On generators of transition semigroups associated to semilinear stochastic partial differential equations

Let X be a real separable Hilbert space. Let Q be a linear, bounded, positive and compact operator on X and let A : Dom(A) subset of X -> X be a linear, self-adjoint operator generating a strongly continuous semigroup on X. Let F : X -> X be a (smooth enough) function and let {W(t)}(t >= 0) be a X-valued cylindrical Wiener process. For any alpha >= 0, we are interested in the mild solution X(t, x) of the semilinear stochastic partial differential equation{dX(t,x) = (AX (t, x) + F(X (t, x)))dt + Q(alpha)dW (t), t > 0; X(0, x) = x is an element of X,and in its associated transition semigroupP(t)phi(x) := E[phi(X(t, x))], phi is an element of B-b(X), t >= 0, x is an element of X; (0.1)where B-b(X) denotes the space of the real-valued, bounded and Borel measurable functions on X. In this paper we study the behavior of the semigroup P(t) in the space L-2(X, nu), where nu is the unique invariant probability measure of (0.1), when F is dissipative and has polynomial growth. Then we prove the logarithmic Sobolev and the Poincare inequalities and we study the maximal Sobolev regularity for the stationary equationlambda u - N(2)u = f, lambda > 0, f is an element of L-2(X, nu);where N-2 is the infinitesimal generator of P(t) in L-2(X, nu).