Large deviations principle for stationary solutions of stochastic differential equations with multiplicative noise

TL;DR

This paper establishes a large deviations principle for stationary solutions and invariant measures of certain stochastic differential equations with multiplicative noise, linking the rate function to the quasi-potential.

Contribution

It introduces a novel approach connecting the LDP for stationary solutions with the quasi-potential, providing new insights into the invariant measures of SDEs.

Findings

LDP for stationary solutions established using weak convergence.

LDP for invariant measures derived via contraction principle.

Equivalence shown between rate functions and quasi-potential.

Abstract

We study the large deviations principle (LDP) for stationary solutions of a class of stochastic differential equations (SDE) in infinite time intervals by the weak convergence approach, and then establish the LDP for the invariant measures of the SDE by the contraction principle. We further point out the equivalence of the rate function of the LDP for invariant measures induced by the LDP for stationary solutions and the rate function defined by quasi-potential. This fact gives another view of the quasi-potential introduced by Freidlin and Wentzell.