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

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.