Log-Harnack Inequality for Mild Solutions of SPDEs with Strongly Multiplicative Noise

TL;DR

This paper establishes a log-Harnack inequality for semi-linear SPDEs with strongly multiplicative, possibly unbounded noise coefficients, extending previous results to more general noise conditions and applying to stochastic reaction-diffusion equations.

Contribution

It introduces new techniques to derive gradient estimates and log-Harnack inequalities for SPDEs with unbounded multiplicative noise, broadening the scope of existing results.

Findings

Derived gradient estimates for SPDEs with unbounded noise

Established log-Harnack inequalities under new conditions

Applied results to stochastic reaction-diffusion equations

Abstract

Due to technical reasons, existing results concerning Harnack type inequalities for SPDEs with multiplicative noise apply only to the case where the coefficient in the noise term is an Hilbert-Schmidt perturbation of a fixed bounded operator. In this paper we investigate a class of semi-linear SPDEs with strongly multiplicative noise whose coefficient is even allowed to be unbounded which is thus no way to be Hilbert-Schmidt. Gradient estimates, log-Harnack inequality and applications are derived. Applications to stochastic reaction-diffusion equations driven by space-time white noise are presented.