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

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

2010-06-16

arxiv.org/abs/1006.3143v2

Journal of Theoretical Probability, 2011

Mathematics

Probability

23 pages

Scientific paper

10.1007/s10959-011-0345-8

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.

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