L2 -theory for transition semigroups associated to dissipative systems

Let X be a real separable Hilbert space. Let C be a linear, bounded, non-negative self-adjoint operator on X and let A be the infinitesimal generator of a strongly continuous semigroup in X. Let { W(t) } t≥ be a X-valued cylindrical Wiener process on a filtered (normal) probability space (Ω,F,{Ft}t≥0,P). Let F: Dom (F) ⊆ X→ X be a smooth enough function. We are interested in the generalized mild solution { X(t, x) } t≥ of the semilinear stochastic partial differential equation {dX(t,x)=(AX(t,x)+F(X(t,x)))dt+CdW(t),t>0;X(0,x)=x∈X.We consider the transition semigroup defined by P(t)φ(x):=E[φ(X(t,x))],φ∈Bb(X),t≥0,x∈X.If O is an open set of X, we consider the Dirichlet semigroup defined by PO(t)φ(x):=E[φ(X(t,x))I{ω∈Ω:τx(ω)>t}],φ∈Bb(O),x∈O,t>0where τx is the exit time defined by τx=inf{s>0:X(s,x)∈Oc}.We study the infinitesimal generator of P(t), PO(t) in L2(X, ν) , L2(O, ν) respectively, where ν is the unique invariant measure of P(t).