Some properties of the rate function of quenched large deviations for random walk in random environment

TL;DR

This paper investigates properties of the rate function in quenched large deviations for random walks in random environments, confirming a conjecture and comparing rate functions for diffusions and Brownian motions.

Contribution

It proves that the second derivative of the rate function diverges at zero in the recurrent case and compares rate functions for different stochastic processes.

Findings

The second derivative of the rate function tends to infinity as the parameter approaches zero in the recurrent case.

Confirmed a conjecture of Greven and den Hollander regarding the rate function.

Established a comparison between rate functions of diffusions and Brownian motions.

Abstract

In this paper, we are interested in some questions of Greven and den Hollander about the rate function $I\_{\eta}^q$ of quenched large deviations for random walk in random environment. By studying the hitting times of RWRE, we prove that in the recurrent case, $\lim\_{\theta\to 0^+}(I\_{\eta}^q)''(\theta)=+\infty$, 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.