Averaging principle and normal deviation for multi-scale SDEs with polynomial nonlinearity

TL;DR

This paper studies multi-scale stochastic differential equations with polynomial nonlinearities, proving averaging principles, normal deviations, and the existence of quasi-periodic solutions as the scale parameter approaches zero.

Contribution

It establishes strong convergence, fluctuation limits, attractor convergence, and the second Bogolyubov theorem for multi-scale SDEs with polynomial nonlinearities, extending existing theory.

Findings

Strong convergence of solutions to averaged equations

Normal deviations converge to Ornstein-Uhlenbeck processes

Attractors of original and averaged systems converge

Abstract

We investigate three types of averaging principles and the normal deviation for multi-scale stochastic differential equations (in short, SDEs) with polynomial nonlinearity. More specifically, we first demonstrate the strong convergence of the solution of SDEs, which involves highly oscillating components and fast processes, to that of the averaged equation. Then we investigate the small fluctuations of the system around its average, and show that the normalized difference weakly converges to an Ornstein-Uhlenbeck type process, which can be viewed as a functional central limit theorem. Additionally, we show that the attractor of the original system tends to that of the averaged equation in probability measure space as the time scale $\varepsilon$ goes to zero. Finally, we establish the second Bogolyubov theorem; that is to say, we prove that there exists a quasi-periodic solution in a neighborhood of the stationary solution of the averaged equation when the $\varepsilon$ is small.