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.
Wang, F. Y., Wu, J. L., & Xu, L. (2011). Log-Harnack inequality for stochastic Burgers equations and applications. Journal of Mathematical Analysis and Applications, 384(1), 151-159. https://doi.org/10.1016/j.jmaa.2011.02.032