Log-Harnack Inequality for Stochastic Burgers Equations and Applications

TL;DR

This paper establishes a log-Harnack inequality for stochastic Burgers equations using gradient estimates, leading to results like strong Feller property, irreducibility, and entropy bounds for the associated semigroup.

Contribution

It introduces a novel approach to prove the log-Harnack inequality for stochastic Burgers equations via Galerkin approximations and gradient estimates.

Findings

Proved the log-Harnack inequality for the semigroup of stochastic Burgers equations.

Established the strong Feller property and irreducibility of the solution.

Derived entropy-cost inequalities and transition density bounds.

Abstract

By proving an $L^2$-gradient estimate for the corresponding Galerkin approximations, the log-Harnack inequality is established for the semigroup associated to a class of stochastic Burgers equations. As applications, we derive the strong Feller property of the semigroup, the irreducibility of the solution, the entropy-cost inequality for the adjoint semigroup, and entropy upper bounds of the transition density.