# Averaging principle for stochastic differential equations in the random periodic regime

## TL;DR

This paper establishes a stochastic averaging principle for non-autonomous SDEs with fast motions exhibiting random periodic solutions, addressing challenges in multi-scale stochastic systems with time-dependent dynamics.

## Contribution

It introduces a novel approach combining Lyapunov methods, coupling, and Feller properties to prove ergodicity and averaging limits in time-periodic stochastic systems.

## Key findings

- Proves ergodicity of time periodic measures for fast motions.
- Identifies the averaging limit in non-autonomous stochastic systems.
- Extends averaging principles to systems with random periodic solutions.

## Abstract

We present the validity of stochastic averaging principle for non-autonomous slow-fast stochastic differential equations (SDEs) whose fast motions admit random periodic solutions. Our investigation is motivated by some problems arising from multi-scale stochastic dynamical systems, where configurations are time dependent due to nonlinearity of the underlying vector fields and the onset of time dependent random invariant sets. Averaging principle with respect to uniform ergodicity of the fast motion is no longer available in this scenario. Lyapunov second method together with synchronous coupling and strong Feller property of Markovian flows of SDEs are used to prove the ergodicity of time periodic measures of the fast motion on certain minimal Poincare section and consequently identify the averaging limit.