Logarithmic Harnack inequalities for transition semigroups in Hilbert spaces

Abstract

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