we will use different length scales L k , ˜ L k ≥ k = 0 , , . . . , andthe following shortcut notation [cf. (1.5) for the definition of R ]: B k = B ( R , L k − R − , L k + 1 , ˜ L k ) for k ≥ , (2.6) p k = p B k , q k = q B k , ρ k = ρ B k . Finally let us set for k ≥ N k = L k +1 L k , n k = ⌊ N k ⌋ , ˜ N k = ˜ L k +1 ˜ L k . (2.7)We start with an easy lemma, introducing the counterpart of a discreteellipticity constant. Lemma 2.1.
Let C L be the tube { z ∈ R d : − < z · e < L, sup ≤ j ≤ d | z · e j | < L } . There exists a constant < κ ≤ , such that for any L ≥ , ω ∈ Ω ,and any rotation R , P ,ω [ T R ( e ) L < T R ( C L ) ] ≥ κ L +1 and (2.8) P ,ω [ ˜ T R ( e ) − L < T R ( − C L ) ] ≥ κ L +1 . Proof.
We define the function ψ ( s ) = R ( e ) s , for 0 ≤ s ≤
1. Withthe support theorem (see [2], page 25), we obtain that there is a constant c > x ∈ R d , ω ∈ Ω, P x,ω [sup ≤ s ≤ | X s − X − ψ ( s ) | < ] L. GOERGEN and P x,ω [sup ≤ s ≤ | X s − X + ψ ( s ) | < ] are both larger than c . Then weset κ = min { c, } . The claim follows by applying the Markov property ⌈ L ⌉ times. (cid:3) We are now ready to prove the main induction step which in essencebounds moments of ρ in terms of moments of ρ . Proposition 2.2. ( d ≥ ) There exist c > R + 2 , c , c > , such thatwhen N ≥ , L ≥ c , ˜ N ≥ N , ˜ L ≥ R + 2 , for any a ∈ (0 , : E [ ρ a ] ≤ c ( κ − L (cid:18) c ˜ L ( d − L L ˜ L E [ q ] (cid:19) ˜ N / (12 N ) (2.9) + X ≤ m ≤ n +1 ( c ˜ L ( d − E [ ρ a ]) ( n + m − / ) . Proof.
For i ∈ Z and L > R + 2, we introduce the slabs of width R : S i = (cid:26) x ∈ R d : iL − R ≤ x · ℓ ≤ iL + R (cid:27) (2.10)and denote by I ( · ) the function on R d such that I ( x ) = i if x · ℓ − iL ∈ [ − L , L ), i ∈ Z . In particular, I takes the value i on S i , for all i ∈ Z . Wedefine the successive times of visit to the different slabs S i as the iterates V k , k ≥
0, of the stopping time V = inf { t ≥ X t ∈ S I ( X ) − ∪ S I ( X )+1 } . (2.11)We also need the stopping time˜ T = inf { t ≥ | X t | ⊥ ≥ ˜ L } . (2.12)In a first step we obtain a control on E [ ρ a ] using the following quantities:for ω ∈ Ω , i ∈ Z , ˆ ρ ( i, ω ) = sup (cid:26) ˆ q ( x, ω )ˆ p ( x, ω ) : x ∈ S i , | x | ⊥ < ˜ L (cid:27) , (2.13)where ˆ q ( x, ω ) = P x,ω [ X V ∈ S I ( x ) − ] = 1 − ˆ p ( x, ω ) . (2.14)The first step then comes with the following lemma. Lemma 2.3.
Under the assumptions of Proposition 2.2, E [ ρ a ] ≤ κ − a ( L +1) P [ G c ] + 2 X ≤ m ≤ n +1 Y − n +1
2, the stopping times σ ± ,ju = inf { t ≥ ± X t · R ( e j ) ≥ u } , (2.18) ¯ L = 2( n + 2)( ˜ L + 1) + R, J = (cid:22) ˜ L ¯ L (cid:23) . (2.19)Since ˜ L = ˜ N ˜ L ≥ n ˜ L , n ≥ L ≥ R , it follows that J ≥ { ˜ T ≤ ˜ T − L + R +1 ∧ T L +1 } , P -a.s., at least one of the projections | X t · R ( e j ) | , j ≥
2, reaches the value J ¯ L before X t exits the box B . Hence: P [ ˜ T ≤ ˜ T − L + R +1 ∧ T L +1 ] ≤ X j ≥ P [ σ + ,jJ ¯ L ≤ T B ] + P [ σ − ,jJ ¯ L ≤ T B ] . (2.20)Let us write σ u in place of σ + , u and bound the term P [ σ J ¯ L ≤ T B ], theother terms being treated similarly. The strong Markov property yields that P [ σ J ¯ L ≤ T B ] ≤ E E ,ω [ σ ( J − L ≤ T B , P X σ ( J − L ,ω [ σ J ¯ L ≤ T B ]] . (2.21)We define the auxiliary box B ′ = B ( R , L − R, L , ˜ L + 1);(2.22)see (2.2) for the notation and let H i , i ≥
0, denote the iterates of the stoppingtime H = T B ∧ T X + B ′ . Then for any ω ∈ Ω, x ∈ B with x · R ( e ) = ( J −
1) ¯ L , we have P x,ω [ σ J ¯ L > T B ] ≥ P x,ω " n +1) − \ k =0 θ − H k { H < T ∂ − B ′ + X } , (2.23)because on the event in the right-hand side, the trajectory either exits B before σ J ¯ L right away on { H < T ∂ − B ′ + X } or it exits the box B through“the right,” since for every k ≥
0, on θ − H k { H < T ∂ − B ′ + X } the trajectory P x,ω -a.s. moves between time H k and H k +1 at most a distance ˜ L + 1 into L. GOERGEN
Fig. 1.
Graphical explanation of (2.25). direction R ( e ) and at least a distance L into direction ℓ until it leaves B ,and since 2( n + 1)( ˜ L + 1) = ¯ L −
2( ˜ L + 1) − R < ¯ L (2.24)and 2( n + 1) L > L − R , the width of B in direction ℓ . In order toobtain a lower bound on the right-hand side of (2.23) with the help of thestrong Markov property, we cover the set G ( J − def = { x ∈ B : | x · R ( e ) − ( J −
1) ¯ L | ≤ n + 1)( ˜ L + 1) } , which contains the trajectories up to T B described by the event in the right-hand side of (2.23), with a collection ofdisjoint and rotated unit cubes C m with centers x m . The cardinality of thiscollection is proportional to the volume of G ( J − k, m ≥ ω ∈ Ω, we have that on { X H k ∈ C m } , P ,ω -a.s., P X Hk ,ω [ X T B ′ + X ∈ ∂ + B ′ + X ] ≥ P X Hk ,ω [ X T B xm ∈ ∂ + B + x m ] , (2.25)as for any x ∈ C m , it follows from the definitions of B ′ [see (2.22)] and B [see (2.6)] that ∂ + B + x m ⊂ ( B ′ + x ) c , ∂ − B ′ + x ⊂ B c + x m and x ∈ B + x m ;see Figure 1. Here U denotes the closure of U ⊂ R d . Therefore any piece oftrajectory contained in B + x m , connecting x ∈ C m to ∂ + B + x m has toexit B ′ + x , but cannot touch ∂ − B ′ + x .As a consequence, we deduce from (2.23) using the strong Markov prop-erty that for any ω ∈ Ω , x ∈ B with x · R ( e ) = ( J − L , P x,ω [ σ J ¯ L > T B ] ≥ (cid:18) inf m inf x ∈ C m P x,ω [ X T B xm ∈ ∂ + B + x m ] (cid:19) n +1) (2.26) def = 1 − φ ( J − , ω ) , and thus, in view of (2.21), we find P [ σ J ¯ L ≤ T B ] ≤ E [ P ,ω [ σ ( J − L ≤ T B ] φ ( J − , ω )] . (2.27) N EFFECTIVE CRITERION AND A NEW EXAMPLE From (2.24), we see that G ( J − ⊂ { x ∈ B : x · R ( e ) ≥ ( J −
2) ¯ L + 2( ˜ L +1) + R } , and therefore the random variable φ ( J − , · ) is H { z ·R ( e ) ≥ ( J − L + R } -measurable whereas P , · [ σ ( J − L ≤ T B ] is H { z ·R ( e ) ≤ ( J − L } -measurable. Thusthe finite range dependence property implies that P [ σ J ¯ L ≤ T B ] ≤ P [ σ ( J − L ≤ T B ] E [ φ ( J − , ω )] . (2.28)Using the notation (2.26) and observing that 1 − p k ≤ k (1 − p ) for k ≥ p ≥
0, we obtain E [ φ ( J − , ω )] ≤ n + 1) E (cid:20) sup m sup x ∈ C m P x,ω [ X T B xm ∈ ∂ − B + x m ] (cid:21) . (2.29)We now observe that the cardinality of the collection of cubes C m is propor-tional to 2 L · n + 1)( ˜ L + 1) · (2 ˜ L ) d − ≤ c ˜ L d − L L ˜ L . Then translationinvariance and an application of Harnack’s inequality to the harmonic func-tion x P x,ω [ X T B ∈ ∂ − B ] yield that E [ φ ( J − , ω )] ≤ c ′ ˜ L d − L L ˜ L E [ q ] , (2.30)where we used the notation (2.4). Coming back to (2.28), we see that P [ σ J ¯ L ≤ T B ] ≤ P [ σ ( J − L ≤ T B ] × c ˜ L d − L L ˜ L E [ q ] , and by induction(2.31) ≤ (cid:26) c ˜ L d − L L ˜ L E [ q ] (cid:27) m for all 0 ≤ m ≤ (cid:22) J (cid:23) . Similar bounds hold for each term in the right-hand side of (2.20) and since ⌊ J ⌋ ≥ ˜ N N from our assumptions on N , ˜ N and ˜ L we conclude from (2.17),(2.20) and (2.31) that P [ G c ] ≤ κ − L d − (cid:26) c ˜ L d − L L ˜ L E [ q ] (cid:27) ˜ N / (12 N ) . (2.32)So far we found an upper bound for the first term of the right-hand sideof (2.15). To complete the proof of (2.9), we are now going to bound thesecond term.For any i ∈ Z , we cover the set { x ∈ S i : | x | ⊥ < ˜ L } [appearing in thedefinition of ˆ ρ ( i, ω ); see (2.13)] with a collection of disjoint and rotated unit ⌊ J ⌋ ≥ J − ≥ ˜ N ˜ L n +2)(˜ L +1)+2 R − ≥ ˜ N ˜ L / n )(3 / L + n ˜ L − ≥ ˜ N N − ≥ ˜ N N ( − ) ≥ ˜ N N . L. GOERGEN cubes ˜ C k with cardinality at most ( R + 1)(2 ˜ L + 1) d − . As a result, for0 < a < E [ˆ ρ ( i, ω ) a ] ≤ X k E (cid:20) sup x ∈ ˜ C k ˆ q ( x, ω ) a inf x ∈ ˜ C k ˆ p ( x, ω ) a (cid:21) . (2.33)By Harnack’s inequality, there is a constant c ≥ x ∈ ˜ C k ˆ q ( x, ω )inf x ∈ ˜ C k ˆ p ( x, ω ) ≤ c ˆ q ( x k , ω )ˆ p ( x k , ω ) for every 1 ≤ k, ω ∈ Ω . Moreover, observe that ˆ q ( x k , ω ) ≤ q ◦ t x k ω ; see (2.6) for the notation. Usingtranslation invariance, we see that the second term on the right-hand sideof (2.15) is less than or equal to2 X ≤ m ≤ n +1 (( R + 1)(2 ˜ L + 1) d − c a E [ ρ a ]) ( m + n − / . (2.34)Choosing c ≥ ( R + 1)3 d − c sufficiently large completes the proof of Propo-sition 2.2. (cid:3) Similarly to [19], we are going to iterate (2.9) along an increasing sequenceof boxes B k , which tend to look like infinite slabs transversal to the direction ℓ . For the definition of these boxes, we consider u ∈ (0 , , v = 8 , α = 240 , (2.35)and choose two sequences L k , ˜ L k , k ≥
0, such that L ≥ c , R + 2 ≤ ˜ L ≤ L , and for k ≥ , (2.36) L k +1 = N k L k , with N k = αu v k and ˜ L k +1 = N k ˜ L k . As a consequence we see that for k ≥ L k = (cid:18) αu (cid:19) k v k ( k − / L , (2.37) ˜ L k = (cid:18) L k L (cid:19) ˜ L . (2.38) Lemma 2.4.
There exists c ≥ c , such that when for some L ≥ c , R + 2 ≤ ˜ L ≤ L , a ∈ (0 , , u ∈ [ κ L /d , , ϕ = c ˜ L ( d − L E [ ρ a ] ≤ κ u L , (2.39) then for all k ≥ , ϕ k def = c ˜ L ( d − k +1 L k E [ ρ a k k ] ≤ κ u k L k with a k = a − k , u k = u v − k . (2.40) N EFFECTIVE CRITERION AND A NEW EXAMPLE As the proof is purely algebraic and hence identical to the proof of [19],Lemma 2.2, we omit it here. We now use the induction result to control theexit behavior from a slab.
Proposition 2.5.
There exists c ≥ c , c > , such that when for some L ≥ c , R + 2 ≤ ˜ L ≤ L , c (cid:18) log 1 κ (cid:19) d − ˜ L ( d − L d − inf a ∈ (0 , E [ ρ a ] < , (2.41) with B , ρ as in (2.6), then for some c > , lim sup L →∞ L − exp { c (log L ) / } log P [ ˜ T − bL < T L ] < for all b > . (2.42) Proof.
In view of (2.37), (2.38), we see that (2.39) is equivalent to u − d − κ − u L c α d − ˜ L ( d − L E [ ρ a ] ≤ . (2.43)The minimum of the function [ κ L /d , ∋ u u − d − κ − u L is c ′ ( L × log κ ) d − , provided L ≥ c . Hence choosing c = 2 c ′ c α d − , we canmake sure that whenever (2.41) holds, for some L ≥ c , R + 2 ≤ ˜ L ≤ L ,then (2.39) holds for some a ∈ (0 , , u ∈ [ κ L /d , k ≥
0. For any b >
0, we are now looking for a bound on P [ ˜ T − bL < T L ] when L is large. For every large enough L , we can find aunique k with L k < bL ≤ L k +1 . (2.44)We then introduce the auxiliary box B ′ k = B ( R , L k − R, L k , ˜ L k + 1), and usean argument similar to (2.23)–(2.26) to find a lower bound for P ,ω [ ˜ T − bL >T L ]; that is, we require in essence that the trajectory successively exits cer-tain translates of the box B ′ k through the “right” side, ⌊ LL k ⌋ + 1 times.We therefore cover the set G ′ def = { x ∈ R d , | x | ⊥ ≤ ( LL k + 1)( ˜ L k + 1) , x · ℓ ∈ ( − bL, L ) } , playing the role of former set G ( J − C ′ j with centers x ′ j . The cardinality of this collection is at most m k def = (( b + 1) L + 1)(2( LL k + 1)( ˜ L k + 1) + 1) ( d − . For L large, we introducethe eventΓ def = (cid:26) ω ∈ Ω : sup j sup x ∈ C ′ j P x,ω [ X T Bk + x ′ j ∈ ∂ − B k + x ′ j ] ≥ κ (1 / u k L k (cid:27) . (2.45)Then, for any ω ∈ Γ c , we obtain by arguments as before that P ,ω [ ˜ T − bL > T L ] ≥ (1 − κ (1 / u k L k ) ( L/L k +1) . (2.46) L. GOERGEN
On the other hand, using translation invariance, Harnack’s inequality andChebyshev’s inequality, we find that there is a c > P [Γ] ≤ cm k κ ( − / u k L k E [ q k ] . (2.47)Since q k ≤ ρ a k k and because of (2.40), we obtain that P [Γ] ≤ c ′ m k c ˜ L ( d − k +1 L k κ (1 / u k L k , (2.48)and a simple computation using (2.44) and (2.36) shows that for large L , m k ˜ L ( d − k +1 L k ≤ (( b + 1) L + 1)(2( L/L k + 1)( ˜ L k + 1) + 1) d − N d − k ˜ L d − k L k ≤ (1 /b + 2) L k +1 ( d − ( L k +1 / ( bL k ) + 1) d − ˜ L d − k N d − k ˜ L d − k L k (2.49) ≤ c ( b ) (cid:18) b + 1 N k (cid:19) d − N dk N − d − k ≤ c ′ ( b ) , since d ≥
2. As a consequence, we obtain from (2.48) that P [Γ] ≤ c ( b ) κ (1 / u k L k . (2.50)Assembling (2.46), (2.50) and using 1 − p m ≤ m (1 − p ), p, m ≥
0, we see thatfor large L : P [ ˜ T − bL < T L ] ≤ (cid:18) c ( b ) + LL k + 1 (cid:19) κ / u k L k . (2.51)From (2.40), (2.36), we obtain u k L k = u α v − k L k +1 ( ) ≥ u α v − k bL , andso the right-hand side of (2.51) is less than c ′ ( b ) N k κ / u /α ) v − k bL . Fromthe inequality L k ≤ bL and (2.37), we deduce that if L is large, then k ≤ c (log bL ) / , and we obtain P [ ˜ T − bL < T L ] ≤ c ′ ( b ) exp (cid:26) − u α (cid:18) log 1 k (cid:19) bL exp( − c (log( bL )) / ) (cid:27) , (2.52)for large L . This implies the claim (2.42). (cid:3) We are now ready to prove the main result of this section.
Theorem 2.6.
There exists a constant c ( d ) > , such that for ℓ ∈ S d − the following conditions are equivalent: N EFFECTIVE CRITERION AND A NEW EXAMPLE (i) There exist a ∈ (0 , and a box B = B ( R , L − R − , L + 2 , ˜ L ) with R ( e ) = ℓ, L ≥ c , R + 2 ≤ ˜ L < L with c (cid:18) log 1 κ (cid:19) d − ˜ L d − L d − E [ ρ aB ] < , (2.53)(ii) ( T ′ ) holds with respect to ℓ [see (1.11)], (iii) ( T ) γ holds with respect to ℓ for some γ ∈ ( , [see (1.10)]. Proof.
The implication (i) implies (ii) is proved in the same way asthe corresponding statement in [19], Theorem 2.4. Indeed, we define c =2 ( d − c and observe that as a result of (2.53), c (cid:18) log 1 κ (cid:19) d − ˜ L ′ ( d − L d − E [ ρ aB ] < , (2.54)with ˜ L ′ = ( ˜ L + 2) ∧ L ∈ ( ˜ L, L ). If B ′ denotes the box B ( R ′ , L − R − , L +1 , ˜ L ′ ) and if the rotation R ′ is close enough to R , p B ≤ p B ′ and hence ρ B ′ ≤ ρ B . (2.55)As a result, whenever R ′ is sufficiently close to R , we can apply Proposition2.5 to the box B ′ , and find thatlim sup L − γ log P [ ˜ T ℓ ′ − bL < T ℓ ′ L ] < γ ∈ (0 , , b > ℓ ′ = R ( e ) . This proves (ii). It is plain that (iii) follows from (ii).We now show that (iii) implies (2.53). The neighborhood appearing inthe definition of ( T ) γ contains for some small α > j = 2 , . . . , d , thevectors ℓ ′ j = cos( α ) ℓ + sin( α ) R ( e j ), ℓ ′′ j = cos( α ) ℓ − sin( α ) R ( e j ). For large L ′ and 0 < b <
1, we choose L + 2 = L ′ − b α ) and ˜ L = L ′ b α ) . (In particular˜ L ≤ L if L ′ is large enough depending on α and b .) As a consequence, if weset B = B ( R , L − − R, L + 2 , ˜ L ), then ∂ − B is included in the region where x · ℓ ′ j ≤ − bL ′ or x · ℓ ′′ j ≤ − bL ′ for some 2 ≤ j ≤ d (see also Figure 2). In otherwords, E [ q B ] ≤ P " there exists ℓ ′ ∈ d [ j =2 { ℓ ′ j , ℓ ′′ j } : ˜ T ℓ ′ − bL ′ < T ℓ ′ L ′ , (2.57)and from (1.10), we see that for some c > E [ q B ] ≤ d − e − cL γ if L is large enough . (2.58)Hence for large L , for a ∈ (0 ,
1) and c ′ > E [ ρ aB ] ≤ E [ ρ aB , p B ≥ e − c ′ L γ ] + E [ ρ aB , p B < e − c ′ L γ ] , (2.59) L. GOERGEN
Fig. 2. so that using the definition (2.5) and Jensen’s inequality to bound the firstterm and ρ B ≤ κ − ( L +3) because of Lemma 2.1 to control the second term,we find for large L, a ∈ (0 ,
1) and c ′ > E [ ρ aB ] ≤ e c ′ aL γ E [ q B ] a + κ − a ( L +3) P [ q B ≥ − e − c ′ L γ ](2.60) ( ) ≤ (2( d − a e ( c ′ − c ) aL γ + 2 κ − a ( L +3) d − e − cL γ . If we choose a = L − / and c ′ > L →∞ L − ( γ − / log E [ ρ L − / B ] < . (2.61)This implies (2.53) and thus finishes the proof of Theorem 2.6. (cid:3) Remark 2.1.
As mentioned in the Introduction, it is conjectured thatthe conditions ( T ), ( T ′ ) and ( T ) γ for a γ ∈ (0 ,
1) are all equivalent andTheorem 2.6 proves part of it. An improvement of the rather crude boundon the second term on the right-hand side of (2.59) is likely to yield theequivalence of ( T ′ ) and ( T γ ) also for γ smaller than 1 /
2. Moreover, thelatter theorem together with Proposition 2.5 strengthen the belief that ( T )and ( T ′ ) are equivalent. Indeed, we have in fact obtained the equivalence of( T ′ ) | ℓ andlim sup L →∞ L − exp { c (log L ) / } log P [ ˜ T ℓ ′ − bL < T ℓ ′ L ] < b > ∗ ) N EFFECTIVE CRITERION AND A NEW EXAMPLE and ℓ ′ close to ℓ [cf. (2.42)], which is just slightly weaker than ( T ) | ℓ , sinceexp { c (log L ) / } grows more slowly than any polynomial. Also note thatusing [11], (3.36) therein, ( ∗ ) actually holds for ℓ ′ ∈ S d − satisfying ℓ ′ · v > v = lim t →∞ X t t denotes the limiting velocity which has been shown tobe deterministic and nonzero (ballistic behavior) under ( T ′ ) | ℓ when d ≥ The one-dimensional case.
We introduce here the one-dimensionalcounterpart of the effective criterion and show that condition ( T ) is equiv-alent to ( T ′ ) and to P -a.s. transience; see Proposition 2.7. Unlike the mul-tidimensional case, condition ( T ′ ) does not imply ballistic behavior when d = 1, since one can construct one-dimensional diffusions in random envi-ronments that tend to infinity, hence satisfy ( T ′ ), and have zero limitingvelocity. A natural question to ask is then whether directional transience,that is, convergence to ∞ into some direction, or at least ballistic behaviorimplies ( T ′ ) also in higher dimensions.We first adapt the definitions (2.4), (2.5), (2.10), (2.13) to the one-dimen-sional setting. Instead of boxes or slabs, we now consider intervals. For any L > ρ B [see (2.5)] is replaced by ρ L = P ,ω [ ˜ T − L < T L ] P ,ω [ ˜ T − L > T L ] . (2.62)For L ≥ , i ∈ Z , we redefine S i [see (2.10)] as S i = iL . The definition ofthe stopping times V k , k ≥ ω ∈ Ω, i ∈ Z : ˆ ρ ( i, ω ) = ˆ q ( iL , ω )ˆ p ( iL , ω ) , whereˆ q ( x, ω ) = P x,ω [ X V ∈ S I ( x ) − ] = P x,ω [ ˜ T − L + x < T L + x ] = 1 − ˆ p ( x, ω ) . Proposition 2.7. ( d = 1 ) The following conditions are equivalent: (i) There exists a ∈ (0 , , L > R , such that E [ ρ aL ] < . (ii) There exists
L > R , such that E [log ρ L ] < . (iii) Condition ( T ) holds relative to e . (iv) Condition ( T ′ ) holds relative to e . (v) lim t →∞ X t · e = ∞ , P -a.s. Proof.
The fact that (i) implies (ii) follows from Jensen’s inequalitysince by Lemma 2.1 E [ ρ aL ] ≤ κ − a ( L +1) < ∞ . We now show that (ii) implies L. GOERGEN (iii). We have from (ii) that − µ def = E [log ρ L ] < L > R . We aregoing to use a similar argument as in Appendix A.1 or as in [18], Proposition2.6 therein. For any b > L > L /b , we define n ′ = (cid:22) bLL (cid:23) and set n = (cid:22) LL (cid:23) . (2.63)[In the spirit of Appendix A.1, − n ′ plays the role of − n + 1; see, e.g., (A.2)or (A.3).] We define the function f on {− n ′ , − n ′ + 1 , . . . , n + 2 } by (A.1)and modify the definition of τ [see (A.3)] as follows: τ = inf { k ≥ X V k ∈ S n +2 ∪ S − n ′ } . (2.64)Since − ˆ p ( X V m ) + ˆ q ( X V m ) ρ ( I ( X V m )) − vanishes P ,ω -a.s. for all m ≥
0, weobtain by an argument similar to (A.4)–(A.7), that for all ω ∈ Ω and
L > L b : P ,ω [ ˜ T − bL < T L ](2.65) ≤ f (0) f ( − n ′ ) Q − − n ′ ,n +1 Q − − n ′ ,n +1 = Q − − n ′ , + Q − − n ′ , + · · · + Q − − n ′ ,n +1 Q − − n ′ , − n ′ +1 + · · · + Q − − n ′ ,n +1 ≤ . We then take the expectation with respect to P of the left-hand side and splitit according to the sets where sup ≤ k ≤ n +1 Q − − n ′ ,k is smaller, respectivelylarger, than n +2 e − c µ L with c µ def = µb L . As a consequence: P [ ˜ T − bL < T L ] ≤ e − c µ L + (cid:18) LL + 2 (cid:19) (2.66) × sup ≤ k ≤ n +1 P " k X j = − n ′ +1 log ˆ ρ ( j ) ≥ − c µ L − log( n + 2) . Then we decompose the sum appearing in the second term into three sums ofindependent random variables ˆ ρ ( j ) , j = i mod 3 where i = 0 , n ≤ n ′ +1 b and by the choice of c µ , we observe for all n ′ large enoughthat for any 0 ≤ k ≤ n + 1, we have n ′ + k ( c µ L + log( n + 2)) ≤ µ/
4. Hencethe probability on the right-hand side of (2.66) is less than X i =0 P " n ′ + k X − n ′ +1 ≤ j ≤ kj = i mod 3 ( − log ˆ ρ ( j ) − µ ) ≤ − µ/ . (2.67)As for any j ∈ Z , ω ∈ Ω, | log ˆ ρ ( j ) | ≤ ( L + 1) log( κ ), by (2.8), it follows froman Azuma-type inequality (see, e.g., [1]) that for any 0 ≤ k ≤ n + 1, (2.67) N EFFECTIVE CRITERION AND A NEW EXAMPLE is less than ≤ X i =0 exp (cid:26) − (cid:18) µ n ′ + k (cid:19) |{ j ∈ [ − n ′ + 1 , k ] | j = i mod 3 }| − × (( L + 1) log κ − + µ ) − (cid:27) (2.68) ≤ (cid:26) − c ( µ, L ) (cid:18) bLL − (cid:19)(cid:27) . In view of (2.66), this implies condition ( T ); see (1.12).The implication (iii) ⇒ (iv) is clear. To show (iv) ⇒ (i), we follow theargument of the corresponding multidimensional statement [see Theorem2.6, (iii) ⇒ (i)], that is, in place of (2.58) and (2.60), we have P [ ˜ T − L 0, under P ,ω for an L > R in fact showsthe equivalence of (ii) and (v). For the reader’s convenience, we extract andpresent here the ideas which are relevant for the implication (v) ⇒ (ii). For ω ∈ Ω, L > R , n ∈ Z , we introduce the shortcut notation p n = ˆ p ( nL , ω ) =1 − q n and δ n def = P nL ,ω [ ˜ T ( n − L = ∞ ] def = 1 − η n . We claim that P [ δ > 0] = 1 . (2.69)Indeed, let us assume by contradiction that there exists some ω ′ in the setof full measure { ω ∈ Ω : P − L ,ω [ X t → ∞ ] = 1 } such that δ ( ω ′ ) = 0. Thenrepeated use of the strong Markov property shows that P ,ω ′ [lim inf t X t ≤− L ] = 1, a contradiction.Next, we see that for n ∈ Z , η n = P nL ,ω [ ˜ T ( n − L < ∞ ](2.70) = q n + p n η n +1 η n and thus η n = q n − p n η n +1 . As a consequence, for all ω ∈ Ω , n ≤ − δ n = 1 − η n ( ) = 1 − − p n − p n η n +1 = ˆ ρ ( n ) − η n δ n +1 , and by induction(2.71) = ( ˆ ρ ( n )ˆ ρ ( n + 1) · · · ˆ ρ ( − − η n η n +1 · · · η − δ . Taking the logarithm of the latter expression and splitting the resulting suminto sums of i.i.d. random variables [similarly as below (2.66)], we obtainfrom the law of large numbers:lim n →∞ n log δ n = E [ − log ˆ ρ (0)] + E [log η ] , P -a.s. , (2.72) L. GOERGEN since lim n n log δ = 0 , P -a.s. by (2.69).On the other hand, by translation invariance of P , we see that for any ε > P [ | log δ n | > εn ] = P [ | log δ | > εn ] n →∞ −→ P [ δ = 0] ( ) = 0 . (2.73)In other words, n log δ n converges to 0 in probability, so the right-handside of (2.72) vanishes and E [log ˆ ρ (0)] = E [log η ] which is strictly negativebecause of (2.69). This proves the implication (v) ⇒ (ii).To show the converse implication, we use the fact that (ii) implies condi-tion ( T ). Following [11] [see the proof of (3.1) ⇒ (3.2) therein], we observethat P [ T L = ∞ ] ≤ P [ ˜ T − L < T L ], since P [ ˜ T − L = T L = ∞ ] = 0 as in everytime unit, the trajectory can escape from the interval [ − L, L ] with a proba-bility bounded away from 0. Observe that the left-hand side increases with L while the right-hand side tends to 0 by condition ( T ) | e . Hence P -a.s.,lim sup t →∞ X t = ∞ . From the strong Markov property and translation in-variance of P , we obtain for any L > P [ ˜ T L/ ◦ θ T L < T L/ ◦ θ T L ] = P [ ˜ T − L/ < T L/ ] . (2.74)Under condition ( T ) | e , the right-hand side decreases exponentially andhence an application of Borel–Cantelli’s lemma yields that P -a.s. for largeinteger L , T L/ < ˜ T L/ ◦ θ T L + T L . As a result, we can P -a.s. constructan integer-valued sequence L k ↑ ∞ , with L k +1 = ⌊ L k ⌋ and T L k +1 < ˜ T L k / ◦ θ T Lk + T L k , k ≥ 0. This shows (v). (cid:3) Remark 2.2. Let us mention that for any L > E [log ρ L ] = − L E [ b (0) /a (0)] , (2.75)and as a consequence, if conditions (i) or (ii) above are satisfied for some L ≥ R , they are in fact satisfied for all L > 0. Indeed, using the scale func-tion s ( x, ω ) = R x exp {− R y b ( u, ω ) /a ( u, ω ) du } dy , for x ∈ R , ω ∈ Ω (see, e.g.,[2], pages 78 and 88), we can write ρ L = s ( L ) − s ( − L ) . It follows that for L > E [log ρ L ] equals E (cid:20) log Z L e − R y b ( u,ω ) /a ( u,ω ) du dy (cid:21) (2.76) − E (cid:20) log Z − L e − R y − L b ( u,ω ) /a ( u,ω ) du e R − L b ( u,ω ) /a ( u,ω ) du dy (cid:21) . Because of translation invariance of P , the second term becomes E (cid:20) log Z L e − R y b ( u,ω ) /a ( u,ω ) du dy (cid:21) + E (cid:20)Z − L b ( u, ω ) a ( u, ω ) du (cid:21) , N EFFECTIVE CRITERION AND A NEW EXAMPLE so that the first term of (2.76) is canceled out. Fubini’s theorem then yields E [log ρ L ] = − R − L E [ b ( u,ω ) a ( u,ω ) ] du , and the claim follows from translation invari-ance of P . 3. An example of a ballistic diffusion. Main result and preliminaries. In this section, we use the effectivecriterion to show that a Brownian motion perturbed by a small random driftwhich is bounded by ε > ℓ = e is oforder ε − η with η > T ′ ) | e . The interest of this classof diffusions stems from the fact that it contains new examples of ballisticdiffusions which in particular do not fulfill the criterion of [11], Theorem 5.2therein, which states that there exists a constant c e > E [( b (0 , ω ) · e ) + ] > c e E [( b (0 , ω ) · e ) − ] , (3.1)then ( T ) | e holds. Before we give further explanations on this matter (seeRemark 3.1 below), we introduce the family of perturbed Brownian motionsstudied in this section. For any ε ∈ (0 , ¯ K ], η > ω ∈ Ω, we consider theclass of diffusions attached to an operator of the form L = ∆ + b ( x, ω ) · ∇ , (3.2)where we require that for all x ∈ R , ω ∈ Ω, | b ( x, ω ) | ≤ ε, λ def = E [ b (0 , ω ) · e ] ≥ ε − η . (3.3)Note that the constant ¯ K , the ellipticity constant ν and the dependencerange R [see (1.2)–(1.5)] do not depend on ε . We keep the convention con-cerning constants stated at the end of the Introduction. Moreover, whenwe write that an expression holds “for large enough L ” we mean that theexpression holds for all L larger than some c ( η ).The main result of the section is Theorem 3.1. When d ≥ , for any η ∈ (0 , there is ε ( η, d ) > suchthat whenever (3.3) holds for < ε ≤ ε , then condition ( T ′ ) | e is satisfied. Remark 3.1. Clearly, (3.1) is equivalent to E [ b (0 , ω ) · e ] > ( c e − E [( b (0 , ω ) · e ) − ] . (3.4)An inspection of the proof of Theorem 5.2 in [11], reveals that c e > 1, andhence (3.4) fails when ε > E [ b (0 , ω ) · e ] is of order ε − η with 0 <η < ω ∈ Ω ( b (0 , ω ) · e ) − is of order ε under an adequate choice of P .With this observation one can rather straightforwardly produce exampleswhere (3.3) holds with ε < ε ( η, d ), but (3.1) or (3.4) fails. L. GOERGEN The rest of the section is devoted to the proof of Theorem 3.1. We willverify the effective criterion (2.53) when ε is smaller than some ε ( η, d ) for a = 1 / B = B (Id , N L ′ − R − , N L ′ + 2 , ( N L ′ ) ) [see (2.2)],where N = L and L = L ′ − R L = (cid:22) ε (cid:23) . (3.5)The starting point to estimate E [ ρ / B ] is (2.15). Here we set [cf. (2.6)] L = N L ′ , ˜ L = ( N L ′ ) , L = L ′ , n = N and a = 1 / 2. With these choices, thebox B defined above, on which we want to check (2.53), equals B + e . Inorder to apply (2.15) we use the following. Lemma 3.2. For a ∈ (0 , and B a box as in (2.6) with ℓ = e , L ≥ R + 3 and R = Id , E [ ρ aB + e ] ≤ c a E [ ρ aB ] . (3.6) Proof. Since for every ω ∈ Ω , P x,ω [ X T B ∈ ∂ ± B ] is harmonic on ( − , d ,Harnack’s inequality implies that P − e ,ω [ X T B ∈ ∂ − B ] P − e ,ω [ X T B ∈ ∂ + B ] ≤ cρ B ( ω ) . The claim then follows from translation invariance of P . (cid:3) For the purpose of this section, we need a bound on P [ G c ] appearing in(2.15) which differs from (2.32) and which is essentially the same as theestimate in [20], Theorem 1.1. We now follow [20] to introduce the notationused for this bound. Let h, H, M be positive integers with2 h ≤ H ≤ ( N L ′ ) 32 and M = (cid:22) ( N L ′ ) H (cid:23) . (3.7)Later on [see (3.51)], we will choose H and h to be of order ( N L ′ ) and L ,respectively. We introduce the exit time S from a tube: S = inf (cid:26) t > | ( X t − X ) · e | ≥ L or sup j ≥ | ( X n − X ) · e j | ≥ h (cid:27) (3.8)and the expected displacement∆( x, ω ) = E x,ω [ X S ] − x, x ∈ R d , ω ∈ Ω . (3.9)Moreover, for 0 < γ ≤ 1, later chosen to be of order ε − η [see (3.51)], wedefine p L = inf j ≥ P [for all z ∈ ˜ B j , ∆( z, ω ) · e ≥ γL ] , (3.10) N EFFECTIVE CRITERION AND A NEW EXAMPLE where for 2 ≤ j ≤ d , ˜ B j = { y ∈ B, | y · e j | < H } . (3.11)Let us now state the analogue of Theorem 1.1 in [20]. Proposition 3.3. There exists a constant c > R + 3 such that when L ≥ c and δ − = exp (cid:26) − γN (cid:27) + 10 Nγ exp (cid:26) − γN (cid:18) H hN − γ (cid:19) (cid:27) < , (3.12) then for any < a ≤ E [ ρ aB ] ≤ c a κ − aNL ′ d exp (cid:26) − M (cid:18) p L − N LM log κ − log δ (cid:19) (cid:27) (3.13) + c a E [ˆ ρ (0 , ω ) a ] N/ (1 − E [ˆ ρ (0 , ω ) a ] / ) + . Since the proof is very similar to the one of Theorem 1.1 in [20], weonly make a few comments here. Because of (3.6), we can estimate E [ ρ aB ]with the help of (2.15). We bound the second term on the right-hand sidein the latter expression using translation invariance of P and obtain thesecond term on the right-hand side of (3.13). The intuitive idea behind theestimate on P [ G c ] in the first term on the right-hand side of (2.15), leadingto the first term on the right-hand side of (3.13), is to consider nested boxesˆ B k = ( − N L ′ + R + 2 , N L ′ + 2) × ( − k H, k H ) d − for 0 ≤ k ≤ M containedin the big box B . Then in order not to exit through “the left or right” of B , the trajectory has to reach the boundary of box ˆ B k before exiting B andthen move from box ˆ B k to box ˆ B k +1 without exiting B . The probability ofthis last step is related to the quantity 1 − p L .Note that the coefficient in the first term of δ − differs from the resultin [20] as the width of B is a multiple of L ′ while the definition of thetime S uses the quantity L = L ′ − R . This affects the right-hand side of theexpression below (1.24) in [20].Despite the finite range dependence, the remark in [20] below (1.29) stillholds since (in the notation of [20]) the random variables Z k ( e ) and Z k − ( e )are measurable respectively in H { z ∈ R d : z · e ≥ kH − H − h } and in H { z ∈ R d : z · e ≤ k − H + H + h } . The involved half-spaces are separated by a dis-tance 2( H − h ) which is larger than H by (3.7). Hence ( Z k ) ≤ k ≤ M are inde-pendent if H and thus L are large enough. L. GOERGEN Bounds on the Green operator. The main Theorem 3.1 will followafter choosing h and H as in (3.51) once we show exponential decay in L ∝ ε − of both terms on the right-hand side of (3.13) for a = 1 / 2. Thereforethe goals of this section are to find a tractable expression for ˆ ρ (0 , ω ) (seeLemma 3.5) that involves the Green operator of the diffusion killed whenexiting the open slab S def = { x ∈ R d : | x · e | < L } , and then investigate itsrelation with the Green operator of killed Brownian motion; see Proposition3.8. Certain deterministic estimates on the latter operator and its kernel (seeLemmas 3.7 and 3.9) will then be instrumental in the proof of the desiredexponential decay of E [ ρ / B ]; see Proposition 3.10.Throughout this section, we use the shortcut notation b = b · e andwe set k f k ∞ = sup x ∈S | f ( x ) | , for any function f on S . For any boundedmeasurable function f on S and any x ∈ S , ω ∈ Ω, let us denote with G ω S f ( x ) def = E x,ω (cid:20)Z T S f ( X s ) ds (cid:21) , respectively(3.14) G S f ( x ) def = E (cid:20)Z T S f ( x + W s ) ds (cid:21) , the Green operator of the diffusion, respectively Brownian motion, killedwhen exiting the slab S . (Here E denotes the expectation with respect tosome measure under which W s is a Brownian motion.) Note that by (3.16)below, these operators acting on L ∞ have norm bounded by 2 L . Moreover,the semi-group P ωt of the diffusion in environment ω killed when exiting S is defined as P ωt f ( x ) = E x,ω [ f ( X t ) , t < T S ] for x ∈ S , t ≥ . (3.15)In a similar fashion, we denote with P t the semi-group of a Brownian motionkilled when exiting S .The following lemma states basic bounds on the expected exit time fromthe slab S and on the supremum-norm of the operator P ωt . Lemma 3.4. For ω ∈ Ω , ε ∈ (0 , / , x ∈ S , under the assumption (3.3)and with the definition (3.5), ( L − ( x · e ) ) ≤ E x,ω [ T S ] ≤ L − ( x · e ) ) . (3.16) For any bounded measurable function f and any ω ∈ Ω , k P ωt f k ∞ ≤ c k f k ∞ exp( − c t/L ) for t > . (3.17) Proof. To show (3.16), we consider for x ∈ S , ω ∈ Ω the P x,ω -martingale( X t ∧ T S · e ) − ( X · e ) − Z t ∧ T S b ( X s , ω )( X s · e ) ds − t ∧ T S . (3.18) N EFFECTIVE CRITERION AND A NEW EXAMPLE After taking expectations and using the monotone convergence theorem,we obtain (3.16) from our assumption | b ( · , · ) | ≤ ε [see (3.3)] and the choice L ≤ ε [see (3.5)].We now turn to (3.17). By the support theorem (see [2]) applied to therescaled diffusion L X L t and the fact that | Lb | ≤ , the probability under P x,ω that the trajectories leave the slab within time L when starting in x ∈ S is bounded away from 0 by some constant c . Hence the strongMarkov property yields for any t > x ∈ S , ω ∈ Ω, that P x,ω [ t ≤ T S ] ≤ c exp( − c t/L ), and (3.17) follows from the definition (3.15). (cid:3) Remark 3.2. 1. For Brownian motion starting at x ∈ S , the expected exit time from theslab S equals L − ( x · e ) . The analogue of (3.17) for Brownian motionis also valid.2. We point out that since T S has a finite moment under P x,ω by (3.16),Fubini’s theorem applied to (3.14) yields for any bounded measurablefunction f and any ω ∈ Ω , x ∈ S that G ω S f ( x ) = Z ∞ P ωt f ( x ) dt. (3.19)Of course, the same relation holds for the killed Brownian motion.Let us now introduce the following shortcut notation for the set appearingin the definition of ˆ ρ (0 , ω ) [see (2.13)]: V def = (cid:26) x ∈ R d ; | x · e | ≤ R , | x | ⊥ ≤ 14 ( N L ′ ) (cid:27) . (3.20)For later purposes, we observe that (3.16) and our assumption (3.3) on λ imply that there are constants c > L ( c , η ) such that when L ≥ L ,then for any x ∈ V , ω ∈ Ω, G ω S λ ( x ) = λE x,ω [ T S ] ≥ c L η . (3.21)The next lemma provides a tractable expression of ρ (0 , ω ) in terms of theGreen operator G ω S . Lemma 3.5. For L ≥ R, ω ∈ Ω , with (3.3) and (3.5), ˆ ρ (0 , ω ) = sup x ∈V L − x · e − G ω S ( b ( · , ω ))( x ) L + x · e + G ω S ( b ( · , ω ))( x ) ≤ . (3.22) [See (2.13), (3.20) for the notation.] L. GOERGEN Proof. For any x ∈ S , ω ∈ Ω, X t ∧ T S · e − X · e − R t ∧ T S b ( X s , ω ) ds is a P x,ω -martingale. Hence, after taking expectations, we obtain from thedominated convergence theorem that [see (2.14) for the notation]ˆ p ( x, ω ) = x · e + L + G ω S ( b ( · , ω ))( x )2 L . (3.23)Inserting this expression into the definition (2.13) of ˆ ρ (0 , ω ) yields the claimedequality. Using (3.16), (3.5), we see that for all L > | G ω S ( b ( · , ω ))( x ) | ≤ L , (3.24)and thus the inequality in (3.22) follows when L ≥ R . (cid:3) In order to explore the relationship between the Green operators of thediffusion and Brownian motion [see (3.40)], we need to collect a few factsabout the semi-group of Brownian motion. From [16], Theorem 8.1.18, wehave that whenever f is a continuous and bounded function, then ( t, x ) P t f ( x ) is bounded and in C , ([0 , ∞ ) × S , R ). Moreover, ∂∂t P t f = 12 ∆ P t f in (0 , ∞ ) × S , (3.25) lim t → P t f ( x ) = f ( x ) , x ∈ S . (3.26)Since every point on the boundary of S is regular according to [16], (8.1.16)therein, we have the following continuity property at the boundary (see [16],Theorem 8.1.18):lim ( t,x ) → ( s,a )( t,x ) ∈ (0 , ∞ ) ×S P t f ( x ) = 0 for ( s, a ) ∈ (0 , ∞ ) × ∂ S . (3.27)Our next step is to express P t and G S in terms of kernels using “themethod of images” from electrostatics. Proposition 3.6. Let f be a bounded measurable function on R d . If wedefine for t > x, y ∈ S p ( t, x, y ) = ∞ X k = −∞ p d ( t, x, y + 2 k Le ) − p d ( t, x, y ∗ + (2 k + 1)2 Le ) , (3.28) where p d ( t, x, y ) def = (2 πt ) − d/ exp {| x − y | / t } is the d -dimensional heat ker-nel and y ∗ is the image of y under reflection with respect to { z ∈ R d : z · e =0 } , then P t f ( x ) = Z S p ( t, x, y ) f ( y ) dy. (3.29) N EFFECTIVE CRITERION AND A NEW EXAMPLE Moreover, when d ≥ , if we define Green ’ s function for distinct x, y ∈ S by g ( x, y ) def = ∞ X k = −∞ g d ( x, y + 2 k Le ) − g d ( x, y ∗ + (2 k + 1)2 Le ) , (3.30) where g d ( x, y ) def = R ∞ p d ( t, x, y ) dt = γ d | x − y | − d for x = y and an appropriateconstant γ d , then G S f ( x ) = Z S g ( x, y ) f ( y ) dy. (3.31) Proof. The fact that p ( t, x, y ) in (3.28) satisfies the equality in (3.29)follows from [8], Proposition 8.10, after mapping the interval [0 , a ] to [ − L, L ]and after multiplying with p d − . It is well known that g d ( x, y ) equals γ d | x − y | − d for an appropriate constant γ d when d ≥ x = y (see, e.g., [16],(8.4.10)). To see that the expression in (3.30) is indeed the kernel of G S , weobserve that p ( t, x, y ) is integrable over t for x = y , since by the monotoneconvergence theorem, we have R ∞ p ( t, x, y ) dt ≤ P ∞ k = −∞ g d ( x, y + 2 k Le ) + g d ( x, y ∗ + (2 k + 1)2 Le ), and since the latter series converges absolutely when d ≥ 4. Moreover, with dominated convergence, g ( x, y ) = Z ∞ p ( t, x, y ) dt for x = y. (3.32)Then we insert (3.29) into (3.19) and since ( t, y ) p ( t, x, y ) f ( y ) is productintegrable by Tonelli’s theorem and (3.17), we obtain (3.31) from Fubini’stheorem and (3.32). (cid:3) The next lemma provides gradient estimates on the semi-group and theGreen operator of killed Brownian motion which play an important rolein the derivation of the perturbation equality (3.40) and in the proof ofProposition 3.10. Lemma 3.7. ( d ≥ For any bounded, continuous function f , there exist c , c > such that for all x ∈ S , t > and L > , |∇ P t f ( x ) | ≤ (cid:18) c L + c √ t (cid:19) exp (cid:18) − c t/L (cid:19) k f k ∞ , (3.33) |∇ G S f ( x ) | ≤ c k f k ∞ L. (3.34) Proof. We first show (3.33). Let ( x (1) , . . . , x ( d ) ) denote the coordinatesof a point x in R d . We estimate the partial derivatives ∂ i , i = 1 , . . . , d , of P t f ( x ) separately. As a consequence of the semi-group property, we havethat for t > x ∈ S , P t f ( x ) = Z S p ( t/ , x, z ) P t/ f ( z ) dz. (3.35) L. GOERGEN We let U ⊂ S be a neighborhood of x . To compute ∂ i P t ( x ) by interchangingderivation and integration, we need to show that | ∂ i p ( t/ , x, z ) P t/ f ( z ) | is dx × dz integrable over U × S . After an application of (3.17), we see thatsup x ∈ U Z S | ∂ i p ( t/ , x, z ) P t/ f ( z ) | dz (3.36) ≤ exp( − c t/L ) k f k ∞ sup x ∈ U Z S | ∂ i p ( t/ , x, z ) | dz. For i = 1, according to (3.28), | ∂ p ( t/ , x, z ) | is smaller than d Y j =2 p ( t/ , x ( j ) , z ( j ) ) ∞ X k = −∞ | ∂ p ( t/ , x (1) , z (1) + 4 kL ) | (3.37) + | ∂ p ( t/ , x (1) , − z (1) + (2 k + 1)2 L ) | . The integral over R d of the first d − x ∈ U , t > 0, theintegral on the right-hand side of (3.36) is smaller than X k =0 p ( t/ , x (1) , L + 4 kL ) + p ( t/ , x (1) , − L + 4 kL )+ X k = − p ( t/ , x (1) , L + (2 k + 1)2 L )+ X k =0 p ( t/ , x (1) , − L + (2 k + 1)2 L )(3.38) + Z L − L √ πt e ( − /t )( x (1) − z ) (cid:12)(cid:12)(cid:12)(cid:12) x (1) − zt (cid:12)(cid:12)(cid:12)(cid:12) dz + p ( t/ , x (1) , − L ) + p ( t/ , x (1) , L ) . For any x ∈ U , the function z p ( t/ , x (1) , z ) is monotone on ( −∞ , − L ]and on [2 L, ∞ ). Therefore the first sum in (3.38) is less than L R ∞−∞ p ( t/ , x (1) ,z ) dz = L . A similar argument yields that the second and third sums in (3.38)are less than cL . The integral in (3.38) is less than 2 R ∞ √ πt e − u u du = √ πt and the last two terms can also be bounded by c √ t . Collecting our estimates,we obtain for i = 1 that the left-hand side of (3.36) is less thanexp( − c t/L ) k f k ∞ (cid:18) cL + c ′ √ t (cid:19) . (3.39)Hence we can interchange the derivative ∂ with the integral in (3.35), andfor any x ∈ S , | ∂ P t ( x ) | is bounded by (3.39). Similar bounds on | ∂ i P t ( x ) | , N EFFECTIVE CRITERION AND A NEW EXAMPLE for 2 ≤ i ≤ d , follow from an easier version of the above arguments. Indeed,in an expression corresponding to (3.37), the last factor containing the sum,which was more delicate to treat, will not be affected by the derivative ∂ i andthus its integral over [ − L, L ] equals 1. This proves (3.33). Since the latterestimate shows that ∇ P t f ( x ) is integrable with respect to t > 0, (3.34) is animmediate consequence of (3.19). (cid:3) The link between the Green operators of the killed diffusion and the killedBrownian motion is expressed by the following perturbation equality: Proposition 3.8. Let f be a bounded, continuous function on R d . Thenwe have for all x ∈ S , ω ∈ Ω that G ω S f ( x ) = G S f ( x ) − G ω S ( b ( · , ω ) · ∇ ) G S f ( x ) . (3.40) Proof. The classical idea of the proof is to take the derivative of P ωt P u − t f ( x ) with respect to t , which yields P ωt (( L − ∆) P u − t f )( x ). Thenone integrates both sides with respect to t from 0 to u and with respect to u from 0 to infinity. The result then follows from Fubini’s theorem. Let usnow present the details of the proof. For ω ∈ Ω , u > , x ∈ S , we claim that P ωu f ( x ) − P u f ( x ) = Z u P ωt ( b ( · , ω ) · ∇ P u − t f )( x ) dt. (3.41)To prove the claim, we define for h > e ( t, x ) def = P u + h − t f ( x ) with 0 ≤ t ≤ u, x ∈ S . (3.42)According to [16], Theorem 8.1.18, e is in C , ((0 , u ) × S ). Hence we canapply Itˆo’s formula to a function e n ∈ C , ((0 , u ) × R d ) such that e n ( t, · ) = e ( t, · ) on D n def = { x ∈ S , dist( x, ∂ S ) ≥ /n } and e n ( t, · ) = 0 on S c . Because of(3.25), we obtain for all ω ∈ Ω, x ∈ D n after taking expectations: E x,ω (cid:20) e ( u ∧ T D n , X u ∧ T Dn ) − e ( h ∧ T D n , X h ∧ T Dn )(3.43) − Z u ∧ T Dn h ∧ T Dn b ( X s , ω ) · ∇ e ( s, X s ) ds (cid:21) = 0 . When n tends to ∞ , t ∧ T D n increases to t ∧ T S , and it follows from thedominated convergence theorem and (3.27) that for any ω ∈ Ω , x ∈ S , E x,ω [ e ( u ∧ T D n , X u ∧ T Dn ) , u ≥ T S ] n →∞ −→ . (3.44)The same result holds for h in place of u . From (3.33), we have thatsup ≤ t ≤ u,x ∈S |∇ e ( v, x ) | is finite. Thus coming back to (3.43) and letting L. GOERGEN n → ∞ , we obtain with dominated convergence that for any x ∈ S , E x,ω [ e ( u, X u ) , u < T S ] − E x,ω [ e ( h, X h ) , h < T S ](3.45) = E x,ω (cid:20)Z u ∧ T S h b ( X s , ω ) · ∇ e ( s, X s ) ds, h < T S (cid:21) . We now insert the definition (3.42) into the above expression and let h tendto 0 using dominated convergence. This concludes the proof of (3.41).The integral with respect to u > G ω S f ( x ) − G S f ( x ); see (3.19). On the right-hand side, (3.33) and (3.17) implythat the iterated integral Z ∞ Z ∞ | P ωt ( b ( · , ω ) · ∇ P u − t f ) | ( x )1 { t
Lemma 3.9. ( d ≥ ) For all x, y ∈ S and L > we have g ( x, y ) ≤ c | x − y | − d exp( − c | x − y | ⊥ /L ) , (3.48) |∇ g ( x, y ) | ≤ ( c | x − y | − d + c L − d ) exp( − c | x − y | ⊥ /L ) . (3.49) Moreover, for any bounded H¨older continuous function f , G S f is twice con-tinuously differentiable on S and ∆ G S f ( x ) = − f ( x ) for x ∈ S . (3.50)The proof is included in Appendix A.2 and the arguments showing (3.48)and (3.49) are similar to the proof of [20], (2.11), (2.13) therein. N EFFECTIVE CRITERION AND A NEW EXAMPLE Proof of Theorem . The starting point for the proof is (3.13)with a = . We first specify the quantities h, H, γ involved in the first termon the right-hand side of (3.13) [see (3.7), (3.10)]: h def = L ′ , H def = ⌊ ( N L ′ ) ⌋ , (3.51) γ def = c L η − . It is clear that the main Theorem 3.1 follows from the effective criteriononce we show exponential decay in L ∝ ε − of both terms on the right-handside of (3.13). We first examine the second term. It suffices to show that forlarge enough L E [ˆ ρ (0 , ω )] ≤ exp (cid:18) − c L − (cid:19) , (3.52)where c is defined in (3.21). Indeed, since we assumed N = L [see (3.5)],the second term of (3.13) then becomes smaller than cL exp( − c L ), whichwill be more than sufficient for the application of the effective criterion(2.53).To prove (3.52), we use (3.22) and write E [ˆ ρ (0 , ω )] as E (cid:20) sup x ∈V L − x · e − G ω S ( b ( · , ω ))( x ) L + x · e + G ω S ( b ( · , ω ))( x ) , inf x ∈V G ω S ( b ( · , ω ))( x ) ≥ c L η (cid:21) (3.53) + 5 P (cid:20) inf x ∈V G ω S ( b ( · , ω ))( x ) < c L η (cid:21) . When L is larger than some c ( η ), the first term becomes smaller than 1 − c L η − ≤ exp( − c L η − ). Hence (3.52) follows from the next propositionwhich estimates the second term of (3.53). Proposition 3.10. ( d ≥ ) For any η ∈ (0 , , under the assumption(3.3) and with (3.5), we have that lim sup L →∞ L − / η log P (cid:20) inf x ∈V G ω S ( b ( · , ω ))( x ) < c L η (cid:21) < , (3.54) where V and c are defined in (3.20) and (3.21). Before proving the proposition, we show that (3.54) together with ourchoices in (3.51) also yield exponential decay of the first term on the right-hand side of (3.13), which then finishes the proof of the main theorem. Using(3.5), we find that δ − ≤ exp( − cL η ) + c ′ L − η exp {− c ′′ L η ( L − c ′′′ L − η ) } , (3.55) L. GOERGEN which tends to 0 as L goes to ∞ , so that (3.12) holds when L is large. If inaddition, we know that [see (3.10) for the notation]lim inf L →∞ p L = 1 , (3.56)an easy calculation using (3.51) and M ≥ c N L [see (3.7) for the definition]shows that for L large enough, the first term on the right-hand side of (3.13)is less than c exp( − cN L ), and the effective criterion (2.53) is satisfied forlarge L .We now prove that Proposition 3.10 implies (3.56). First we cover the sets˜ B j , ≤ j ≤ d [see (3.11)] with a collection of disjoint cubes of side length R .The cardinality of this collection is for large L at most L ν where ν onlydepends on d . Translation invariance then yields p L ≥ − sup ≤ j ≤ d c ′ L ν P (cid:20) inf x ∈ [ − R/ ,R/ d ∆( x, ω ) · e < γL (cid:21) . (3.57)In this expression we will in essence replace ∆( x, ω ) · e with G ω S ( b ( · , ω ))( x ).More precisely, we claim that for large L and for all ω ∈ Ω, x ∈ [ − R , R ] d , | ∆( x, ω ) · e − G ω S ( b ( · , ω ))( x ) | ≤ c . (3.58)Then with our choice of γ [see (3.51)] and with (3.57), Proposition 3.10implies (3.56) since [ − R , R ] d ⊂ V . We now prove (3.58). The martingaleargument leading to (3.23) also shows that for any x ∈ S , ω ∈ Ω G ω S ( b ( · , ω ))( x ) = E x,ω [ X T S · e ] − x · e . (3.59)The support theorem (see [2]) applied to the rescaled diffusion L X L t yieldsa lower bound c > x ∈ R d , ω ∈ Ω) for the probability under P x,ω that X exits a cube of side length L centered at x through the “left orright.” Hence with the strong Markov property, for all ω ∈ Ω, x ∈ R d , P x,ω [ S < ˜ T − L + x · e ∧ T L + x · e ] ≤ d − − c ) L , (3.60)which becomes smaller than L − for large enough L . Since | X S · e | ≤ L + | x · e | , P x,ω -a.s. we obtain from (3.59) and (3.60) that for large enough L and for all ω ∈ Ω , x ∈ [ − R , R ] d , the left-hand side of (3.58) is less than | E x,ω [( X S − X T S ) · e , S = ˜ T − L + x · e ∧ T L + x · e ] | + c. (3.61)On the event { S = ˜ T − L + x · e ∧ T L + x · e } ∩ { ( X S · e )( X T S · e ) > } , the trajec-tory P x,ω -a.s. leaves the slab S and the box [ − L, L ] × [ − h, h ] d − + x “throughthe same side.” Hence on this event, | ( X S − X T S ) · e | ≤ R , P x,ω -a.s. for x ∈ [ − R , R ] d . It remains to show that for all ω ∈ Ω, x ∈ [ − R , R ] d , | E x,ω [( X S − X T S ) · e , (3.62) S = ˜ T − L + x · e ∧ T L + x · e , ( X S · e )( X T S · e ) < | ≤ c. N EFFECTIVE CRITERION AND A NEW EXAMPLE When x · e = 0 the above quantity vanishes. We now consider the case where0 < x · e ≤ R . The remaining case is treated analogously. We find that for0 < x · e ≤ R , P x,ω [ S = ˜ T − L + x · e ∧ T L + x · e , ( X S · e )( X T S · e ) < ≤ P x,ω [ T L < ˜ T − L + x · e < T L + x · e ] + P x,ω [ ˜ T − L + x · e < T L < ˜ T − L ] . We estimate the first term on the right-hand side. The strong Markov prop-erty implies that for all ω ∈ Ω , < x · e ≤ R , P x,ω [ T L < ˜ T − L + x · e < T L + x · e ](3.64) ≤ E x,ω [ T L < ˜ T − L + x · e , P X TL ,ω [ ˜ T − L + R/ < T L + R/ ]] . The function e ( x ) def = − e εx · e + e ε ( L + R/ satisfies L e ( x ) < | b ( · , · ) | ≤ ε . Hence e ( X t ) is a supermartingale under P x,ω for any x ∈ R d , ω ∈ Ω. Since e ( x ) is nonnegative when x · e ≤ L + R , Chebyshev’s inequality and thestopping theorem yield for any y ∈ R d with y · e = L , P y,ω [ ˜ T − L + R/ < T L + R/ ] ≤ E y,ω [ e ( X ˜ T − L + R/ ∧ T L + R/ )] e ε ( L + R/ − e ε ( − L + R/ (3.65) ≤ − e − εR/ − e − εL ≤ cε ≤ c ′ L − , for large enough L . Inserting this bound into (3.64) and repeating the sametype of argument for the second term on the right-hand side of (3.63), weobtain that its left-hand side is of order L − . This finishes the proof of (3.62)since ( X S − X T S ) · e is of order L , P x,ω -a.s. for x ∈ [ − R , R ] d . Thus (3.58)follows in view of (3.61). As a consequence, Proposition 3.10 implies (3.56)and the main theorem follows as we explained below (3.56). Proof of Proposition 3.10. The idea of the proof is to decomposethe e projection of the drift b ( x, ω ) into its expectation E [ b · e ] = λ anda mean-zero term ˜ b ( x, ω ). As a consequence, the Green operator applied to b splits into two terms: a leading term G ω S λ which is larger than twice thebound imposed on the Green operator in the event of interest in (3.54) byour choice of constants and by (3.21); an error term G ω S ˜ b that we decomposeusing the perturbation equality (3.40) and which turns out to make no sub-stantial contribution to the leading term with high probability. Hence theevent of interest in (3.54) is very unlikely. We now give the details of theproof. Let us introduce the box U def = { x ∈ R d ; | x · e | ≤ L − , | x | ⊥ ≤ ( N L ′ ) + L } (3.66) L. GOERGEN which will be useful later in a discretization step where we need to restrictourselves to points located at a constant distance of ∂ S . As mentioned abovewe define [see (3.3)] ˜ b def = b − λ. (3.67)[For the sake of simplicity we drop the ω dependence of b , ˜ b from the no-tation.] Then the perturbation equality (3.40) applied to G ω S ˜ b together with(3.21) yields that for large enough L , P (cid:20) inf x ∈V G ω S b ( x ) < c L η (cid:21) ≤ P (cid:20) inf x ∈V G S ˜ b ( x ) − G ω S ( b · ∇ ) G S ˜ b ( x ) ≤ − c L η , (3.68) sup y ∈U |∇ G S ˜ b ( y ) | ≤ L − η/ (cid:21) + P (cid:20) sup y ∈U |∇ G S ˜ b ( y ) | > L − η/ (cid:21) . The proposition obviously follows once we prove the following three claims:there exist ν ′ , ν ′′ ≥ d such that for large enough L ,on the set (cid:26) ω ∈ Ω; sup y ∈U |∇ G S ˜ b ( y ) | ≤ L − η/ (cid:27) , (3.69) sup x ∈V | G ω S ( b · ∇ ) G S ˜ b ( x ) | ≤ cL η/ , P (cid:20) sup y ∈U |∇ G S ˜ b ( y ) | > L − η/ (cid:21) ≤ L ν ′ exp( − c ′ L / η ) , (3.70) P (cid:20) inf x ∈V G S ˜ b ( x ) ≤ − c L η (cid:21) ≤ L ν ′′ exp( − c ′ L c +2 η )(3.71) where c = 1 when d = 4 and c = 2 when d ≥ . We now show (3.69). In view of (3.34) and (3.5), we have that sup x ∈S |∇ G S ˜ b ( x ) | ≤ c εL ≤ c / 2. Therefore for any ω ∈ Ω satisfying sup y ∈U |∇ G S ˜ b ( y ) | ≤ L − η/ and any x ∈ V we find that | G ω S ( b · ∇ ) G S ˜ b ( x ) | ≤ εL − η/ G ω S U ( x )+ ε c G ω S { z ∈S ;dist( z,∂ S ) ≤ } ( x )(3.72) + ε c G ω S { z ∈S ; | z | ⊥ ≥ / NL ′ ) + L } ( x ) . N EFFECTIVE CRITERION AND A NEW EXAMPLE The first term on the right-hand side is smaller than L − η/ E x,ω [ T S ] ≤ L η/ by (3.16).To bound the second term on the right-hand side of (3.72), we define for L ≥ R ) the auxiliary set ˆ S = { x ∈ S ; dist( x, ∂ S ) < } . With a martin-gale argument similar to (3.18), (3.16), we obtain that for any ω ∈ Ω and x ∈ S , E x,ω [ T ˆ S ] ≤ (1 − ε ) − ≤ 2. Then we introduce the successive times ofentrance in { x ∈ R d ; | x · e | ≥ L − } and departure from { x ∈ R d ; | x · e | >L − } : R = T L − ∧ ˜ T − L +1 , D = T { x ∈ R d ; | x · e | >L − } ◦ θ R + R , and by induction for k ≥ R k = R ◦ θ D k − + D k − , D k = T { x ∈ R d ; | x · e | >L − } ◦ θ R k + R k . With the help of these definitions we now express the Green operator appear-ing in the second term on the right-hand side of (3.72): for any ω ∈ Ω , x ∈ V ,we have G ω S ( x )1 { z ∈S ;dist( z,∂ S ) ≤ } = X k ≥ E x,ω (cid:20)Z D k ∧ T S R k { z ∈S ;dist( z,∂ S ) ≤ } ( X s ) ds, R k < T S (cid:21) (3.74) ≤ X k ≥ E x,ω [ E X Rk ,ω [ T ˆ S ] , R k < T S ] ≤ X k ≥ P x,ω [ R k < T S ] . The sum is bounded by a constant since the strong Markov property andthe support theorem imply that for k ≥ , x ∈ V , P x,ω [ R k < T S ] ≤ (1 − c ) k − .Hence the second term on the right-hand side of (3.72) is less than c ′ L − .We now examine the last term on the right-hand side of (3.72). We call˜ U the set { z ∈ S ; | z | ⊥ ≥ / N L ′ ) + L } appearing in that term. For any ω ∈ Ω , x ∈ V , the Markov property yields G ω S ˜ U ( x ) = E x,ω (cid:20) E X H ˜ U ,ω (cid:20)Z T S ˜ U ( X s ) ds (cid:21) , H ˜ U < T S (cid:21) (3.75) ≤ sup z ∈S E z,ω [ T S ] P x,ω [ H ˜ U < T S ] . Using (3.16) and a scaling argument similar to the one leading to (3.60), wefind that the latter expression is smaller than cL e − c ′ L . As a consequence,the last term on the right-hand side of (3.72) is smaller than L − for largeenough L . This proves (3.69). L. GOERGEN Next we turn to the proof of (3.70). In order to deal with the supremumover the set U , we cover U with disjoint cubes of side-length ε and cen-ters y i , i ∈ I , where |I| ≤ cL d − . If Q is such a cube with center y i , thenaccording to Lemma 3.9, − G S ˜ b ( y ) is twice continuously differentiable on Q ′ def = y i + ( − , ) d ⊂ S and satisfies the equation ∆ u = ˜ b on Q ′ . Therefore[5], (3.20), page 41, applies and we find that for any y ∈ Q |∇ G S ˜ b ( y ) − ∇ G S ˜ b ( y i ) | (3.76) ≤ c | y − y i | (cid:18) sup z ∈ Q ′ | G S ˜ b ( z ) | + sup z ∈ Q ′ | ˜ b ( z ) | (cid:19)(cid:18)(cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) c ′ | y − y i | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + 1 (cid:19) . Since the bounds in (3.16) also hold for Brownian motion, we have thatsup z ∈ Q ′ | G S ˜ b ( z ) | ≤ L ε ≤ L . Thus the right-hand side of (3.76) is less than cL − η for large enough L . With this discretization step we obtain for largeenough L : P (cid:20) sup y ∈U |∇ G S ˜ b ( y ) | > L − η/ (cid:21) (3.77) ≤ X i ∈I P [ |∇ G S ˜ b ( y i ) | > L − η/ ] . To bound the terms of the sum on the right-hand side of (3.77), we separatelyestimate P [ ∂ j G S ˜ b ( y i ) > d L − η/ ] and P [ ∂ j G S ˜ b ( y i ) < − d L − η/ ] for j =1 , . . . , d with the help of an Azuma-type inequality. Therefore we cover theslab S with disjoint cubes of side-length R and assign these cubes to 2 d disjoint families of cubes that are spaced by a distance R . We denote with Q mk = x m,k + [ − R , R ) d , 1 ≤ m ≤ d , k ≥ 1, the cubes associated to the m thfamily and define for i ∈ I , 1 ≤ j ≤ d , ω ∈ Ω, Y mi,k ( ω ) = Z Q mk ∩S ∂ j g ( y i , z )˜ b ( z, ω ) dz, k ≥ . (3.78)For fixed m ∈ { , . . . , d } and i ∈ I , ≤ j ≤ d , these random variables are P -independent (as k varies) and have mean 0 by Fubini’s theorem. Moreover,it follows from (3.49) that for all ω ∈ Ω; m, i, j, k ≥ | Y mi,k ( ω ) | ≤ cL − ( | x m,k − y i | − d ∧ L − d ) exp( − c | x m,k − y i | ⊥ /L )(3.79) def = γ m,k . Indeed, either | y i − x m,k | ≤ √ dR and using polar coordinates we obtain that | Y mi,k | ≤ cε R B √ dR ( y i ) ( r − d + L − d ) r d − dr ≤ c ′ L − (1 + L − d ), or | y i − x m,k | ≥√ dR and we can bound the integral by the supremum of the integrand times N EFFECTIVE CRITERION AND A NEW EXAMPLE the constant volume of Q mk . Using a slight variation of the proof of Azuma’sinequality, we find for 1 ≤ j ≤ d , i ∈ I , P " ∂ j G S ˜ b ( y i ) > d L − η/ ≤ d X m =1 P "X k ≥ Y mk ( ω ) > d d +1 L − η/ ≤ d X m =1 exp (cid:18) − d − − d +1) L − / η P k ≥ ( γ m,k ) (cid:19) (3.80) ≤ d exp( − cL / η ) , since the following easy computation and (3.79) show that P k ≥ ( γ m,k ) isof order L − for all m ≥ L X k ≥ ( γ m,k ) ≤ c X | x m,k − y i |≤ L ( | x m,k − y i | − d +1 ∧ L − d +1 ) + X | x m,k − y i | ⊥ ≥ L L − d +2 exp( − c ′ | x m,k − y i | ⊥ /L ) ≤ c Z L ( r − d +2 + L − d +2 ) r d − dr (3.81) + L − d +3 Z ∞ L e − c ′ r/L r d − dr ≤ c + L − d +2 Z ∞ e − c ′ u u du ≤ c ′′ . The same bound as in (3.80) holds for the terms P [ ∂ j G S ˜ b ( y i ) < − d L − η/ ],1 ≤ j ≤ d . Collecting the estimates (3.77), (3.80) and recalling that the car-dinality of I is polynomial in L , we have proved the claim (3.70).Finally we come to (3.71). The argument is similar to the previous one.First we handle the infimum over V by covering V with disjoint cubes ofthe form x i + [ − R , R ] d , for some adequate points x i , i ∈ I ′ where x i · e = 0and |I ′ | ≤ cL d − . Then it follows from (3.34) that for all ω ∈ Ω and | x − x i | ≤ R , | G S ˜ b ( x ) − G S ˜ b ( x i ) | ≤ c εL R √ d ≤ c. (3.82)Hence the discretization step implies that the left-hand side of (3.71) is lessthan X x i P (cid:20) G S ˜ b ( x i ) ≤ − c L η (cid:21) . (3.83) L. GOERGEN Then we use the same 2 d R -disjoint families of boxes Q mk as before to coverthe slab S and we define for i ∈ I ′ , m ≥ ω ∈ Ω,˜ Y mi,k ( ω ) = Z Q mk ∩S g ( x i , z )˜ b ( z, ω ) dz, k ≥ . (3.84)Again we observe that for fixed m ∈ { , . . . , d } and i ∈ I ′ , these randomvariables are P -independent and have mean 0. Moreover, it follows from(3.48) that for all ω ∈ Ω; m, i, k ≥ | ˜ Y mi,k ( ω ) | ≤ cL − ( | x m,k − x i | − d ∧ 1) exp( − c | x m,k − x i | ⊥ /L ) def = ˜ γ m,k . (3.85)A computation as in (3.81) shows that for large enough L and for all 1 ≤ m ≤ d : X k ≥ (˜ γ m,k ) ≤ L − (cid:26) c log L, d = 4, c, d ≥ L that each term in (3.83) is less thanexp( − cL η / log( L )) when d = 4 , respectively(3.87) exp( − cL η ) when d ≥ . This completes the proof of (3.71) and thus of Proposition 3.10. (cid:3) APPENDIX A.1. Proof of Lemma 2.3. We now give the proof of Lemma 2.3. In orderto bound ρ ( ω ) on G [see (2.16)], we first construct a function which—afterappropriate normalization—dominates P x,ω [ ˜ T − L + R +1 < ˜ T ∧ T L +1 ]. For theconstruction, we divide the box B into slabs of width L and consider anexpression inspired from the solution of a discrete one-dimensional Dirichletproblem for the exit probability of a Markov chain whose states correspondin essence to the slabs S i , i ∈ Z .Indeed, we recall (2.13) and for integers a < b , we consider the products Q a,b = Q bj = a +1 ˆ ρ ( j, ω ) − and set Q a,a = 1. Then we define the function f on {− n + 1 , − n + 2 , . . . , n + 2 } × Ω via f ( n + 2 , ω ) = 0 , f ( n + 1 , ω ) = 1 , (A.1) f ( i, ω ) = X i ≤ m ≤ n +1 Y m,n +1 for i ≤ n . For simplicity we drop the ω -dependence from the notation. We now showthat for ω ∈ Ω, P ,ω [ ˜ T − L + R +1 < ˜ T ∧ T L +1 ] ≤ f (0) f (1 − n ) . (A.2) N EFFECTIVE CRITERION AND A NEW EXAMPLE Let us introduce the ( F V m ) m ≥ -stopping time τ = inf { m ≥ X V m ∈ S n +2 ∪ S − n } . (A.3)Observe that P ,ω -a.s. on the event which appears in (A.2), X V τ ∈ S − n and V τ < ˜ T , and thus for ω ∈ Ω, P ,ω [ ˜ T − L + R +1 < ˜ T ∧ T L +1 ] ≤ E ,ω [ f ( I ( X V τ )) , V τ < ˜ T ] f (1 − n ) . (A.4)As we will see now, the numerator on the right-hand side is less than f (0):for ω ∈ Ω, m ≥ E ,ω [ f ( I ( X V ( m +1) ∧ τ )) , V ( m +1) ∧ τ ≤ ˜ T ] ≤ E ,ω [ f ( I ( X V m ∧ τ )) , V m ∧ τ ≤ ˜ T , τ ≤ m ](A.5) + E ,ω [ f ( I ( X V m +1 )) , V m ≤ ˜ T , τ > m ]and by the strong Markov property, the second term on the right-hand sideequals E ,ω [ V m ≤ ˜ T , τ > m, E X Vm ,ω [ f ( I ( X V ))]] . (A.6)However on { V m ≤ ˜ T , τ > m } , P ,ω -a.s.: E X Vm ,ω [ f ( I ( X V ))]= f ( I ( X V m )) + ˆ p ( X V m )[ f ( I ( X V m ) + 1) − f ( I ( X V m ))]+ ˆ q ( X V m )[ f ( I ( X V m ) − − f ( I ( X V m ))](A.7) ( A.1 ) = f ( I ( X V m )) + Y I ( X Vm ) ,n +1 [ − ˆ p ( X V m ) + ˆ q ( X V m ) ρ ( I ( X V m )) − ] . Note that P ,ω -a.s., X V m ∈ S I ( X Vm ) , for m ≥ 0. Hence the expression insidethe square brackets is nonpositive; see (2.13). As a result, we obtain that theleft-hand side of (A.5) is smaller than or equal to E ,ω [ f ( I ( X V m ∧ τ )) , V m ∧ τ ≤ ˜ T ]. The latter expression is hence nonincreasing with m . Since τ is P ,ω -a.s.finite, it follows from Fatou’s inequality that for ω ∈ Ω, E ,ω [ f ( I ( X V τ )) , V τ ≤ ˜ T ] ≤ f (0) . (A.8)Together with (A.4), this implies (A.2).We now derive a bound on ρ . Let us define for ω ∈ Ω, A = P ,ω [ ˜ T − L + R +1 < ˜ T ∧ T L +1 ] + P ,ω [ ˜ T < ˜ T − L + R +1 ∧ T L +1 ] . (A.9)Observe that q (0 , ω ) ≤ A and since q − q is nondecreasing in q , we obtain for ω ∈ Ω that ρ ( ω ) ≤ A (1 − A ) + . Using (A.2) and (2.16), it follows for ω ∈ G that ρ ( ω ) ≤ f (0) + f (1 − n ) κ L ( f (1 − n ) − f (0) − f (1 − n ) κ L ) + . (A.10) L. GOERGEN Let us for the time being assume that there is a c > R + 2 such that for L ≥ c and ω ∈ Ω, f (0) + f (1 − n ) κ L ≤ f (0) , (A.11) f (1 − n ) − f (0) − f (1 − n ) κ L ≥ Y − n +1 ,n +1 . (A.12)Then in view of (A.10) and the definition of f (0), for L ≥ c , ω ∈ G , ρ ( ω ) ≤ X ≤ m ≤ n +1 Y − n +1 We now prove Lemma 3.9. We start with theproof of (3.49). A similar and easier argument also shows (3.48). Since for d ≥ 4, we have | ∂ i g d ( x, y ) | ≤ c | x − y | − d and | ∂ i ∂ j g d ( x, y ) | ≤ c ′ | x − y | − d , (A.18)the sum of the first and second derivatives of the terms with k ≥ x, y ∈ S . Hence g ( x, y ) is twice continu-ously differentiable for x, y ∈ S , x = y , and interchanging differentiation andsummation yields for all x, y ∈ S|∇ g ( x, y ) | ≤ |∇ g d ( x, y ) | + c (2 L ) − d +1 and(A.19) | ∂ i ∂ j g ( x, y ) | ≤ | ∂ i ∂ j g d ( x, y ) | + c ′ (2 L ) − d , as well as(A.20) ∆ g ( x, y ) = 0 for x = y .(A.21)For any x ∈ S , we consider a small vector h with x + h ∈ S and an point y ∈ S with | x − y | ⊥ ≥ L . Moreover, we denote with W a d -dimensional Brownianmotion starting at y under some measure P and with T the stopping timeinf { t ≥ | W t − y | ⊥ ≥ | x − y |} . Since g ( x, y ) is symmetric in x and y , it isalso harmonic in y and thus g ( x, W t ∧ T ∧ T S ) is a bounded martingale under P . The stopping theorem thus implies that1 | h | | g ( x + h, y ) − g ( x, y ) | (A.22) = 1 | h | | E P [ g ( x + h, W T ∧ T S ) − g ( x, W T ∧ T S )] | . Direct inspection of g ( x, y ) shows that it vanishes on the boundary of S .Hence using the mean value theorem, the latter expression is smaller thansup {|∇ g ( x ′ , y ′ ) | ; x ′ ∈ B ( x, h ) , | y ′ − y | ⊥ = | x − y |} P [ T < T S ] . (A.23)Because of (A.19) the first factor above is less than c | x − y | − d + c ′ L − d anda scaling argument similar to the one leading to (3.60) yields that P [ T According to [5], Lemma 4.2, the first term on the right-hand side is twicecontinuously differentiable on U , and its Laplacian equals − f ( x ). With thesame argument as below (A.18), we see that ˜ g ( · , y ) is harmonic on U forany y ∈ S . Hence Fubini’s theorem together with the mean value theorem(see [5], Theorem 2.7) yield that the second term on the right-hand side of(A.24) is harmonic on U . 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