A conditional quenched CLT for random walks among random conductances on Z d

Transcription

1 A conditional quenched CLT for random walks among random conductances on Z d Christophe Gallesco Nina Gantert Serguei Popov Marina Vachkovskaia October, 3 Department of Statistics, Institute of Mathematics, Statistics and Scientific Computation, University of Campinas UNICAMP, rua Sérgio Buarque de Holanda 65, , Campinas SP, Brazil gallesco@ime.usp.br; popov@ime.unicamp.br; URL: popov marinav@ime.unicamp.br; URL: marinav Fakultät für Mathematik, Technische Universität München, Boltzmannstr. 3, Garching, Germany gantert@ma.tum.de Abstract Consider a random walk among random conductances on Z d with d. We study the quenched limit law under the usual diffusive scaling of the random walk conditioned to have its first coordinate positive. We show that the conditional limit law is a linear transformation of the product law of a Brownian meander and a d -dimensional Brownian motion. random conductance model, uniform heat kernel bounds, Brownian meander, re- Keywords: versibility AMS subject classifications: 6J, 6K37 Introduction and results In this paper we study random walks on a d-dimensional integer lattice with random conductances. One can briefly describe the model in the following way: initially, weights i.e., some nonnegative numbers are attached to the edges of the lattice at random. The transition probabilities are then defined to be proportional to the weights, thus obtaining a reversible Markov chain; due to a wellknown correspondence between reversible Markov chains and electric networks, the weights are also called conductances. We refer to the collection of all conductances as environment. This model attracted considerable attention recently, and, in particular, quenched i.e., for fixed environment functional central limit theorems and heat kernel estimates were obtained in rather general situations, see e.g., 3, 6, 6 and references therein. We also refer to the survey paper 5. To prove the quenched functional CLT, one usually uses the so-called corrector approach, described in the following way. First, one constructs an auxiliary random field which depends only on the environment, with the following property: the sum of the corrector and the random walk is a martingale, for which it is not difficult to show the CLT. Then, using the Ergodic Theorem, one shows that the corrector is likely to be small in comparison to the random walk itself. While this approach has been quite fruitful, it also has its limitations, mainly due to the fact that the construction of the corrector is not very explicit. For example, it was understood only quite

2 recently how to prove the quenched CLT for the random walk with i.i.d. conductances in half-space, see 7, 9. It is therefore important to go beyond the usual setup, proving other types of limit laws. In this paper, we continue the line of research of and which were, by their turn, mainly motivated by 8, 9, where a one-dimensional model with random conductances but with unbounded jumps was considered. We now define the model formally. For x, y Z d with d, we write x y if x and y are neighbors in the lattice Z d and we let B d be the set of unordered nearest-neighbor pairs x, y of Z d. Let ω b b Bd be non-negative random variables; P stands for the law of this family. We assume that P is stationary and ergodic with respect to the family of shifts θ x, x Z d. The quantity ω b is usually called the conductance of the edge b. The collection of all conductances ω = ω b b Bd is called the environment. If x y, we will also write ω x,y to refer to the conductance between x and y. For a particular realization ω of our environment, we define π x = y x ω x,y. Given that π x, for all x Z d which is P-a.s. the case by Condition UE below, the random walk X in the environment ω is defined through its transition probabilities ωx,y p ω x, y = π x, if y x,, otherwise, that is, if P x ω is the quenched law of the random walk starting from x, we have P x ωx = x =, P x ωxk + = z Xk = y = p ω y, z. Clearly, this random walk is P-a.s. reversible with the reversible measure π x, x Z d. Also, we denote by E x ω the quenched expectation for the process starting from x. When the random walk starts from, we use the shorter notations P ω, E ω. In order to prove our results, we need to make the uniform ellipticity assumption on the environment: Condition UE. There exists κ > such that, P-a.s., κ < ω,x < κ for x. For all n, we define the continuous map Z n t, t, as the natural polygonal interpolation of the map k/n n / Xk. In other words nz n t = X nt + nt nt X nt + with the integer part. Also, we denote by W d = W,..., W d the d-dimensional standard Brownian motion. Now, let us embed the graph Z d in R d. Denote by B = e,..., e d } the canonical basis of R d and by x,..., x d the vector coordinates in R d. By Condition UE and as our environment is stationary and ergodic there exists an invertible linear transformation D : R d R d letting the hyperplane x = } invariant and such that the sequence DZ n n tends weakly to W d. Indeed, by Condition UE and ergodicity of the environment, it is well known cf. 5 that Z n n tends weakly to a d-dimensional Brownian motion with a positive definite covariance matrix Σ. This implies that Σ has positive eigenvalues λ i and is diagonalizable in an orthonormal basis. If the law of the environment is also invariant under the symmetries of Z d, it is known that Σ = σ I for some constant σ, where I is the identity matrix. Thus, there exists a rotation T such that T Z n n tends weakly to Brownian motion with diagonal covariance matrix Σ = λ i i d in the basis B. This implies that Σ / T Z n n tends weakly to W d. Finally, by some unitary transformation R, we can rotate the hyperplane Σ / T x = } to make it coincide with the hyperplane x = }. Now, using the isotropy of W d we obtain that RΣ / T Z n n tends weakly to W d. For convenience, in the rest of the paper, we will choose R such that De e >. R can also involve a reflection. In

3 the case that the law of the environment is also invariant under the symmetries of Z d, then the last statement is true with D = σ I where σ is from the quenched CLT. Denoting X = X,..., X d in the basis B, we define and ˆτ = infk : X k = } Λ n = ˆτ > n} = X k > for all k =,..., n}. Consider the conditional quenched probability measure Q n ω := P ω Λ n, for all n. Denote by C, the space of continuous functions from, into R d and by B the Borel σ-field on C,. For each n, the random map DZ n induces a probability measure µ n ω on C,, B : for any A B, µ n ωa := Q n ωdz n A. Let us next recall the formal definition of the Brownian meander W +. For this, define τ = sups, : W s = } and = τ. Then, W + s := / W τ + s, s. We denote by P W + P W d the product law of Brownian meander and d -dimensional standard Brownian motion on the time interval,. Now, we are ready to formulate the quenched invariance principle for the random walk conditioned to stay positive, which is the main result of this paper: Theorem. Under Condition UE, we have that, P-a.s., µ n ω tends weakly to P W + P W d as n as probability measures on C,. The next result, referred as Uniform Central Limit Theorem UCLT, will be useful in order to prove Theorem.. Let W Σ be a d-dimensonal Brownian motion with covariance matrix Σ defined above. Denoting by C b C,, R respectively, C u b C,, R the space of bounded continuous respectively, bounded uniformly continuous functionals from C, into R, we have the following result: Theorem. Under Condition UE, the following statements hold and are equivalent: i we have P-a.s., for all H > and any F C b C,, R, lim sup E θxωf Z n EF W Σ = ; x H n,h n d ii we have P-a.s., for all H > and any F C u b C,, R, lim sup E θxωf Z n EF W Σ = ; x H n,h n d iii we have P-a.s., for all H > and any closed set B, lim sup sup x H n,h P θxωz n B P W Σ B; n d 3

4 iv we have P-a.s., for all H > and any open set G, lim inf inf x H n,h n d P θxωz n G P W Σ G; v we have P-a.s., for all H > and any A B such that P W Σ A =, lim sup P θxωz n A P W Σ A =. x H n,h n d As mentioned, our approach does not involve the corrector in a direct way although, of course, we use the classical invariance principle which relies on the corrector approach. Instead, the key ingredients are the following. We use the uniform heat kernel bounds of to prove a uniform CLT see Theorem.. In addition, we rely on several auxiliary results giving uniform bounds for exit times and hitting times of hyperplanes, leading to statements which say, roughly speaking, that the conditioning does not change the size of fluctuations see the beginning of Section for a more detailed description. These auxiliary results are shown in Section. They are of independent interest and might be applied to study other fine questions for the random conductance model than the conditional quenched CLT considered here. To prove that the conditioning does not change the size of fluctuations, we use an iteration method relying on the Markov property and the uniform bounds shown before. In Section 3, we give the proof of Theorem.. Finally, in Section 4, we give the proof of Theorem.. We will denote by C, C,... the global constants, that is, those that are used all along the paper and by γ, γ, γ,... the local constants, that is, those that are used only in the subsection in which they appear for the first time. For the local constants, we restart the numeration in the beginning of each subsection. Also, whenever the context is clear, to avoid heavy notations, we will not put the integer part symbol. For example, for δ, we will write Xδn instead of X δn. Auxiliary results In this section, we will prove several auxiliary results that will be needed later to prove Theorem.. Before going to the technical side, let us give a short description. Lemma. gives a uniform bound on the upper tail of the exit time of a strip as well as on the lower tail of the hitting time of a set sufficiently far away from the starting point. Lemma. provides a uniform lower bound for the probability of progressing in the direction e before backstepping to the hyperplane of the origin. Lemma.3 says that the probability that the hitting time of a hyperplane is larger than it should be, conditioned on the first coordinate being positive, decays fast enough. Lemma.4 says that the probability that the transversal fluctuations are larger than they should be, conditioned on the first coordinate being positive, decays fast enough. Instead of considering the process X in the canonical basis B of R d it is also convenient to introduce the embedded graph Z d := DZ d with the basis B = e,..., e d} := DB and consider the process DX in this new basis. All the results obtained in this section concern the original random walk X expressed in B but they remain valid for DX expressed in B with the -norm replaced by the graph distance in Z d. Let us introduce the following notations. First, for a, b Z, a < b, we denote by a, b the set a, b Z. Vectors of Z d will be denoted by x, y or z. For x Z d we denote by x,..., x d its coordinates 4

5 in B. For l R, we denote l} j = for j, d. If F Z d, let us define x = x,..., x d Z d : x j = l }, if l, x = x,..., x d Z d : x j = l }, if l <, τ F = infn : Xn F } and τ + F = infn : Xn F }. At this point we mention that under Condition UE, we can apply Theorem.7 of to the random walks Y n := Xn and Y n := Xn +, to obtain that uniform heat kernel lower and upper bounds are available for this model. That is, there exist absolute constants C, C, C 3 and C 4 such that P-a.s., for n N, p n ωx, y C n d/ exp x y } C n and if x y n with the -norm on Z d and has the same parity as n, p n ωx, y C 3 n d/ exp x y } C 4. n We denote by d the distance induced by the -norm. The heat kernel upper bound has two simple consequences gathered in the following Lemma. Estimate implies that there exist positive constants C 5 and C 6 such that P-a.s., for h > and δ >, the following holds. i Let H and H be two parallel hyperplanes in Z d orthogonal to e i for some i, d and let us denote by S the strip delimited by H and H. If d H, H hn / then there exists n = n δ, h such that sup P x ωτ H H > δ h n C 5 x S δ for all n n ; ii Let x Z d. If A Z d is such that d x, A > hn / then there exists n = n δ, h such that P x ωτ A δ n C 6 δ h for all n n. Proof. Let us denote by S the strip delimited by H and H. To prove i, we just notice that P x ωτ H H > δ n P x ωxδ n S and apply. More precisely, suppose that H and H are orthogonal to e. With a slight abuse of notation, we also denote by H and H the coordinates where the hyperplanes H and H cross the first axis. We have P x ωxδ n S y S C δ n d/ exp C δ n d/ y H,H x y } C δ n y x } d y i x i exp C δ exp C n δ n i= y i Z }. 3 5

6 Using 3, we can see that there exist positive contants γ, γ and n = n δ, h such that h γ P x ωxδ δ n S γ exp C t }dt for all n n. We deduce that there exists a constant γ 3 > such that P x ωxδ n S γ 3 h δ for all n n. To prove ii we use an argument by Barlow cf. Chapter 3. First, if we denote by Bx, r the -ball of center x and radius r := hn / we have that Then, we have P x ωτ B c x,r δ n P x ω P x ωτ A δ n P x ωτ B c x,r δ n. Xδ n x > r + P x ω τ B c x,r δ n, Xδ n x r. 4 Writing S = τ B c x,r, by the Markov property, the second term of the right-hand side of 4 equals E x ω S δ n}p X S ω X δ n S x r sup sup P y ω y Bx,r+ m δ n X δ n m y > r where Bx, r := y Z d : y x = r}. Combining this last inequality with 4 we obtain, P x ωτ B c x,r δ n sup sup P y ω X δ n m y > r y Z d m δ n sup sup P y ω X δ n m y > r d y Z d m δ n where is the -norm on Z d. Applying to bound the last term of the above equation from above and performing the same kind of computations as in the proof of i, we obtain ii. Next, we prove the following lemma, which gives a uniform lower bound for the probability of progressing in direction e before backstepping to the hyperplane }. Lemma. Let v >, then there exist a constant C 7 inf y l} P y ωτ v+l} < τ } C 7, for all integers l. = C 7 v > such that we have P-a.s., Proof. We are going to show that we can choose v > small enough in such a way that the statement of Lemma. is true for this v. The generalization to all v > is then a direct consequence of the elliptic Harnack inequality, see 3. For the moment, let v, 4 and fix l such that vl. Then, consider w v,. We start by writing P y ωτ v+l} < τ } P y ωx wl v + l, τ } > wl P y ωx wl v + l P y ωτ } wl. 5 6

7 Next, let us define ν := wl if wl is even or ν := wl + otherwise. In the same way, we define ρ := vl if vl is even or vl + otherwise. Observe that in any of these cases, P y ωx wl v + l P y ωx ν > l + ρ. 6 We will bound the term of the right-hand side of 6 from below. For y l}, we denote by Py the non-empty set of vectors z Z d that satisfy the following conditions: z y > ρ, y z is even and y z ν. Applying, we obtain P y ωx ν > l + ρ C 3 ν d/ C 3 ν d/ u Py ν v = ρ+ C ν 3 ν d/ ρ+ exp ν v v = ν v u y } C 4 ν ν v + +v d v d = exp ν v + +v d... exp γ ν v γ + + v } d ν d v i } dv d... dv with γ a positive constant depending only on d. Now making the change of variables u i = v i ν, i,..., d}, in the last multiple integral, we obtain P y ωx ν > l + ρ C 3 d ν ν u ρ+ ν ν P d u i... exp γ 4 d u i } du d... du. Now, by definition of ρ, ν and the way we chose v, l and w we have that ρ+ ν 4 vw / and ν. This implies that P y ωx ν > l + ρ C 3 d C 3 d ρ+ ν u 4 vw / P d u i... exp γ 4 u d u i P d u i... exp γ 4 } du d... du d u i } du d... du =: C 3 d Jvw /. 7 By ii of Lemma. we obtain P y ωτ } 5, 6 and 7 we obtain wl C 6 w /. Combining this last inequality with P y ωτ v+l} < τ } C 3 d Jvw / C 6 w /. 8 Observe that for fixed w, we have Jvw / J > as v, since the integrated function is positive and the domain of integration of J has Lebesgue measure equal to d!. Let w = s > : C 6 s / C } 3 4 d J 7

8 that is, w = C3. J d+ C Letting v < w /4, we can choose a sufficiently small w in such 6 a way that the second term of the right-hand side of 8 is smaller than C 3 4 J. Once we have d chosen w, we can choose v sufficiently small in such a way that Jvw / > J. We obtain that This shows Lemma.. P y ωτ v+l} < τ } 4 C 3 J >. d For ε,, we denote N := ε n. We next prove an upper bound for the probability that the hitting time of the hyperplane N} is larger than ε / n, given Λ n. Lemma.3 There exists a function f = fε with lim ε ε fε = such that we have P-a.s. lim sup P ω τ N} > ε / n Λ n fε. Proof. Let us begin the proof by sketching the main argument. Consider α,, we will show that lim sup P ω τ N} > ε / n Λ n lim sup P ω τ N} > αε / n Λ n + o ε when ε. Then, iterating the argument using hyperplanes of the form j N} cf. Figure we N 3 N N N Figure : Iteration method. will have that for all j, lim sup P ω τ j N} > α j ε / n Λ n lim sup P ω τ j+ N} > α j+ ε / n Λ n + o j ε when ε. Finally, restricting α to the interval 4,, we will show that the o jε are decreasing fast enough. Now, let us start the formal argument. Fix α 4, and let A l := τ l} < τ + } }. We have P ω τ N} > ε / n Λ n = P ω τ P ω Λ n N} > ε / n, τ N} > αε / n, Λ n + P ω τ N} > ε / n, τ N} αε / n, Λ n 8

9 P ω τ N} > αε / n Λ n + P ω Λ n P ωτ N} > ε n, τ N} αε / n, A N, Λ n. 9 Then, we have by the Markov property P ω τ N} > ε / n, τ N} αε / n, A N, Λ n = P ω Xτ N} = y, τ N} = k, τ N} > ε / n, A N, Λ n y N} k αε / n y N} k αε / n P y ωτ N} > ε / n k, Λ n k P ω A N. Now, let us bound from above the term P y ωτ N} > ε / n k, Λ n k uniformly in y N} and in k αε / n. Observe that, since ε,, we have P y ωτ N} > ε / n k, Λ n k P y ωτ N} > αε / n, Λ αn P y ωτ } N} > αε / n. Let δ := β ε, where β is a positive constant to be determined later. Then, consider ε small enough in such a way that δ < αε /. Then, divide the time interval, αε / n into intervals of size δ n. Denoting S, N = N i}, we obtain by the Markov property P y ωτ } N} > αε / n P y ω τ } N} / αε / n δ n z S,N Pz ωτ } N} > δ n for large enough n. Using i of Lemma., we have for all z S, N, P z ωτ } N} > δ n C 5 ε δ i δ n, i δ n αε / δ for sufficiently large n. Since ε/δ = β, let us choose the constant β such that C 5 β /. Thus, for ε sufficiently small such that β ε < αε /, we obtain by 3 From 9, we deduce αε 3 P y ωτ } N} > αε / β n 4. αε 3 P ω τ N} > ε / n Λ n P ω τ N} > αε / β P ω A n Λ n + 4 N. 4 P ω Λ n Then, we will find an upper bound for the ratio in the second term of the right-hand side of 4. By the Markov property we have P ω Λ n P ω A N P ωλ n A N min P y ωτ } > n. 5 y N} 9

10 Let K ε and let N = K n. We start by noting that for any y N} we have by the Markov property P y ωτ } > n P y ωτ } > n, τ N } < τ } min P z ωτ } > np y ωτ N z N } } < τ }. 6 Let us now bound from below both terms in the right-hand side of 6. We first show that we can choose a sufficiently large K in such a way that P z ωτ } > n / uniformly in z N }. Using ii of Lemma., we have P z ωτ } n C 6 /K for sufficiently large n. Choosing K sufficiently large so that C 6 /K / we obtain P z ωτ } > n 7 uniformly in z N }. Now going back to equation 6, we now show that with probability of order ε γ with γ >, starting from the line N}, the random walk reaches the line N } before reaching the line }. By Lemma., there exists C 7 > such that for every l >, P u ωτ l} < τ } C 7, with u l}. Now consider, the following sequence U j j of hyperplanes defined by U = N } U j+ = U j }. Let j the smallest j such that U j K n. Using the induction relation, we obtain that for some constant γ >, j γ ln K ε for large enough n. By convention, set U = N}. By the Markov property, we obtain that uniformly in y N}, P y ωτ N } < τ } P y ω j τ Ui < τ } } j ε min P u γ ωτ Ui < τ } 8 u U i K for some constant γ > and large enough n. Combining 6, 7, and 8 we deduce min P y ωτ } > n ε γ 9 y N} K for large enough n. Then by 4, 5 and 9 we obtain P ω τ N} > ε / n Λ n P ω τ N} > αε / n Λ n + 6K γ ε γ αε 3 β. By the same argument, we can deduce that for all j we have P ω τ j N} > α j ε / n Λ n P ω τ j+ N} > α j+ ε / n Λ n + 6K γ ε γ αβ ε 3 4α j j for large enough n. Iterating using, we deduce lim sup P ω τ N} > ε / n Λ n 6K γ ε γ αβ ε 3 4α j j. As α 4,, the last series is convergent. Define the function f in the statement of Lemma.3 as fε := 6K γ j= ε γ αβ ε 3 4α j j. j=

11 Using the dominated convergence theorem, it is straightforward to show that lim ε ε fε =. This proves Lemma.3. In the next lemma, N still stands for ε n. However, the quantities like α, δ, β,... defined in the proof of the lemma are not related to the corresponding quantities defined in the proof of Lemma.3. The next lemma controls the transversal fluctuations of X,..., X d, given Λ n. Lemma.4 We have P-a.s., with lim ε ε gε =. lim sup P ω i,d sup X i j > ε / N Λ n gε j τ N} Proof. First, observe that, by symmetry, it suffices to show that there exists g = g ε such that lim sup P ω sup X i j > ε / N Λ n g ε 3 j τ N} with lim ε ε g ε = for some i, d. For the sake of simplicity, let us take i = in the rest of the proof. Fix α, and let ε / := α α ε / >. We introduce the following sequence of events cf. Figure, } G k = sup j τ k N},τ k+ N} X j X τ k N} ε / α k N for k, with the convention that sup j } =. Then, we denote ε α k N N k N k Figure : On the definition of G k. B δ k = τ k N} δn} τ k N} < τ } } for δ, and k. Now, observe that on the event B δ k G k we have that sup j τn} X j ε / N since α,. This implies that P ω sup X j ε / N Λ n P ω B δ G k Λ n. j τ N} k

12 In order to prove Lemma.4, we will show that lim inf P ω B δ k G k Λ n tends to when ε. We start by writing P ω B δ G k Λ n = P ω B δ Λ n P ω B δ G k c Λ n k k P ω B δ Λ n ln N ln k= P ω B δ G c k Λ n. 4 From now on, we dedicate ourselves to bounding from above the terms P ω B δ Gc k Λ n for k ln N ln. We have by the Markov property, P ω B δ G c k Λ n P ω Bk δ Gc k Λ n = P ω Λ n P ωbk δ, Gc k, Λ n = Bk δ P ω Λ n, Gc k, Λ n, τ k N} = j, Xτ k N} = y P ωb k P ω Λ n P ωb k P ω Λ n P ω j δn y k N} P y j δn y k ω N} P y y k ω N} Using again the Markov property, we obtain sup Xi y e > ε / α k N, Λ n j i τ k+ N} sup Xi y e > ε / α k N, Λ δn. i τ k+ N} P ω Λ n P ω Bk P ωλ n Bk min P y ωτ } > n. y k N} By the same argument which we used in Lemma.3 to treat the term min y N} P y ωτ } > n cf. the derivation of 9, we obtain, for large enough n and all k ln N ln, P ω Bk K k P ω Λ n γ γ 5 ε for some positive constants γ, γ and K from Lemma.3. Now, we need to bound the terms P y ω sup Xi y e > ε / α k N, Λ δn i τ k+ N} from above, uniformly in y k N}. In order not to carry on heavy notations we treat the case y =. However, as one can check, the bound we will obtain is uniform in y k N}. Let E k = x,..., x d Z d : x = ± ε / α k N }. We start by writing P y ω sup X i > ε / α k N, Λ δn = P y ωτ Ek < τ k+ N}, τ } > δn i τ k+ N} P y ωτ Ek < τ k+ N} } + P y ωτ Ek > δn. 6

13 Let us bound the first term of the right-hand side of 6 from above. To do so, we first write P y ωτ Ek < τ k+ N} } P y ωτ ε / α k N} < τ k+ N} } + P y ωτ ε / α k N} < τ k+ N} }. 7 We treat the first term of the right-hand side of 7 the method for the second term is similar. Let L, ε / and divide the interval, ε / α k N into intervals of size L k N. Furthermore, let We have by the Markov property, F k = k+ N j= j}. P y ωτ ε / γα k N} < τ k+ N} } Let us show that P y ω ε / α k N L k N j= L ε / α k j= τ j L k N } < τ k+ N} } } P z ωτ j L z j L k N } F k N } < τ k+ N} }. 8 k z j L k N } F k P z ωτ j L k N } < τ k+ N} } for ε sufficiently small and L sufficiently large belonging to, ε /. Consider w 4, L, we have for z j L k N } F k, P z ωτ j L k N } > τ k+ N} } P z ωτ k+ N} } w k N, τ j L k N } > w k N Using i of Lemma., we deduce Using ii of Lemma., we obtain for all j, P z ωτ k+ N} } w k N P z ωτ j L k N } w k N. 9 P z ωτ k+ N} } w k N C 5 w /. 3 P z ωτ j L k N } w k N C 6 w / Combining 9, 3 and 3 we obtain for all j, P z ωτ jl k N} > τ k+ N} } C 5 w / C 6 w / L. 3 L. 3 First, choose w sufficiently large such that C 5 w / /4 and thus choose L sufficiently large in such a way that C 6 w / /L /4. We obtain P z ωτ jl k N} > τ k+ N} }. 33 3

14 Now using 7, 8 and 33 we have since ε / > L, L P y ε / α k ωτ Ek < τ k+ N} } Next, let us treat the term P y ωτ Ek > δn. Let η = β ε where β is a positive constant to be chosen later. Then suppose that ε is sufficiently small such that η ε / α k < δ and divide the time interval, δn into intervals of size η ε α k n. Denoting by HE k = ε / α k N j= ε / α k N + j}, we obtain by the Markov property P y ωτ Ek > δn P y ω τ Ek / δn η ε α k n z HE k Pz ωτ Ek > η ε α k n i η ε α k n, i η ε α k n δη ε / α k 35 for n sufficiently large. We now bound the term P z ωτ Ek > η ε α k n from above uniformly in z HE k. Using i of Lemma., we have P z ωτ Ek > η ε α k n C 5 ε η. Since εη = β, choose β small enough such that C 5 β /. η ε / α k < δ, we obtain using 35, For ε sufficiently small such that δβ P y ε / α k ωτ Ek > δn Combining 6, 7, 34 and 36, we deduce that, P-a.s., for all large enough n and k ln N P y y k ω sup Xi y e > ε / α k N, Λ δn N} i τ k+ N} Using 5 and 37, we obtain for all large enough n and k ln N ln, ln, L ε / α k δβ ε α k P ω B δ G c k Λ n γ K γ kγ+ ε γ L ε / α k δβ ε α k. We finally deduce that, P-a.s., for large enough n, ln N ln k= P ω B δ G c k Λ n k= γ K γ kγ+ ε γ L ε / α k δβ ε α k. 4

15 Observe that since α,, the series above converges. Let δ = ε/, we have for ε < /4, Let ln N ln k= P ω B δ G c k Λ n hε := k= k= γ K γ kγ+ ε γ α 6 α L ε / α k / α α + 4 β ε α k. γ K γ kγ+ ε γ α 6 α L ε / α k / α α + 4 β ε α k. By the Lebesgue dominated convergence theorem, we have ε hε as ε. Using 4 and Lemma.3 since δ = ε / we have for ε < /4, lim inf P ω B δ G k Λ n fε hε. k This last term tends to as ε. Lemma.4. Now, take g ε := fε + hε to show 3 and therefore 3 Proof of the UCLT In this section we prove Theorem.. The proof is similar in spirit to the proof of Theorem. of, but it is greatly simplified in the present case by the use of the heat kernel upper bounds. As in, we will consider good sites see Definition 3., where uniform estimates hold for the distance of Z n and W, and then show that the random walk hits, with high probability, a good site in small distance from its starting point. In order to take advantage of the natural left shift on the space CR + of continuous functions from R + into R d, we will rather prove Theorem. for Z n assuming values in CR + instead of C,. Then, the result for Z n assuming values in C, will be easily obtained by the mapping theorem cf. 4. Let C u b CR +, R be the space of bounded uniformly continuous functionals from CR + into R. In this section, we write W for the d-dimensional Brownian motion with covariance matrix Σ from section. The first step is to prove the following Proposition 3. For all F C u b CR +, R, we have P-a.s., for every H >, lim sup E θxωf Z n EF W =. x H n,h n d Fix F C u b CR +, R. We will prove that, P-a.s., for every ε, H >, sup E θxωf Z n EF W ε 38 x H n,h n d for n large enough. Before this, we need to introduce some definitions and prove an intermediate result. Let d be the distance on the space C R+ defined by df, g = n= } n+ min, sup s,n fs gs 5

16 with the euclidian norm on R d. Now, for any given ε >, let h ε := h, : P W s > ε + P dθ s W, W > ε ε sup s h sup s h }. 39 Observe that h ε > for ε > and h ε when ε. Next, adapting section 3 of we introduce the following Definition 3. For a given realization of the environment ω and N N, we say that x Z d is ε, N-good, if min n : E ω F Z m EF W } ε, for all m n N; P θxω sup s hε Z m s ε, sup s hε dθ s Z m, Z m ε ε, for all m N. We now show that starting from a site x H n, H n d, with high probability, the random walk X will meet a ε, n-good site at a distance at most h n before time hn unlike as in, there is no need here to introduce the notion of a nice site since by, every point in H n, H n d is nice. We denote by G the set of ε, n-good sites in Z d. Proposition 3. Fix h >. For any ε >, we can choose ε small enough in such a way that we have P-a.s., for all sufficiently large n and all x H n, H n d : i P x ωτ G > h ε n ε ; ii P x ω sup j hεn Xj X > h n ε. Proof. Fix ε. Then, for any ε > there exists N such that P is ε, N-good > ε. By the Ergodic Theorem, we have P-a.s. for all n > n ω, Let us define x H n, H n d and x is not ε, N-good} < 5 d ε H d n d. 4 Bad := x H n, H n d and x is not ε, N-good} and Cub := H n, H n d. In order to show i we observe that for all x H n, H n d, P x ωτ G > h ε n P x ωxh ε n Bad + P x ωτ Cub c h ε n. 4 For the second term of the right-hand side of 4, we apply ii of Lemma. to obtain that P x ωτ Cub c h ε n γ h ε /. Thus, we can choose ε small enough in such a way that P x ωτ Cub c h ε n ε /. Then, using and the fact that Bad < 5 d ε H d n d for large n, we can show that uniformly in x Bad H n, H n d we have P x ωxh ε n Bad γ ε /h ε for n sufficiently large. Thus, choosing ε sufficiently small in such a way that γ ε /h ε ε / we obtain P x ωxh ε n Bad ε /. To show ii, we notice that P x ω sup j h εn Xj X > h n = P x ωτ B c x,h n h ε n 4 6

17 with Bx, r the euclidian ball of center x and radius r. Now, we can apply ii of Lemma. to the right-hand term of 4 to obtain that P x ω sup j h εn Xj X > h hε / n γ 3 h. Finally, choosing ε sufficiently small such that γ 3 h / /h ε we obtain ii. This concludes the proof of Proposition 3.. Proof of Proposition 3.. Let us prove 38. Consider x H n, H n d. We start by writing E θxωf Z n EF W E θxω F Z n E θxτg ωf Z n Eθxω + E θxτg ωf Z n EF W := U + V. 43 First, taking ε ε we obtain V ε/ by definition of a ε, n-good site. It remains to treat the first term of the right-hand side of 43. Denote X := X x. Now, observe that by the Markov property U = E θxω F Z n E θx θ τg xωf Z n Eθxω F Z n F θ n τ G Z n n / X τ G. 44 We are going to show that for n sufficiently large we have uniformly in x H n, H n d, E θxω F Z n F θ n τ G Z n n / X τ G ε for small enough ε. Let M n := Z n n / X τ G. Since F is uniformly continuous, we can choose η > in such a way that if df, g η then F f F g ε 4. Then, we have E θxω F Z n F θ n τ G M n F = Eθxω Z n F θ n τ G M n dz n, θ n τ G M n η} F + E θxω Z n F θ n τ G M n dz n, θ n τ G M n > η} ε 4 + F P θxω dz n, θ n τ G M n > η. 45 Since h ε, we have P θxω dz n, θ n τ G M n > η P θxω dz n, θ n τ G M n > η, τ G hn + P θxωτ G > h ε n P θxω sup Z n θ n τ G M n > η, τ G h ε n + P θxω t,n τ G dθ n τ G Z n, θn τ G M n > η, τ G h ε n + P θxωτ G > h ε n. 46 Let F τg be the σ-field generated by X until time τ G. We first decompose the first term of the right-hand side of 46 in the following way: P θxω sup Z n θ n τ G M n > η t,n τ G, τ G h ε n P θxω sup t,n τ G Z n > η 4 + P θxω 7 sup t,h ε θ n τ G M n > η 4

18 = P θxω sup Z n > η t,n τ G 4 = P θxω sup Z n > η t,n τ G 4 + E θxω P θxω + E θxω P θxτg ω sup t,h ε θ n τ G M n > η 4 F τ G sup Z n > η. 47 t,h ε 4 We now deal with the second term of the right-hand side of 46: P θxω dθ n τ G Z n, θn τ G M n > η, τ G h ε n P θxω X τg > η 4 n + P θxω dθ n τ G M n, θn τ G M n > η 4, τ G h ε n P θxω sup Z n > η + E θxω τ G h ε n}p θxω dθ t,n τ G 4 n τ G M n, θn M n > η τg 4 F τ G = P θxω sup Z n > η + E θxω τ G h ε n}p θxτg ω dz n, θ t,n τ G 4 n τ G Z n > η Combining 46, 47 and 48, we obtain P θxω dz n, θ n τ G M n > η P θxωτ G > h ε n + P θxω sup Z n > η t,n τ G 4 + E θxω P θxτg ω sup Z n > η t,h ε 4 + τ G h ε n}p θxτg ω dz n, θ n τ G Z n > η On one hand, by definition of a ε, n-good point, choosing small enough ε >, we have uniformly in x H n, H n d, E θxω P θxτg ω sup t,h ε Z n > η 4 + τ G h ε n}p θxτg ω dz n, θ n τ G Z n > η 4 ε 3 F 5 for all sufficiently large n. On the other hand, by Proposition 3., for sufficiently small ε, we have uniformly in x H n, H n d, P θxωτ G > h ε n ε and P θxω sup Z n > η 3 F t,n τ G 4 ε 3 F 5 for sufficiently large n. Combining 5, 5 with 49, 46, 45 and 44, we have U ε/. Together with V ε/, this leads to the desired result. Denote by C b CR +, R the space of bounded continuous functionals from CR + into R and by B the Borel σ-field on CR +. The next step is the following proposition, its proof follows essentially the proof of Theorem. of 4 cf. also Proposition 3.7 of. Proposition 3.3 The first statement implies the second one: i for any F C u b CR +, R, we have P-a.s., lim sup E θxωf Z n EF W = ; x H n,h n d 8

19 ii for any open set G, we have P-a.s., lim inf inf x H n,h n d P θxωz n G P W G. Finally, we have Proposition 3.4, which is similar to Proposition 3.8 of. Proposition 3.4 The following statements are equivalent: i we have P-a.s., for every open set G, lim inf ii for every open set G, we have P-a.s., lim inf inf x H n,h P θxωz n G P W G; n d inf x H n,h P θxωz n G P W G. n d Proof. i ii is trivial. Let us show that ii i. Suppose that there exists a countable family H of open sets such that for every open set G there exists a sequence O n n=,,... H such that On G pointwise as n. By ii, since the family H is countable we would have, P-a.s., for all O H, lim inf inf x H n,h P θxωz n O P W O. 5 n Then, the same kind of reasoning as that used in the proof of Proposition 3.3 to prove i ii would provide the desired result. The fact that H exists, follows from the fact that the space CR + is second-countable. Proof of Theorem.. One can check that it is straightforward using the same arguments as in the proof of Proposition 3.3 to deduce that i, ii, iii and v of Theorem. are equivalent to statement i of Proposition 3.4. That is, one can prove the equivalence of items i-v of Theorem.. To conclude the proof of Theorem., it remains to show that ii of Proposition 3.4 holds. By Proposition 3.3, ii of Proposition 3.4 is equivalent to i of Proposition 3.3. Since by Proposition 3., i of Proposition 3.3 holds, the proof of Theorem. is complete. 4 Proof of Theorem. For the sake of brevity, let us denote in this section, the process DZ n resp. DX by Z resp. X. We also recall that W d = W,..., W d is a d-dimensional standard Brownian motion. In order to prove Theorem., we first show convergence of the finite-dimensional distributions and then, in Section 4., we prove the tightness of the sequence P ω Z n Λ n n. For ε,, we recall that N := ε n. In this section for any set F R d we denote β F = infn : X n F } and β + F = infn : X n F }. We start by recalling the transition density function of the Brownian meander see 4 from, to t, x q, ; t, x = t 3/ x exp x Ñx t / 53 t 9

20 for x >, < t and from t, x to t, x for x, x >, < t < t, where for v and for x, x > and < t. qt, x ; t, x = gt t, x, x Ñx t / Ñx t / / v Ñv = π e u du gt, x, x = π / exp x x exp x + x t t 4. Convergence of finite-dimensional distributions In this subsection, we prove the convergence of the finite-dimensional distributions. A key ingredient is the decomposition according to the event A R, see 55, which says that the random walk progresses enough in the desired direction, without big fluctuations of the other coordinates, before it returns to the hyperplane of the origin. In order to show that this event has large probability, Lemma.3 and Lemma.4 come into play, see 6 and 8. First, let us consider then marginal for t =. Proposition 4. We have P-a.s., lim P ωz n > u,..., Zd n > u d Λ n = exp u / for all u = u,..., u d R + R d. Proof. First, we introduce some notations. Let and D u = x R d : x > u,..., x d > u d } d i= R ε,n = x R d : x = N, x i ε / N, ε / N, i, d}. Let us denote R ε,n = DR ε,n, and define the event u i e t dt, 54 π A R = β Rε,n < β + } }. 55 We start by bounding the term P ω Z n D u Λ n from above. Fix ε, u and consider the following decomposition P ω Z n D u Λ n P ω Λ n = P ω Λ n P ω Z n D u, A R, Λ n + P ω A c R, Λ n P ω Z n D u, A R, Λ n, β Rε,n ε / n

21 Since ε /,, we have + P ω Z n D u, A R, Λ n, β Rε,n > ε / n + P ω A c R Λ n P ω Λ n P ω Z n D u, A R, Λ n, β Rε,n ε / n + P ω β Rε,n > ε / n Λ n + P ω A c R Λ n. 56 P ω A c R Λ n = P ω β Rε,n > β + } Λ n P ω β Rε,n > n Λ n P ω β Rε,n > ε / n Λ n. 57 Then, using the Markov property at time β Rε,n we deduce P ω Λ n P ωz n D u, A R, Λ n, β Rε,n ε / n P ωa R P ω Λ n P y y R ε,n j ε / ω n X n j n D u, Λ n j. 58 Again, using the Markov property at time β Rε,n we obtain Combining 56, 57, 58 and 59 we obtain P ω Λ n P ω A R min y R ε,n P y ωλ n. 59 P ω Z n D u Λ n y R ε,n j ε / n Py ωx n j D u n, Λn j min y Rε,n P y ωλ n Now, to bound the term P ω β Rε,n > ε / n Λ n from above we notice that + P ω β Rε,n > ε / n Λ n. P ω β Rε,n > ε / n Λ n = P ω τ Rε,n > ε / n Λ n P ω sup X i j > ε / N Λ n + P ω τ N} > ε / n Λ n. 6 i,d j τ N} By Lemmas.3 and.4 we have lim sup P ω β Rε,n > ε / n Λ n fε + gε. 6 By definition of Z n, we have P y ωλ n = P y ω Z n >, t,. Thus, from Theorem. we obtain, recalling that W is the first component of W d, lim min Z n t >, t, = P εσ min W t > = P W < εσ t y R ε,n P y ω 6 = εσ + oε 63 π as ε, where P x is law of W d starting at x and σ := De e > cf. Section. Now, let us treat the term P y ωx n j D u n, Λn j. y R ε,n j ε / n

22 Fix δ > and let sgnx = if x and if x >. Denote by U i := X i n ε / n > u i sgnu i δ } n and V i := j ε / n X i n ε / n X i n j δ } n for i =,..., d. Observe that we have for y R ε,n and j ε / n P y d ωx n j D u n, Λn j P y ω U i V i Λ n ε / n. Let us consider the set I = U,..., U d, V,..., V d } and denote by J the set formed by all intersections of d distinct elements of I: J contains d d elements. Let us denote by J,..., J d d all the elements of J. Therefore, we obtain P y ωx n j D u n, Λn j P y j ε / ω J i, Λ n ε n / n. 66 i d d Let us treat the term P y ω d U i, Λ n ε / n. We have by definition of Zn P y ω d d U i, Λ n ε / n P y ω By Theorem. we deduce lim sup y R ε,n P y ω d U i, Λ n ε / n P } Z n ε/ n i > u i sgnu i δ, Z n ε/ n t >, t,. εσ ε W / > u sgnu δ, min W t > t d P i= γ ε / ε / W i > u i sgnu i δ 67 for some constant γ. Abbreviate ε := σ ε ε / / and let us compute the first term of the right-hand side of 67 for sufficiently small ε. By the reflection principle for the Brownian motion, we have P ε W > u sgnu δ, min t W t > Therefore, we obtain, as ε lim sup y R ε,n P y ω = P ε W > u sgnu δ = P W > u sgnu δ ε P = u sgnu δ +ε e x dx. π u sgnu δ ε d U i, Λ n ε / n P ε W < u sgnu δ W < u sgnu δ ε

23 εσ e u sgnu δ π ε / d + oε i= u i sgnu i δ γ ε/ ε / e t π dt. 68 The other terms P y ωj i, Λ n ε / n necessarily contain a term V j for some j, d. Thus, we have for J i d U i, d P y ωj i, Λ n ε / n P y ωv j. 69 Let us bound the terms lim sup y Rε,n P y ωv j for j, d. We start by writing By Theorem., we obtain lim P y ω y R ε,n P y ωv j = P y ω X j n ε / n X j n i δ n i ε / n = P y ω X j k X j n ε / n δ n n ε / n k n P y ω Z n ε / j t min Z n t ε / j s δ s t + P y ω Zj n t Zj n s δ. ε / t min ε / t Zj n t = P min ε / s t j= ε / t ε / s t Zj n s δ W j t min ε / s t W j s δ 7 and lim P y ω y R ε,n min ε / t Zj n t = P ε / s t min ε / t Zj n s δ W j t ε / s t W j s δ. 7 Observe that the right-hand sides of 7 and 7 are equal since W j is a Brownian motion. Thus, let us compute for example the right-hand side term of 7. By Lévy s Theorem cf. 8, Chapter VI, Theorem.3, we have P W j t min W js t ε / s t δ = P t ε / W j t δ. Then, P W j t δ P t ε / t ε / W j t δ = 4P W j ε / δ. Using an estimate on the tail of the Gaussian law cf. 7, Appendix B, Lemma.9 we obtain P W j t δ 4ε/4 t ε / δ π exp δ ε / }. 3

24 We finally obtain lim sup d y R ε,n To sum up, combining 63, 66, 68, 69, and 7, we have P-a.s. lim sup P ω Z n D u Λ n εσ εσ e u sgnu δ + oε π π ε / + P y ωv i 8dε/4 δ π exp δ ε / }. 7 d + oε i= u i sgnu i δ γ ε/ ε / e t π dt d 8dε /4 d δ π exp δ } ε / + fε + gε. 73 Let us now bound the term P ω Z n D u Λ n from below. We have by the Markov property P ω Z n D u Λ n P ωa R, β Rε,n ε / n P ω Λ n min min y R ε,n j ε / n P y ωx n j D u n, Λn j. 74 We first decompose the term P ω Λ n P ω A R, β Rε,n ε / n in the following way P ω A R, β Rε,n ε / n P ω Λ n = P ωa R P ω Λ n P ωa R, β Rε,n > ε / n P ω Λ n = P ωa R P ω β Rε,n > ε / n A R. 75 P ω Λ n Then, we write P ω β Rε,n > ε / n A R = P ωβ Rε,n > ε / n, A R P ω A R P ωβ Rε,n > ε / n, Λ ε / n P ω A R, Λ ε / n = P ωβ Rε,n > ε / n Λ ε / n P ω A c R Λ ε / n. 76 For the term P ω β Rε,n > ε / n Λ ε / n, we have, recalling that N = ε n, P ω β Rε,n > ε / n Λ ε / n = P ωτ Rε,n > ε / n Λ ε / n P ω sup X i j > ε / N Λ ε i,d / n + P ω τ N} > ε / n Λ ε / n. j τ N} By Lemmas.3 and.4 we deduce For the term P ω A c R Λ ε / n, we write 77 lim sup P ω β Rε,n > ε / n Λ ε / n gε3/4 + fε 3/4. 78 P ω A c R Λ ε / n = P ωβ Rε,n > β + } Λ ε / n P ωβ Rε,n > ε / n Λ ε / n. 4

25 Hence, by 78 we obtain lim sup P ω A c R Λ ε / n fε3/4 + gε 3/4. 79 Going back to the term P ω Λ n P ω A R in 75, we write P ω A R P ω Λ n P ω A R = P ω Λ n, A R + P ω Λ n, A c R = P ω Λ n A R + P ω Λ n, A c RP ω A R P ω Λ n A R + P ω Λ n, A c RP ω Λ n, A R = P ω Λ n A R + P ω A c R Λ n P ω A c R Λ n. 8 By 6, we have lim sup Then, we have by the Markov property P ω A c R Λ n lim sup P ω β Rε,n > ε / n Λ n fε + gε. 8 P ω Λ n A R y R ε,n P y ωλ n ε / n + P ωβ Rε,n > ε / n A R. 8 Thus, by 8, 8, 76, 78, 79, and 8, we deduce lim inf P ω A R P ω Λ n lim sup P y ωλ n ε y R / n + fε3/4 + gε 3/4 ε,n fε 3/4 gε 3/4 Combining 74, 75, 78, 79, and 83, we obtain P-a.s. lim inf P ωz n D u Λ n lim sup P y ωλ n ε y R / n + fε3/4 + gε 3/4 ε,n fε 3/4 gε 3/4 fε3/4 + gε 3/4 fε 3/4 gε 3/4 lim inf min min y R ε,n Analogously to 63 we have j ε / n fε + gε. + fε gε 83 fε + gε + fε gε P y ωx n j D u n, Λn j. 84 lim P y ωλ n ε y R / n = εσ + oε. 85 ε,n π ε / At this point, let us introduce more notations. Let δ > be the constant used in the definitions of V i and U i cf. 64 and 65 and introduce E i = X i n > u i + sgnu i δ } n and F i = X i n X i n j δ } n j ε / n 5

26 for i, d. Observe that for all y R ε,n and j ε / n we have P y d d ωx n j D u n, Λn j P y ω E i F i, Λ n P y ω E i, Λ n d P y ωf c i. 86 By Theorem. and similar computations as those to derive equations 68 and 7, we obtain for some constant γ, lim min P y ω y R ε,n as ε and d E i, Λ n = lim sup εσ π e u +sgnu δ + oε d y R ε,n Combining 84, 85, 87, and 88, we obtain P-a.s. lim inf P ωz n D u Λ n εσ π ε / + oε + fε3/4 + gε 3/4 fε 3/4 gε 3/4 fε3/4 + gε 3/4 fε 3/4 gε 3/4 εσ e u +sgnu δ d + oε π e t i= u i +sgnu i δ γ ε / d π dt 87 P y ωfi c 8dε/4 δ π exp δ } ε /. 88 fε + gε + fε gε e t i= u i +sgnu i δ γ ε / dt 8dε/4 π δ π exp δ ε / }. 89 Finally, take δ = ε /8 and let ε in 73 and 89 to prove 54. The next steps in showing that the f.d.d. s converge are standard and we follow 4 and. First, we will prove the following Proposition 4. We have P-a.s., for u >, < a i < b i <, i, d and < t <, lim P ω Z n t u, Proof. For ε > we have d i= P ω Z n n nt u ε, } u Zi n t a i, b i Λ n = d i= P ω Z n t u, q, ; t, vdv } Zi n n nt a i ε, b i + ε Λ n d i= P ω Z n n nt u + ε, } Zi n t a i, b i Λ n d i= d i= bi a i e v t dv. 9 πt } Zi n n nt a i + ε, b i ε Λ n. 9 6

27 for all sufficiently large n. Now, suppose that we have for all u, a i < b i and < t <, lim P ω Z n n nt u, d i= } u Zi n n nt a i, b i Λ n = q, ; t, vdv d i= bi a i e v t dv. πt 9 Combining 9 and 9, we obtain 9 since the limit distribution q, ; t, x is absolutely continuous. Let us denote by l = lt, n the quantity n nt /. We recall that x is the vector of coordinates x,..., x d. Then, observe that P ω Z n n nt u, = = d i= P ω Λ n P ω Z nt lu, lu lb P ω Λ n la = P ωλ nt P ω Λ n = P ωλ nt P ω Λ n lu lb la lu lb la } Zi n n nt a i, b i Λ n d i= lbd P ω la d lbd P ω la d } Z nt i la i, lb i, Λ nt, X k >, nt < k n Z nt dx, d i= Z nt i dx i }, Λ nt, X k >, nt < k n X k >, nt < k n Z nt dx, P ω Z nt dx, lbd P x nt ω la d d i= d i= Z nt i dx i } Λ nt Z n s >, s n nt P ω Z nt dx, By 59, 63, 83, and 85 we have P-a.s. d i= Z nt i dx i } Z nt i dx i } Λ nt. 93 P ω Λ nt lim P ω Λ n = t /. 94 Using Theorem. and Dini s theorem on uniform convergence of non-decreasing sequences of continuous functions, we obtain lim Pz nt t / ω Z n s >, s n nt = Ñ z t uniformly in z on every compact set of the form, K K, K d. By Proposition 4., we have lim P ω Z nt x, d i= } Z nt i x i Λ nt = exp Now, applying Lemma.8 of 4 to 93, we obtain lim P ω Z n n nt u, d i= } Zi n n nt a i, b i Λ n 7 x d i= xi e v π dv.

28 u t / b t / bd t / =... t / Ñ x a t / a d t / t t / x e x d i= e x i π dx... dx d. Finally, make the change of variables y = t / x to obtain the desired result. The final step in showing convergence of the f.d.d. s is Proposition 4.3 We have P-a.s., for all k, u i >, < a i j < bi j and < t < t < < t k, <, i, k, j, d lim P k } ω Z n t i u i, Z n t i a i, b i,..., Zd n t i a i d, bi d Λ n = d b j j= a j u b k j... e x t e x x t t πt t... e x k x k t k t k πtk t k dx k... dx a k πt j uk... q, ; t, x qt, x ; t, y... qt k, x k ; t k, x k dx k... dx. 95 Proof. The proof is by induction in k. This result holds for k = by virtue of 9. Suppose 95 is true for k = m, we show that it can be extended to k = m. Let t i = n t i n and let D i = x R d : x u, a i j < x i b i j, j, d} for i, m. We mention here that in this proof, y i for i, m are all elements of R d while y i for i, m belong to R. By the same argument as in the beginning of the proof of Proposition 4., observe that lim P m ω Z n t i D i } Λ n = lim P ω m Z n t i D i }, provided that the limits exist. Then, we write for sufficiently large n m P ω Z n t i D i }, = P ω Λ n = P ωλ ntm P ω Λ n D m D m m i=m D m P ω Z n t i D i } Λ n Z n t D,..., Z n t m D m, m i=m Z n t i D i } Λ n Z n t m dy m, Z n t m dy m, X >,..., X n > Z n t D,..., Z n t m D m, Z n t m dy m Λ ntm D m P ω 96 P ym n ω Z n s >, s t m t m, Z n t m t m dy m P ym n ω Z n s >, s t m. 97 By the induction hypothesis we have lim P ω Z n t D,..., Z n t m D m, Z n t m D m Λ ntm 8

29 = d j= b j a j u t / m b m j... a m j... On the other hand, by 94 we have P-a.s. e y t πt um t / m e y y t t πt t... e y m y m t m t m πtm t m dy m... dy q, ; t /t m, y qt /t m, y ; t /t m, y... qt m /t m, y m ;, y m dy m... dy. 98 P ω Λ ntm lim = t / m P ω Λ n. 99 Using Theorem. and Dini s theorem on uniform convergence of non-decreasing sequences of continuous functions, we obtain lim Pym n ω Z n s >, s t m t m, Z n t m t m D m = d j= b m j a m j e ym y m j tm t m πtm t m dy m um gt m t m, y m, vdv uniformly in y m on every compact set of the form, K K, K d, and Z n s >, s t m = Ñym t m / lim n Pym ω uniformly in y m on every compact set of the form, K K, K d. Combining 96, 97, 98, 99,,, and using Lemma.8 of 4 twice, we obtain lim P m ω Z n t i D i } Λ n = d j= b j t m b m j... a m j e x t πt a j um um e x x t t πt t... u t / m e xm x m tm t m πtm t m dx m... dx um t / m... q, ; t /t m, y qt /t m, y ; t /t m, y... qt m /t m, y m ;, y m t / m dy m... dy gt m t m, y m, y m Ñy m t m / dy m. Now, make the change of variables t / m y = x,..., t / m y m = x m in to obtain 95 for k = m. 4. Tightness In this section, to finish the proof of Theorem., we prove that the sequence of measures P ω Z n Λ n n is tight P-a.s. The proof is standard: we consider the modulus of continuity, divide the time 9

30 interval into small subintervals, and have to control the probability of fluctuations of our conditioned process over these small time intervals, where we use the results of the last subsection. First, we define the modulus of continuity for functions f C, : w f δ = sup fs ft } t s δ where s, t, and is the -norm on R d. By Theorem 4.5 of 5 it suffices to show that P-a.s., for every ˆε > lim lim sup P ω w Z nδ ˆε Λ n = 3 δ since Z n =. Now observe that P ω w Z nδ ˆε Λ n = P ω sup Z n t Z n s ˆε Λ n t s δ P ω sup X nt X ns ˆε n Λ n t s δ 4 for n /δ. Let m := /4δ and divide the interval, into intervals I k := k m, k+ m, for k m. Additionally, consider the intervals J l := l+ m, l+3 m, for l m and J m :=. Observe that P ω sup X nt X ns ˆε n Λ n t s δ P ω sup X nt X ns ˆε } n k m s,t I k sup X nt X ns ˆε } n Λ n l m s,t J l m P ω sup X nt X ns ˆε n Λ n k m s,t I k + P ω sup X nt X ns ˆε n Λ n 5 l m s,t J l with the convention that sup s,t } =. Our next step is to bound from above the lim sup of both terms in parentheses in the right-hand side of 5. As an example, let us treat the terms indexed by I k for k, m. The term indexed by I and those indexed by J k, k, m can be treated in a similar way. To do that, we will use the same approach as in the proof of Proposition 4.. Analogously to 56 we have for ε, and δ,, P ω sup X nt X ns ˆε n Λ n s,t I k P ω Λ n P ω sup X nt X ns ˆε n, A R, Λ n, β Rε,n δnm s,t I k + P ω β Rε,n > δnm Λ n + P ω β Rε,n > ε / n Λ n. 6 Analogously to 58, we obtain P ω Λ n P ω sup X nt X ns ˆε n, A R, Λ n, β Rε,n δnm s,t I k 3

31 P ωa R P ω Λ n P y y R ε,n j δn m ω sup X nt j X ns j ˆε n. s,t I k Now, observe that for all sufficiently large n P y j δn m ω sup X nt j X ns j ˆε n s,t I k P y ω sup s,t I k X nt X ns ˆε n 7 with I k = k δ m, k+ m. Now, let I k = k 3δ m, k+ m. By Theorem. and the estimate on the tail of the Gaussian law given in 7, Appendix B, Lemma.9, we have lim sup P y ω X nt X ns ˆε n d P sup W t W s ˆε y R ε,n since δ <. We obtain lim sup P y y R ε,n j δn m ω sup s,t I k 8d P sup X nt j X ns j ˆε n s,t I k s,t I k W + 3δ m 6d ˆε πm exp ˆε m 8 ˆε } 6d ˆε πm exp 8 ˆε m }. 9 8 Thus, we have by 59, 6, 6, and 9 lim sup P ω sup X nt X ns ˆε n Λ n s,t I k εσ 6d + oε π ˆε πm exp ˆε m } + fε + gε + lim sup P ω β Rε,n > δnm Λ n. 8 Combining 5 and we find lim sup P ω sup X nt X ns ˆε n Λ n t s δ 6d m ˆε πm exp ˆε m } εσ + oε 8 π Then, let ε = m 3 and δ = m / in. We have by 6 lim sup Therefore, we obtain + fε + gε + lim sup P ω β Rε,n > δnm Λ n P ω β Rε,n > δnm Λ n = lim sup P ω β Rε,n > ε / n Λ n fm 3 + gm 3. lim m lim sup P ω sup s,t Îk X nt X ns ˆε n Λ n =.. As ˆε is arbitrary and m = /4δ, using 4, this last expression proves 3 and consequently the tightness of the sequence P ω Z n Λ n n. 3

32 Acknowledgements This work was mainly done when N.G. was visiting Brazil in the end of ; this visit was supported by FAPESP / C.G. is grateful to FAPESP grant 9/539 3 for financial support. S.P. and M.V. were partially supported by CNPq grant 3886/8 and 3455/9. C.G., S.P., and M.V. also thank CNPq 4743/9 9 and FAPESP 9/ for financial support. We gratefully acknowledge the John von Neumann guest professor program of Technische Universität München which supported a visit of S.P. and M.V. to Munich. We also thank the referee for helpful comments. References M.T. Barlow 995 St Flour Lecture Notes: Diffusions on Fractals. Lect. Notes Math. 69,. M.T. Barlow, J.-D. Deuschel Invariance principle for the random conductance model with unbounded conductances. Ann. Probab. 38, N. Berger, M. Biskup 7 Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Related Fields 37, P. Billingsley 968 Convergence of Probability Measures st ed.. Wiley, New York. 5 M. Biskup Recent progress on the random conductance model. Prob. Surveys 8, M. Biskup, T.M. Prescott 7 Functional CLT for random walk among bounded random conductances. Elect. J. Probab., paper No. 49, Zhen-Qing Chen, D.A. Croydon, T. Kumagai 3 Quenched invariance principles for random walks and elliptic diffusions in random media with boundary. arxiv: F. Comets, S. Popov, G.M. Schütz, M. Vachkovskaia Quenched invariance principle for Knudsen stochastic billiard in random tube. Ann. Probab. 38 3, F. Comets, S. Popov, G.M. Schütz, M. Vachkovskaia Knudsen gas in a finite random tube: transport diffusion and first passage properties. J. Statist. Phys. 4, T. Delmotte 999 Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoamericana 5, 8 3. C. Gallesco, S. Popov Random walks among random conductances I: Uniform quenched CLT. Elect. J. Probab. 7, paper No. 85,. C. Gallesco, S. Popov Random walks among random conductances II: Conditional quenched CLT. Available on Arxiv: D. Gilbarg, N. S. Trudinger Elliptic partial differential equations of second order. Classics in Mathematics. Reprint of the 998 edition. Springer, Berlin. 4 D. Iglehart 974 Functional central limit theorems for random walks conditioned to stay positive. Ann. Probab. 4,