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

Abstract

Let X be a real separable Hilbert space. Let Q be a linear, bounded, positive and compact operator on X and let A:Dom(A)⊆X→X be a linear, self-adjoint operator generating a strongly continuous semigroup on X. Let F:X→X be a (smooth enough) function and let {W(t)}t≥0 be a X-valued cylindrical Wiener process. For any α≥0, we are interested in the mild solution X(t,x) of the semilinear stochastic partial differential equation{dX(t,x)=(AX(t,x)+F(X(t,x)))dt+QαdW(t),t>0;X(0,x)=x∈X, and in its associated transition semigroup(0.1)P(t)φ(x):=E[φ(X(t,x))],φ∈Bb(X),t≥0,x∈X; where Bb(X) denotes the space of the real-valued, bounded and Borel measurable functions on X. In this paper we study the behavior of the semigroup P(t) in the space L2(X,ν), where ν is the unique invariant probability measure of (0.1), when F is dissipative and has polynomial growth. Then we prove the logarithmic Sobolev and the Poincaré inequalities and we study the maximal Sobolev regularity for the stationary equationλu−N2u=f,λ>0,f∈L2(X,ν); where N2 is the infinitesimal generator of P(t) in L2(X,ν).