FUNCTIONAL CENTRAL LIMIT (RWRE)

Transcription

1 FUNCTIONAL CENTRAL LIMIT THEOREM FOR BALLISTIC RANDOM WALK IN RANDOM ENVIRONMENT (RWRE) Timo Seppäläinen University of Wisconsin-Madison (Joint work with Firas Rassoul-Agha, Univ of Utah)

2 RWRE on Z d RWRE is a Markov chain on Z d with transition matrix created by a spatially ergodic random mechanism. Ω = space of environments = P Zd where P = space of probability vectors on Z d. An environment is ω = (ωx) x Z d Ω. Vector ωx gives probabilities ωx,u of jumps u out of state x. Transition probabilities are πx,y(ω) = ω x, y x. Given ω and initial state z, random walk Xn defined by P ω z (X 0 = z) = 1 P ω z ( X n+1 = y Xn = x ) = πx,y(ω)

3 P ω z is the quenched path distribution. On Ω we put IID probability measure P. Averaged (annealed) distribution Pz = P ω z ( ) P(dω). P ergodic under natural translations (Txω)z = ω x+z on Ω. Transition probabilities obey πx,y(ω) = π0,y x(txω). Usual assumptions Nearest-neighbor jumps and uniform ellipticity: πx,y(ω) κ > 0, x y = 1 πx,y(ω) = 0, x y = 1. GOAL Answer basic questions: recurrence/transience, LLN, CLT, large deviations. Dimension d = 1 fairly well understood, but d 2 not.

4 Examples of basic ignorance about RWRE in d 2 (a) Recurrence/transience No general criteria exist. (b) 0-1 Law For u R d, Au = {Xn u }. Expect P0(Au) {0, 1}. Known only for d = 2 (Zerner and Merkl). (c) Law of large numbers Currently known that Xn/n but limit can be random with two distinct values. In d = law implies LLN. In d 5 limit velocities cannot both be nonzero (Berger).

5 FUNCTIONAL CENTRAL LIMIT THEOREMS Assuming that a limit velocity v = lim Xn/n exists, let Bn(t) = n 1/2 {X [nt] ntv} (t 0) Let W be the distribution of a Brownian motion on R d with some diffusion matrix Γ. There are two kinds of CLT s: Averaged: P0{Bn } W ( ) Quenched: P ω 0 {B n } W ( ) for P-a.e. ω (Quenched) = (Averaged) by integrating out ω.

6 THE BALLISTIC CASE Ballistic walks are those that satisfy Xn/n v 0. We make this assumption via the Sznitman-Zerner regeneration times. Assume directional transience: û Z d such that P0{Xn û } = 1. Then w.p. 1 there exists first time τ1 such that sup n<τ1 Xn û < Xτ1 û inf n τ1 Xn û Iteration gives sequence τ1 < τ2 < τ3 < < Not stopping times, yet {Xτ k Xτ k 1 } k 2 IID under P0.

7 Backtracking time β = inf{n : Xn û < X0 û}. Theorem Assume directional transience Xn û. If E0(τ1) < then a.s. Xn/n v E 0[Xτ1 β = ] E0[τ1 β = ] If E0(τ 1 2 ) < then averaged CLT holds. Limiting Brownian motion has diffusion matrix [(Xτ1 τ 1v)(Xτ1 τ 1v) t ] β = Γ = E 0 E0[τ1 β = ] Proof from IID results and some estimation. Sznitman and Zerner: these hypotheses follow from some more fundamental transience assumptions.

8 HOW TO OBTAIN A QUENCHED CLT? APPROACH I. UPGRADE THE AVERAGED CLT Control E ω 0 [F (B n( ))] E0[F (Bn( ))] for a rich enough class of {F } on path space. With this strategy: Theorem (Bolthausen and Sznitman, 2002) d 4, { non-nestling: P z ω : (z û) π0,z(ω) η > 0 and small noise. Then quenched CLT holds. } = 1 Theorem (Berger and Zeitouni, 2007) d 4, averaged CLT, and some moments on τ1 and τ2 τ1. Then quenched CLT holds.

9 II. A MARTINGALE APPROACH FROM SCRATCH Assumptions Directional transience Xn û E0(τ p 1 ) < for a large p (e.g. p > 176d suffices) Bounded steps (not necessarily nearest neighbor) Instead of ellipticity, assume that walk not restricted to a 1-dimensional subspace, and P{ z : π0,0 + π0,z = 1 } < 1. Theorem (Rassoul-Agha, S. 2007) Under these assumptions quenched CLT holds. Note The regularity assumption is the right one.

10 JUSTIFICATION FOR HIGH MOMENT ASSUMP- TION ON τ1 E0(τ p 1 ) < p < guaranteed by: Non-nestling In the uniformly elliptic nearest neighbor case, E [ ( z z û π0,z ) + ] > κ 1 E [( z z û π0,z ) ]. Sznitman s (T ): E0 [ exp(c sup 0 n τ1 Xn γ ) ] <, 0 < γ < 1 Believed that in uniformly elliptic case, (T ) ballistic Not proved, but if we accept it then this last quenched CLT covers all ballistic uniformly elliptic walks.

11 INGREDIENTS OF THE PROOF Consider environment chain T X n ω with transition kernel Πf(ω) = z π0,z(ω)f(tzω). (Recall translations πx,y(tzω) = π x+z,y+z (ω).) Suppose we have an invariant distribution P for Π. Define drift D(ω) = z z π0,z(ω). Let v = E (D), g(ω) = D(ω) v.

12 Decompose into martingale plus additive functional: Xn nv = Xn n 1 k=0 D(T Xk ω) + n 1 k=0 g(t Xk ω) Mn + Sn(g) Solve (1 + ε)hε Πhε = g with hε = Decompose further: k=1 Π k 1 g (1 + ε) k. Xn nv = Mn + n 1 k=0 ( hε(t Xk+1 ω) Πhε(T Xk ω) ) + εsn(hε) + Rn(ε) = Mn(ε) + εsn(hε) + Rn(ε) HOPE: After ε 0, Xn nv = Mn + Rn. Apply martingale CLT to Mn under P ω 0. Show n 1/2 Rn 0.

13 By adapting arguments of Maxwell-Woodroofe (2000) this can be achieved under the hypothesis Note n 1 k=0 n 1 k=0 Π k g L 2 (P ) Π k g(ω) = E ω 0 (X n) nv. = O(n α ), α < 1/2. Theorem (Rassoul-Agha, S. 2005) Assume P ergodic for Π, z z 2 E (π0,z) < and E ( E ω 0 (X n) nv 2 ) Cn 1 δ, δ > 0. Then Bn(t) = n 1/2 {X [nt] ntv} converges to a Brownian motion under P ω 0 for P -a.e. ω.

14 Comments This is an intermediate, general result. There remains the difficulty of applying it. One wants to work with P but the theorem is in terms of an unknown invariant P. Need to prove existence of P and relate it to P. Hard part is to prove estimate E ( E ω 0 (X n) nv 2 ) Cn 1 δ, δ > 0. This fails for walks in 1 dimension, or if P{ z : π0,0 + π0,z = 1 } = 1. In these cases E 0 ω (X n) behaves diffusively, there is no QCLT for Bn but there is one for Bn(t) = n 1/2 {X [nt] E ω 0 (X [nt] )}

15 APPLYING THE GENERAL THEOREM PRELIMINARY WORK Existence and regularity of P from IID regeneration structure and moments of τ1. Consequence: can work exclusively with P. Remaining goal: E( E ω 0 (X n) E0(Xn) 2 ) Cn 1 δ. TWO MAIN STEPS 1. Decompose E( E 0 ω (X n) E0(Xn) 2 ) into martingale increments to isolate contributions of individual sites. This leads to a bound in terms of E0,0( X [0,n) X [0,n) ), intersections of two independent walks X, X in a common environment.

16 2. To control intersections, introduce joint regeneration times (µ k, µ k ) such that (X, X) regenerate together at position (Xµ k, X µk ) at a common level. E0,0 ( X [0,n) X [0,n) ) n 1 k=0 n 1 k=0 E0,0 E0,0 ( X [µk, µ k+1 ) X [ µk, µ k+1 ) ) [h( X µk Xµ k ) ] by restarting the walks at common regeneration levels, with h(x) = E0,x ( X [0, µ 1) X [0, µ 1) β = β = ), and a Markov chain Y k = X µk Xµ k. Estimation task has been reduced to controlling the Green function of the Markov chain Y k.

17 The idea for the Green function estimation: When X and X are far from each other, they have a high chance of regenerating together before intersecting. Thus far away from the origin Y k can be approximated by a symmetric random walk. The regularity assumptions on the RWRE guarantee that Y k exits large cubes. Excursions of Y k can be related to random walk excursions. Consequence: n 1 k=0 E0,0 [ h( X µk Xµ k ) ] Cn 1 η from which follows E( E ω 0 (X n) E0(Xn) 2 ) Cn 1 δ. Proof is complete.