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

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 a"e 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 a parts per thousand(a)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 wavefront propagation for these type of reaction-diffusion equations.