Large deviation principle for occupation measures of stochastic generalized Burgers-Huxley equation

TL;DR

This paper establishes the existence, uniqueness, and ergodic properties of solutions to the stochastic generalized Burgers-Huxley equation, and proves a large deviation principle for occupation measures, revealing the exponential convergence rate.

Contribution

It provides the first large deviation principle for occupation measures of the stochastic Burgers-Huxley equation, combining existence, ergodicity, and asymptotic analysis.

Findings

Existence of unique mild and strong solutions.

Irreducibility and strong Feller property of the semigroup.

Large deviation principle for occupation measures.

Abstract

The present work deals with the global solvability as well as asymptotic analysis of stochastic generalized Burgers-Huxley (SGBH) equation perturbed by space-time white noise in a bounded interval of $\mathbb{R}$. We first prove the existence of unique mild as well as strong solution to SGBH equation and then obtain the existence of an invariant measure. Later, we establish two major properties of the Markovian semigroup associated with the solutions of SGBH equation, that is, irreducibility and strong Feller property. These two properties guarantees the uniqueness of invariant measures and ergodicity also. Then, under further assumptions on the noise coefficient, we discuss the ergodic behavior of the solution of SGBH equation by providing a Large Deviation Principle (LDP) for the occupation measure for large time (Donsker-Varadhan), which describes the exact rate of exponential convergence.