Strong averaging principle for nonautonomous slow-fast SPDEs driven by $\alpha$-stable processes

TL;DR

This paper establishes an averaging principle for nonautonomous slow-fast SPDEs driven by $oldsymbol{ extit{ extalpha}}$-stable processes, demonstrating strong convergence of the slow component to an averaged equation despite the lack of finite second moments.

Contribution

It introduces a novel averaging principle for nonautonomous SPDEs driven by $ extit{ extalpha}$-stable processes, handling the technical challenges posed by infinite variance.

Findings

Proves strong convergence in $L^p$ sense for $p extless extit{ extalpha}$

Shows convergence to an averaged equation under periodic or asymptotic conditions

Provides a concrete example illustrating the applicability of the theoretical results

Abstract

This paper considers a class of nonautonomous slow-fast stochastic partial differential equations driven by $\alpha$-stable processes for $\alpha\in (1,2)$. By introducing the evolution system of measures, we establish an averaging principle for this stochastic system. Specifically, we first prove the strong convergence (in the $L^p$ sense for $p\in (1,\alpha)$) of the slow component to the solution of a simplified averaged equation with coefficients depend on the scaling parameter. Furthermore, under conditions that coefficients are time-periodic or satisfy certain asymptotic convergence, we prove that the slow component converges strongly to the solution of an averaged equation, whose coefficients are independent of the scaling parameter. Finally, a concrete example is provided to illustrate the applicability of our assumptions. Notably, the absence of finite second moments in the solution caused by the $\alpha$-stable processes requires new technical treatments, thereby solving a problem mentioned in [1,Remark 3.3].