Gradient Estimates and Applications for SDEs in Hilbert Space with Multiplicative Noise and Log-Hölder Drif

Feng-Yu Wang

Published 2014-04-11, updated 2014-10-11Version 3

Consider the stochastic evolution equation in a separable Hilbert space $\H$ with a nice multiplicative noise and a locally log-H\"older continuous drift $b: [0,\infty)\times \H\to \H,$ i.e. for any $n\ge 1$ there exists a constant $K_n>0$ such that $$|b_t(x)-b_t(y)|^2 \le \ff{K_n}{\log (\e + |x-y|^{-1})},\ \ \ t\in [0,n], x,y\in \H, |x|\lor |y|\le n.$$ We prove that for any initial data the equation has a unique (possibly explosive) mild solution. Under a reasonable condition ensuring the non-explosion of the solution, the strong Feller property of the associated Markov semigroup is proved. Gradient estimates and log-Harnack inequalities are derived for the associated semigroup under certain global conditions, which are new even in finite-dimensions.