Some Properties of the Rate Function of Quenched Large Deviations for Random Walk in Random Environment

#### A. Devulder

2006, v.12, Issue 1, 27-42

ABSTRACT

In this paper, we are interested in some questions of Greven and den Hollander [Large deviations for a random walk in random environment. Ann. Prob., 1994, 22, 1381-1428] about the rate function $I^q_{\eta}$ of quenched large deviations for random walk in random environment. By studying the hitting times of RWRE, we prove that in the recurrent case, $\lim_{\theta\to 0^+}(I^q_{\eta})''(\theta )=+\infty$, which gives an affirmative answer to a conjecture of Greven and den Hollander. We also establish a comparison result between the rate function of quenched large deviations for a diffusion in a drifted Brownian potential, and the rate function for a drifted Brownian motion with the same speed.

Keywords: random walk in random environment,large deviations