Small-time annealed large deviations principle for one-dimensional diffusions in a random environment

Abstract

In this paper, we establish a small-time annealed path large deviation principle for one-dimensional diffusions in a random environment associated with the generator ${\mathcal{L}}_{W}f(x)=e^{-\rho (x,W)}(e^{a(x,W)}{f}^{\prime }(x){)}^{\prime }$. The coefficients $\{\rho (x,\cdot ):x\in \mathbb{R}\}$ and $\{a(x,\cdot ):x\in \mathbb{R}\}$ are random. We assume that for each fixed realization of the environment, $\rho$ and $a$ are continuous and locally exponentially integrable, and that the support of the associated intrinsic coordinates is compact and non-collapsing. This framework includes the extensively studied Brox diffusion $d{X}_t=d{B}_t-\frac{1}{2}\dot{W}(X_t)dt$, where $B$ is a standard Brownian motion and $W$ is an independent two-sided Brownian motion representing the environment. The Itô--McKean representation of the diffusions and the estimates of the first exit probabilities derived via Moser iteration play a crucial role.

22 pages