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

TL;DR

This paper investigates semilinear stochastic evolution equations in Hilbert spaces, establishing existence, uniqueness, approximation methods, and limit theorems, with applications to stochastic partial differential equations.

Contribution

It extends previous work by incorporating probability distribution dependence in the nonlinear term and develops Trotter-Kato approximations for these equations.

Findings

Proved existence and uniqueness of mild solutions.

Established weak convergence of probability measures.

Demonstrated classical limit theorem for parameter dependence.

Abstract

Motivated by the work of T.E. Govindan in [5,8,9], this paper is concerned with a more general semilinear stochastic evolution equation. The difference between the equations considered in this paper and the previous one is that it makes some changes to the nonlinear function in random integral, which also depends on the probability distribution of stochastic process at that time. First, this paper considers the existence and uniqueness of mild solutions for such equations. Furthermore, Trotter-Kato approximation system is introduced for the mild solutions, and the weak convergence of induced probability measures and zeroth-order approximations are obtained. Then we consider the classical limit theorem about the parameter dependence of this kind of equations. Finally, an example of stochastic partial differential equation is given to illustrate our results.