Large Asymptotic Behaviour of Kolmogorov Equations in Hilbert Spaces

Let us consider a dynamical system in a separable Hilbert space H (norm |⋅|, inner product 〈 ⋅ , ⋅ 〉 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744376/a3a6af35-efd2-43dc-a017-2efeb7a27be6/content/eq996.tif"/> ) governed by a differential stochastic equation perturbed by a cylindrical Wiener process W in H, (1.1) d X ( t )     = ( A X ( t ) + F ( X ( t ) ) ) d t + d W ( t ) ,         t     ≥     0 ,       X ( 0 )     = x ∈ H .                                                                                                                                             } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744376/a3a6af35-efd2-43dc-a017-2efeb7a27be6/content/eq997.tif"/> where A : D(A) ⊂ H → H is a linear operator infinitesimal generator of a strongly continuous semigroup e^tA in H, and F : H → H is a nonlinear mapping.