Random walks in static and markovian time-dependent random environment

Abstract

Firstly, we introduce ellipticity criteria for random walks in i.i.d. random environments, under which we can extend the ballisticity conditions of Sznitman’s and the polynomial effective criteria of Berger, Drewitz and Ram´ırez originally defined for uniformly elliptic random walks. We prove under these ellipticity criteria the equivalence of Sznitman’s (T 0 ) condition with the polynomial effective criterion (P)M, for M large enough. We furthermore give ellipticity criteria under which a random walk satisfying (P)M for M large enough, is ballistic, satisfies the annealed central limit theorem or the quenched central limit theorem. Secondly, we consider a random walk in a time-dependent random environment on the lattice Z d . Recently, Rassoul-Agha, Sepp¨al¨ainen and Yilmaz [RSY11] proved a general large deviation principle under mild ergodicity assumptions on the random environment for such a random walk, establishing first level 2 and 3 large deviation principles. Here we present two alternative short proofs of the level 1 large deviations under mild ergodicity assumptions on the environment: one for the continuous time case and another one for the discrete time case. Both proofs provide the existence, continuity and convexity of the rate function. Our methods are based on the use of the sub-additive ergodic theorem as presented by Varadhan in [Var03].