Harnack inequalities with power p∈(1,+∞) for transition semigroups in Hilbert spaces

We consider the stochastic differential equation {dX(t)=[AX(t)+F(X(t))]dt+C1/2dW(t),t>0,X(0)=x∈X,where X is a separable Hilbert space, { W(t) } t≥ is a X-cylindrical Wiener process, A and C are suitable operators on X and F: Dom (F) ⊆ X→ X is a smooth enough function. We establish a Harnack inequality with power p∈ (1 , + ∞) for the transition semigroup { P(t) } t≥ associated with the stochastic problem above, under less restrictive conditions than those considered in the literature. Some applications to these inequalities are shown.