Large deviations for random walk in a space--time product environment

Abstract

We consider random walk $(X_n{)}_{n\ge 0}$ on ${\mathbb{Z}}^d$ in a space--time product environment $\omega \in \Omega$. We take the point of view of the particle and focus on the environment Markov chain $(T_{n,X_n}\omega {)}_{n\ge 0}$ where $T$ denotes the shift on $\Omega$. Conditioned on the particle having asymptotic mean velocity equal to any given $\xi$, we show that the empirical process of the environment Markov chain converges to a stationary process ${\mu }_{\xi }^{\infty }$ under the averaged measure. When $d\ge 3$ and $\xi$ is sufficiently close to the typical velocity, we prove that averaged and quenched large deviations are equivalent and when conditioned on the particle having asymptotic mean velocity $\xi$, the empirical process of the environment Markov chain converges to ${\mu }_{\xi }^{\infty }$ under the quenched measure as well. In this case, we show that ${\mu }_{\xi }^{\infty }$ is a stationary Markov process whose kernel is obtained from the original kernel by a Doob $h$-transform.

Published in at http://dx.doi.org/10.1214/08-AOP400 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• The Point of View of the Particle on the Law of Large Numbers for Random Walks in a Mixing Random Environment
• Coincidence of Lyapunov exponents for random walks in weak random potentials

Cited by in corpus (8)

• Differing averaged and quenched large deviations for random walks in random environments in dimensions two and three
• Process-level quenched large deviations for random walk in random environment
• Variational formulas and disorder regimes of random walks in random potentials
• Busemann functions and Gibbs measures in directed polymer models on ${\mathbb{Z}}^2$
• Equality of averaged and quenched large deviations for random walks in random environments in dimensions four and higher
• Harmonic functions, h-transform and large deviations for random walks in random environments in dimensions four and higher
• Random walk in a birth-and-death dynamical environment
• On large deviation rate functions for a continuous-time directed polymer in weak disorder