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Communications in

Mathematical Physics

Commun. Math. Phys. 87, 81-87 (1982)

© Springer-Verlag 1982

Weak Convergence of a Random Walk in a Random Environment G r e g o r y F. Lawler Department of Mathematics, Duke University, Durham, NC 27706, USA

Abstract. Let hi(x), i = 1,...,d, x ~ Z d, satisfy ni(x) > • > 0, and n 1 (x) + . . . + na(x ) = 1. Define a M a r k o v chain on Z" b y specifying that a particle at x takes a j u m p of + 1 in the i th direction with probability ½n~(x) and a j u m p of - 1 in the i th direction with probability ½ni(x ). If the n~(x) are chosen from a stationary, ergodic distribution, then for almost all n the corresponding chain converges weakly to a Brownian motion.

1. Introduction Let Z ~ be the integer lattice and let % i = 1,..., d, denote the unit vector whose i th c o m p o n e n t is equal to 1. Let

S = { ( p l , . . . , Pd) e [~a :Pi > O, p I + " - + Pn = 1 }, and suppose we have a function 7 r : z d ~ s . generated with transition probability

Then a M a r k o v chain X~(j) on Z d is

P { X = ( j + 1) = x + e ilX~(j) -- x} -- ½rri(x),

(1.1)

and generator d

L~g(x) = ~ ½n,(x){g(x + e,) + g(x - el) }. i=1 If the function 7r is chosen from some probability distribution on S, this gives an example of a r a n d o m walk in a r a n d o m environment. F o r any n, we can consider the limiting distribution of the process X~ satisfying X~(0) = 0 and (1.1). Let ~ > 0 and set s

= {(pl ..... p.)es:pl

and let C" be the set of functions n:Zd--*S ~. The main result of this paper is:

Theorem 1. Let # be a stationary ergodic measure on CL T h e n there exists b ~ S ~ such

0010-3616/82/0087/0081/$01.40

82

G . F . Lawler

that for p--almost all rceC ~, the processes X~(t) = ~ n X , ( [ n t ] ) converge in distribution to a Brownian motion with covariance (bi 6o) A special case of this theorem occurs when the x(x) are independent, identically distributed random variables taking values in S t A similar theorem for diffusion processes with random coefficients was proved by Papanicolaou and Varadhan [3], and a considerable portion of this paper is only a restating of their proof in the context of discrete random walk. The crucial new step is Lemma 4, which replaces Lemma 3.1 of their paper. This is a discrete version of an a priori estimate for solutions of uniformly elliptic equations. The ideas of Krylov [2] are used in the proof of Lemma 4; properties of concave functions are used to estimate solutions to a discrete Monge-Ampere equation. 2. An Ergodic Theorem on the Space of Environments

Fix an environment rceC ~, and assume X~(O)=O. Let Zj=(Z) ..... Z].)= and let ~'j=a{Z~ .... ,Zj}. Let Yj=Tr(X,~(j)). Then Yj is measurable with respect to J-j, and

X,,(j)-X,~(j-1),

P{Z~=ei[J-j-1}=P{Zj = - e i l J j - 1 } = ~_y). 1 i n

Then X~(n)= ~ Zj is a martingale and j=l

B(Z}IZ}~[~-j_ 1)

i I • i2 il ~ i~ = i2 Y~_

J'O

n-1

Let Vi, = ~

Y}. Then the invariance principle for martingales (see e.g. Theorem 4.1

j=0

of [1]) states that W~(t)=(W~(t),...,wa(t)) converges in distribution to the standard Brownian motion on Re, where [ntl

W'.(t)=-(Vin) -1/2 ~. Z}. j=l

Now suppose there exists a beS ~ such that lim -1 ,~1

n(X~(j))

= b

a.s..

n ~ c v n j=O

Then by the above argument we can conclude that

X~"l(t) = -~X~([nt]) converges in distribution to a Brownian motion with covariance (bl c~o).Therefore, in order to prove Theorem 1 it is sufficient to prove:

Random Walk in Random Environment

83

Theorem 2. Let # be a stationary ergodic probability measure on C ~. Then there exists b~S ~ such that for #--almost all nEC ~, ln--1

l i m - ~ ~ X ~ ( j ) ) = b a.s..

(2.1)

n ~ oo ? / j = O

This is clearly an ergodic theorem and the idea of Papanicolaou and Varadhan 1-3] is to find a measure on C ~ so that a standard ergodic argument can be used. We define the canonical M a r k o v chain with state space C ~ to be the chain whose generator L,¢ is given by d

~ g ( n ) ~- E ~ni(O)(g(~ein) ~l_g(~- e i n ) } ' i=1

where zxn(y ) = n(y - x). In this chain, the "particle" stays fixed at the origin and allow the environment to change around it (rather than having the particle move around a fixed environment). If we define go : C~ ~ Eu by go(n) = n(0), and let ~ J n denote the (random) environment at thej th step of this chain, then (2. l) is equivalent to ln-1 l i m - ~ go(&ain)= b a.s. #.

(2.2)

n~oo n j=0

By standard ergodic theory we can prove (2.2), and hence (2.1), if we prove: Theorem 3. Let # be a stationary ergodic probability measure on C ~. Then there

exists an ergodic probability measure 2 on C ~ which is mutually absolutely continuous with # and which is invariant under the canonical Markov chain ~C,a. Clearly,

b = ~ go(n)d2(n). C~

To prove Theorem 3 we need some lemmas. For each n > 0, let 7", denote the elements of Z a under the equivalence relation 1

(zl,..., za)~ (wl,..., Wd) if ~nn(Z~- w~)EZ for each i. Then [T, [ = (2n) a. If n: T, ~ S ", we may think of n as a periodic environment in C ". Let C~ denote the set of such periodic environments. For neC], let R~" denote the resolvent operator 1

R~g(x)=j~_o ( 1 - ~ )

J

,

L~g(x).

If g : T. ~ E we define the usual L p norms (with respect to normalized counting measure on T~), Ilgllp = 1(2n)-a ~ X~ Tn

IIg l[~ = sup Ig(x)l x~Z~

(g(x)FI I/p

84

G.F. Lawler

L e m m a 4. There exists a constant c 1 (depending only on d and ~) such that for every neCk, g : T n ~ R , IIR"g IIco 1 ~). 1

Random Walk in Random Environment

85

3. Proof of Lemma 4 It remains to prove Lemma 4. Let O, = {(zl,... ,zd)eZn: dO. =

Izl I + . . . + Izdl _- 2 > z(x) for e v e r y x e D . T h e r e f o r e t h e r e exists a least b ( d e p e n d i n g o n a) s u c h t h a t a.x + b > z(x) for all xED. It is e a s y t o see t h a t a ' x o + b = Z(Xo) for s o m e x o e D , a n d since

a'x o +b=a'2

+ b + a'(x o - 2 ) > = ½ 2 > 0 ,

x o e i n t D. By L e m m a 8, asl(xo). T h e r e f o r e

A =

U

I(x),

xeintD

m e a s ( A ) < m e a s ( U I(x)),