On weak convergence for stochastic evolutionary systems in average principle

TL;DR

This paper proves the weak convergence of stochastic evolutionary systems to their average systems using a novel approach based on singular perturbation, extending existing methods for semimartingales.

Contribution

It introduces a new method employing singular perturbation solutions instead of ergodic theorems for proving weak convergence.

Findings

Weak convergence established for stochastic evolutionary systems.

Method applicable to systems modeled by semimartingales.

Provides a new approach for analyzing stochastic systems in average principle.

Abstract

Weak convergence of the stochastic evolutionary system to the average evolutionary system is proved. The method proposed by R.Liptser in for semimartingales is used. But we apply a solution of singular perturbation problem instead of ergodic theorem.