Large Deviation Principle for Occupation Measures of Stochastic Generalized Burgers–Huxley Equation

Abstract

The present work deals with the global solvability as well as asymptotic analysis of the stochastic generalized Burgers–Huxley (SGBH) equation perturbed by a white-in-time and correlated-in-space noise defined in a bounded interval of $${\mathbb {R}}$$ . We first prove the existence of a unique mild as well as strong solution to the 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 the SGBH equation, that is, irreducibility and the strong Feller property. These two properties guarantee 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 the SGBH equation by providing a large deviation principle for the occupation measure for large time (Donsker–Varadhan), which describes the exact rate of exponential convergence.