Log-Harnack inequality for mild solutions of SPDEs with 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 a Hilbert-Schmidt perturbation of a constant bounded operator. In this paper we obtained gradient estimates, log-Harnack inequality for mild solutions of general SPDEs with multiplicative noise whose coefficient is even allowed to be unbounded which cannot be Hilbert-Schmidt. Applications to stochastic reaction-diffusion equations driven by space-time white noise are presented.