Perturbation of strong Feller semigroups and well-posedness of semilinear stochastic equations on Banach spaces

TL;DR

This paper establishes a perturbation theorem for strong Feller semigroups and applies it to prove well-posedness and invariant measure properties of semilinear stochastic equations on Banach spaces.

Contribution

It introduces a Miyadera-Voigt type perturbation theorem for strong Feller semigroups and uses it to analyze well-posedness of semilinear stochastic equations.

Findings

Proved a Miyadera-Voigt type perturbation theorem for strong Feller semigroups.

Established well-posedness of semilinear stochastic equations on Banach spaces.

Analyzed existence and uniqueness of invariant measures for the transition semigroup.

Abstract

We prove a Miyadera-Voigt type perturbation theorem for strong Feller semigroups. Using this result, we prove well-posedness of the semilinear stochastic equation dX(t) = [AX(t) + F(X(t))]dt + GdW_H(t) on a separable Banach space E, assuming that F is bounded and measurable and that the associated linear equation, i.e. the equation with F = 0, is well-posed and its transition semigroup is strongly Feller and satisfies an appropriate gradient estimate. We also study existence and uniqueness of invariant measures for the associated transition semigroup.