Transition semigroups for stochastic semilinear equations on Hilbert spaces.

*(English)*Zbl 0991.60049Transition semigroups of stochastic equations of the type \(dX_t=[AX_t+F(X_t)]dt+BdW_t\) on a Hilbert space \(H\) are studied. The author considers a large class of measurable nonlinearities \(F\) mapping \(H\) into \(H\) including all functions of linear growth. Assuming the existence of an invariant measure \(\mu\) of the corresponding non-symmetric Ornstein-Uhlenbeck process, the author proves the existence of a transition semigroup \((P_t)\) in \(L^p(H,\mu)\) for the equations. Sufficient conditions are provided for hyperboundedness of \(P_t\) and for the log-Sobolev inequality to hold. Moreover, sufficient and necessary conditions are derived in case of bounded nonlinearities. The hyperboundedness of \(P_t\) is important to verify the existence of invariant measures with density. A further result is the existence, uniqueness, and some regularity of the invariant density for \((P_t)\). The domain of the generator of \(P_t\) is also characterized. The author neither assumes strong Feller properties nor the existence of an associated Dirichlet form. The proofs rely mainly on Girsanov transform and Miyadera perturbations. The first one provides estimates for the norm, and the second one provides information about the domain of the generator.

Reviewer: Dirk Blömker (Aachen)

##### MSC:

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

60J35 | Transition functions, generators and resolvents |

47D06 | One-parameter semigroups and linear evolution equations |

47D07 | Markov semigroups and applications to diffusion processes |