Slow manifold and averaging for slow-fast stochastic differential system

TL;DR

This paper develops intermediate reduced models for multiscale stochastic systems, combining stochastic slow manifold reduction and averaging to capture system behavior with controlled approximation errors.

Contribution

It introduces an intermediate autonomous stochastic model that simplifies analysis while accurately approximating the original slow-fast stochastic system.

Findings

Reduced stochastic system on slow manifold with () error

Averaged deterministic system with () error

Intermediate stochastic model with () error that balances complexity and accuracy

Abstract

We consider multiscale stochastic dynamical systems. In this article an \emph{intermediate} reduced model is obtained for a slow-fast system with fast mode driven by white noise. First, the reduced stochastic system on exponentially attracting slow manifold reduced system is derived to errors of $\mathcal{O}(\epsilon)$. Second, averaging derives an autonomous deterministic system up to errors of $\mathcal{O}(\sqrt{\epsilon})$. Then an intermediate reduced model, which is an autonomous deterministic system driven by white noise up to errors of $\mathcal{O}(\epsilon)$, is derived using a martingale approach to account for fluctuations about the averaged system. This intermediate reduced model has a simpler form than the reduced model on the stochastic slow manifold.