Large Deviations Principle for a Large Class of One-Dimensional Markov Processes

Abstract

We study the large deviations principle for one dimensional, continuous, homogeneous, strong Markov processes that do not necessarily behave locally as a Wiener process. Any strong Markov process $X_t$ in $\mathbb{R}$ that is continuous with probability one, under some minimal regularity conditions, is governed by a generalized elliptic operator $D_v{D}_u$, where $v$ and $u$ are two strictly increasing functions, $v$ is right continuous and $u$ is continuous. In this paper, we study large deviations principle for Markov processes whose infinitesimal generator is $\epsilon D_v{D}_u$ where $0<\epsilon \ll 1$. This result generalizes the classical large deviations results for a large class of one dimensional "classical" stochastic processes. Moreover, we consider reaction-diffusion equations governed by a generalized operator $D_v{D}_u$. We apply our results to the problem of wave front propagation for these type of reaction-diffusion equations.

arXiv comment23 pages