Averaging principles for time-inhomogeneous multi-scale SDEs via nonautonomous Poisson equations

TL;DR

This paper establishes averaging principles and convergence rates for time-inhomogeneous multi-scale stochastic differential equations, analyzing their asymptotic behavior through nonautonomous Poisson equations and measure evolution systems.

Contribution

It introduces new averaging principles for time-inhomogeneous multi-scale SDEs, including explicit convergence rates and handling of periodic or asymptotic coefficients.

Findings

Strong and weak convergence of the slow component to averaged equations

Explicit convergence rates for the averaging principles

Effectiveness demonstrated through two illustrative examples

Abstract

The purpose of this paper is to establish asymptotic behaviors of time-inhomogeneous multi-scale stochastic differential equations (SDEs). To achieve them, we analyze the evolution system of measures for time-inhomogeneous Markov semigroups, and investigate regular properties of nonautonomous Poisson equations. The strong and the weak averaging principle for time-inhomogeneous multi-scale SDEs, as well as explicit convergence rates, are provided. Specifically, we show the slow component in the multi-scale stochastic system converges strongly or weakly to the solution of an averaged equation, whose coefficients retain the dependence of the scaling parameter. When the coefficients of the fast component exhibit additional asymptotic or time-periodic behaviors, we prove the slow component converges strongly or weakly to the solution of an averaged equation, whose coefficients are independent of the scaling parameter. Finally, two examples are given to indicate the effectiveness of all the averaged equations mentioned above.