Książka - szczegóły

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 prove in the $L^{p}(H,μ)$ spaces the existence of a transition semigroup $(P_{t})$ for the equations. Sufficient conditions are provided for hyperboundedness of $P_{t}$ and for the Log Sobolev Inequality to hold; and in the case of a bounded nonlinear term, sufficient and necessary conditions are obtained. We prove the existence, uniqueness and some regularity of an invariant density for $(P_{t})$. A characterization of the domain of the generator is also given. The main tools are the Girsanov transform and Miyadera perturbations.