We consider the stochastic differential equation $$ \left\{ \begin{ array}{ lc} dX(t)=[AX(t)+F(X(t))]dt+C^{ 1/2}dW(t), & t>0;\\ X(0)=x \in \mathcal{ X}; \end{ array}\right. $$ where $\mathcal{ X}$ is a Hilbert space, $\{ W(t)\}_{ t\geq 0}$ is a $\mathcal{ X}$-valued cylindrical Wiener process, $A, C$ are suitable operators on $\mathcal{ X}$ and $F:{ \rm Dom}\,(F)\subseteq \mathcal{ X}\to \mathcal{ X}$ is a smooth enough function. We establish a logarithmic Harnack inequality for the transition semigroup $\{ P(t)\}_{ t\geq 0}$ associated with the stochastic problem above, under less restrictive conditions than those considered in the literature. Some applications to these inequalities are also shown.
Categories: math.PR math.AP