Weak Bernoullicity of random walk in random scenery

Abstract

In this paper) we consider an arbitrary irreducible random walk and an ar~ bitrary stationary and ergodic random scenery on Zd. \Ve find conditions on their distributions such that the associated random walk in random scenery is or is not weak Bernoulli. Our results extend an earlier das~ sification for the special case in which the individual scenery values arc assumed to be independent and identically distributed random variables. 1 Background and main theorems 1.1 Background. For a fixed integer d ~ I, let X = (Xn)nE:J: be independent and identically distributed random variables taking values in Zd according to a common distribution 't11 OIl Z d satisfying (*1) m is irreducible, i.e.) supp(m) = {;r;: m(;r;) > O} generates the group Zd. Let = be the corresponding random walk on Z d, defined by 5 = 0, l\eA'i, let C = be a random scener,1J on Zd taking values in a finite set 'yd F according to a distribution ft on F J • satisfying (*2) ft is stationary with respect to translations in the group Z d; (*3) ft is ergodic with respect to translations in fm, the subgroup of Zd generated by SUpp(lII) SUpp(lII) = { 11' 111()111(11) > O}. The random walk and the random scenery are assumed to be independent. The joint process y= with }~, = (Xn, Csl is called the random walk m random scener,1J associated with m and ft. In ergodic theory, }' is an example of what is called a skew prod'uct. It is obvious that, because of (*2), }' is a stationary process. The role of (*3) is to make }' non-periodic. Indeed, if F = {O, I}, d = I, m is simple random ~ ReKearch :mppoJ1ed in paJ1 by KBN gran1 2 PO:JA 04G22 (2002-2005) j' ReKearch :mppoJ1ed in paJ1 by a gran1 from 1he S\vedh,h Na1 ural Science ReKearch Council and by NSF-gran1 DfvlS-010:JS41.