Some properties of invariant measures of non symmetric dissipative stochastic systems. (English) Zbl 1087.60049

Da Prato, Giuseppe; Debussche, Arnaud; Goldys, Beniamin

Probab. Theory Relat. Fields 123, No. 3, 355-380 (2002).

Summary: We consider transition semigroups generated by stochastic partial differential equations with dissipative nonlinear terms. We prove an integration by part formula and a Logarithmic Sobolev inequality for the invariant measure. No symmetry or reversibility assumptions are made. Furthermore we prove some compactness results on the transition semigroup and on the embedding of the Sobolev spaces based on the invariant measure. We use these results to derive asymptotic properties for a stochastic reaction-diffusion equation.

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
35K57	Reaction-diffusion equations
35R60	PDEs with randomness, stochastic partial differential equations
37L40	Invariant measures for infinite-dimensional dissipative dynamical systems