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

TL;DR

This paper establishes a large deviations principle for a broad class of one-dimensional Markov processes governed by generalized elliptic operators, extending classical results and applying to reaction-diffusion wave front propagation.

Contribution

It generalizes large deviations principles to Markov processes with non-standard generators and applies findings to reaction-diffusion equations.

Findings

Large deviations principle proven for generalized elliptic operators.

Extension of classical results to non-Wiener Markov processes.

Application to wave front propagation in reaction-diffusion systems.

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.