Level 1 quenched large deviation principle for random walk in dynamic random environment

TL;DR

This paper provides two simplified proofs for the level 1 large deviation principle of a random walk in a dynamic random environment, establishing key properties of the rate function under mild ergodicity conditions.

Contribution

It introduces alternative short proofs for the level 1 large deviation principle in dynamic environments, utilizing the sub-additive ergodic theorem.

Findings

Existence of the rate function

Continuity of the rate function

Convexity of the rate function

Abstract

Consider a random walk in a time-dependent random environment on the lattice Zd. Recently, Rassoul-Agha, Seppalainen 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 2003.