Logarithmic Harnack inequalities for transition semigroups in Hilbert spaces

Abstract: 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.