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

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