Trotter–Kato approximations of semilinear stochastic evolution equations in Hilbert spaces

Abstract

This article deeply studies a class of semilinear stochastic evolution equations in real separable Hilbert spaces. The main goal here is to consider Trotter–Kato approximations for mild solutions of such equations. After proving the existence and uniqueness of mild solutions, the Trotter–Kato approximation system is introduced and studied in detail. Based on this, the weak convergence of probability measures induced by mild solutions of Trotter–Kato approximation equations and the approximation error estimator are established. Finally, as an application, the classical limit theorem about the parameter dependence of this kind of equations is exhibited. To deal with the general additive diffusion term, which not only depends on the state but also depends on the probability distribution of the solution process at that time, we adopt some specific techniques of measure theory and random functional analysis.