Harnack inequalities with power p is an element of (1,+infinity) for transition semigroups in Hilbert spaces

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