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

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.