Level 1 Quenched Large Deviations for Random Walk in Dynamic Random Environment
by David Campos   Alexander Drewitz   Alejandro F. Ramirez   Firas Rassoul-Agha   Timo Seppalainen  

Vol. 8 No. 1 (2013) P.1~P.29

ABSTRACT

Consider a random walk in a time-dependent random environment on the lattice $\mathbb{Z}^d$. Recently, Rassoul-Agha, Sepp$\ddot{a}$l$\ddot{a}$inen and Yilmaz [13] 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 [22].

KEYWORDS
Random walk in random environment, large deviations, sub-additive ergodic theorem

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 60F10, 82C41

MILESTONES

Received: 2012-01-30
Revised : 2012-08-21
Accepted: 2012-08-21

Download Full Content