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

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