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Consider the stochastic evolution equation in a separable Hilbert space witha nice multiplicative noise and a locally Dini continuous drift. We prove thatfor any initial data the equation has a unique (possibly explosive) mildsolution. Under a reasonable condition ensuring the non-explosion of thesolution, the strong Feller property of the associated Markov semigroup isproved. Gradient estimates and log-Harnack inequalities are derived for theassociated semigroup under certain global conditions, which are new even infinite-dimensions.

Gradient Estimates and Applications for SDEs in Hilbert Space with Multiplicative Noise and Dini Continuous Drift