Weak averaging principle for multiscale stochastic dynamical systems driven by stable processes

TL;DR

This paper establishes a weak averaging principle for multiscale stochastic systems driven by independent stable Lévy noises, showing convergence of slow components to a Lévy process under certain conditions.

Contribution

It introduces a novel averaging principle for systems with different stable noise indices, expanding understanding of multiscale stochastic dynamics driven by stable processes.

Findings

Slow components converge to a Lévy process as scale parameter approaches zero.

The relation between the homogenizing index and the stable index of fast noise is established.

A new approach using nonlocal Poisson equations and correctors is developed.

Abstract

We study the averaging principle for a family of multiscale stochastic dynamical systems. The fast and slow components of the systems are driven by two independent stable L\'evy noises, whose stable indexes may be different. The homogenizing index $r_0$ of slow components has a relation with the stable index $\alpha_1$ of the noise of fast components given by $0<r_0<2-2/{\alpha_1}$. By first studying a nonlocal Poisson equation and then constructing suitable correctors, we obtain that the slow components weakly converge to a L\'evy process as the scale parameter goes to zero.