Invariant Densities for Stochastic Semilinear Evolution Equations and Related Properties of Transition Semigroups

A large class of stochastic semilinear equations (*) with measurable nonlinear term on a Hilbert space H is considered. Assuming the corresponding nonsymmetric Ornstein-Uhlenbeck process has an invariant measure μ, we study the transition semigroup (P_t ) for (*) in the L^p (H, μ) spaces. The main tools are Girsanov transform and Miyadera perturbations. Sufficient conditions are provided for hyperboundedness of P_t and Log Sobolev Inequality to hold and in the case of bounded nonlinear term the sufficient and necessary conditions are obtained. We prove the existence and uniqueness of invariant density for (P_t ) and we give suitable counterexamples. Related results are reviewed.