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

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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:

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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 <=cl n2 IIg IIa.

The p r o o f of this l e m m a is delayed until Sect. 3. The next l e m m a follows from our assumption that # is stationary (see P a r t h a s a r a t h y [4]). L e m m a 5. For each n, there exists n.~C~ such that if#,, is the probability measure on C ~ which assigns measure (21"/) - d to ZxIrnfor each x e Tn, then #. ~ #

weakly.

Proof o f Theorem 3. Let rc.eC~ be a sequence as in I_emma 5 with #. ~ # . Let ~b. be the density, with respect to normalized counting measure on T., of an invariant probability measure on T~ for n., i.e. L~.~b. = ~. and II~b. II1 = 1. If R~ = R~. is the resolvent corresponding to n., then R~tk. = n E t ~ n . If we consider R. as a m a p from La(T~) to L~(T~), then L e m m a 4 states that the m a p is b o u n d e d by cln 2. Therefore R*:LI(T~)~La/td-1)(Tn) is also b o u n d e d by Cl n2. Since R~* (k. _- n 2the, we get

n2 ][~)n ]1d/(d- 1)~ Cl n2 II q~ fix II~,,

t1,,/(,~-:~--<

=

el n2,

q.

Let 2n be the probability measure on C~, 2.(%n.) = (2n)- adp.(x). Then ;~. is invariant under the canonical M a r k o v chain £a and d2,

~n

< c1"

d/(d-1)

Since #~ ~ tt weakly, standard arguments give that 2, has a subsequence converging to a probability measure 2 which is invariant under Lf. Also 2 ~ # and, in fact, d2 a/ca- ~) < a/ca- 1) -d# - c I . c~ d#

Let E = {d2/d# = 0}. Since 2 is invariant, 2(L,eE) = 2(E) = 0, and hence ~ E c E (a.s. #). Since # is ergodic and 2 ~ #, #(E) = 0, and hence # ~ 2. Since # and 2 are mutually absolutely continuous and # is ergodic, 2 is ergodic. Example. Let d = 2 and # be product measure with #{Nx) = e} = #{n(x) = 1 - ~} =½, where 0 < a < ½. Then # is not invariant under f , ifB = {n(el) = a}, then #(B) =½,

but # ( S B ) = - ~3 + ~. Although it is not easy to describe 2 in this case, s y m m e t r y considerations give that b _- (>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 _-
{zeO,:lz~ l + . . . + Izal = n},

int D, = D,/dD.. Let rceC~. Iff:D,--+ [0, oo) with f ( x ) = 0 for xeOD,, let Q f ( x ) = Ex F, f ( S , ( j ) ) , j=0

where z = inf {j:X,(j)~OD,}, and Ex denotes expectation assuming X.(0)= x. We will prove the following: Lemma 6. There exists a constant c z (dependin O only on d and o0 such that for every f:O,--* [0, co), IIQfll 0o < C2n2 Ilflla, where

1

IlSll~: I~TI x~ (S(x)L To get Lemma 4 from Lemma 6 is routine using the fact that the expected time until hitting OD, is of order n 2. Fix n, and write D = D,. Ifu :D-+ ~, we define the second difference operators on int O by dlu(x ) = u(x + el) + u(x - ei) - 2u(x). We will call u concave on D ifAiu(x) < 0 for all xeint D and all i (note this is weaker than the usual definition of concave). We define the discrete M o n g e - A m p e r e operator M on int D by d M u = ~I Aiu" i=1 we will prove the following: Lemma V. Let f : O - + [ O , oo) be a function with f - O concave function z :D --* r0, oo) such that (i) z - O

on gO. Then there exists a

on ~D,

(ii) ( - 1)dMz = f d on intO. Moreover, there exists a constant c 3 (dependin9 only on d) such that

(iii) IIz II ® --
Suppose that we have Lemma 7, and let us derive Lemma 6. Fix xeint D, and let

86

G.F. Lawler

X,(j) be the M a r k o v chain induced by rc with X~(0) = x. Then d

~(z(x~(1))

-

z(XAO)) = ~ ½~,(x)A,z(x) i=l

=< - ½~1Mz(x)l x/a = --½otf(x). Here we have used the inequality Continuing as above we m a y deduce

(albl + . . . + a n b J > (al... an) (bl... be).

(j- 1) ^* E[z(X,~(j ^ z)) - z(X~(O)) + ½~z ~ f(X~(k))] <=O. k=O

Letting j go to infinity,

½otQf(x) = Ex½ot ~ f(X,~(k)) < z(x). k=0

and L e m m a 7 then gives the required bound. T o prove L e m m a 7, let d be the set of all concave functions u on D satisfying (i) u = 0 on OD, (ii) ( - 1)dMu > f d on intD. We first note that ~¢ is non-empty: let h:D--* [0, ~ ) by

h(x) = n(n + 1) - Ixl(Ixl + 1), where I(xl ..... xd)l = Ixll + . . . + Ixal. One can check that ( - 1)dMh > 2 d and hence / ~ h e d for fl sufficiently large. It is easy to check that if u l , u 2 e d , then min ( u ~ , u 2 ) e d ; in fact, if we let

z(x) = inf u(x), U:EJO'

one can verify that z e ~ . It remains to be shown that ( - 1)aMz =fd. Suppose ( - 1)dMz(x)> (f(x)) d for some x e i n t D, i.e. d

( - 1)a [ I (z(x + ei) + z(x - el) - 2z(x)) > (f(y))a. i=1

Let y < z(x) be such that d

( - 1)a l-I (z(x + e,) + z(x - e,) - 2y) = (f(x)) a. i=1

and set

y=x" Then again one can check that vs~¢, contradicting the minimality of z. We now wish to estimate z. F o r x e i n t D, let

I(x) = {(a I .... , an)~R d :z(x + ei) - z(x) <=a i <=z(x) - z(x - el) }.

Random Walk in Random Environment

87

N o t e t h a t m e a s (I(x)) = ( - t)~Mz(x) = ( f ( x ) ) d. W e state t h e n e x t e a s i l y p r o v a b l e fact as a l e m m a : 8. Let a E R d, b > 0, and let r be the affine function r(x) = a ' x + b. Suppose r(x) > z(x) for every x e D and r(xo)= Z(Xo) for some x o e i n t D. Then aeI(xo). N o w let 2 = II z LI0o a n d let Y ~ i n t D w i t h z ( ~ ) = 5. A s s u m e 2 > 0. Let

Lemma

Z = {aeRa:lal <2/4n}. F i x aEA. If b > 32, t h e n a ' x + b > 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)), <

Z ( ( f ( x ) ) d" xeD

Since m e a s ( A ) = (5d)(c4n) -d for s o m e c4, w e get

g < c4n[ ~ ( f ( x ) ) d ] 1/a x~D

< c~n ~

IIf lid.

Acknowledgements. I would like to thank S. R. S. Varadhan for suggesting this problem and for bringing to my attention the work of Krylov. I would also like to thank Bob Vanderbei for useful discussions. This paper was written while the author was a visiting member at the Courant Institute of Mathematical Sciences. References 1. Hall, P., Heyde, C. C.: Martingale limit theory and its application. New York: Academic Press 1980 2. Krylov, N. V.: An inequality in the theory of stochastic integrals. Theor. Prob. Appl. 16, 438-448 (1971) 3. Papanicolaou, C., Varadhan, S. R. S.: Diffusions with Random coefficients. In: Essays in Honor of C. R. Rao. Amsterdam: North Holland 1982 4. Parthasarathy, K. R.: On the category of ergodic measure. Ill. J. Math. 5, 648-655 (1961) Communicated by T. Spencer Received May 20, 1982