Logarithmic Harnack inequalities for transition semigroups in Hilbert spaces

We consider the stochastic differential equation $$\{\begin{array}{lc}dX(t)=[AX(t)+F(X(t))]dt+C^{1\mathrm{/}2}dW(t),& t>0;\\ X(0)=x\in \mathcal{X};\end{array}$$ where $\mathcal{X}$ is a Hilbert space, ${W(t)}_{t\geq 0}$isa$$\mathcal{X}$$-valuedcylindricalWienerprocess,$A, C$ 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 $\{P(t){\}}_{t\ge 0}$ associated with the stochastic problem above, under less restrictive conditions than those considered in the literature. Some applications to these inequalities are also shown.