Orders of strong and weak averaging principle for multiscale SPDEs driven by $\alpha$-stable process

TL;DR

This paper investigates the averaging principle for multiscale SPDEs driven by $oldsymbol{ ext{alpha}}$-stable processes, establishing convergence orders and extending previous results from Wiener noise to $oldsymbol{ ext{alpha}}$-stable noise in infinite dimensions.

Contribution

It extends the averaging principle and convergence order results from Wiener noise to $oldsymbol{ ext{alpha}}$-stable processes in infinite-dimensional SPDEs.

Findings

Strong convergence order is $1 - 1/\alpha$.

Weak convergence order is $1 - r$ for any $r \in (0,1)$.

Extends previous finite-dimensional results to infinite-dimensional SPDEs.

Abstract

In this paper, the averaging principle is studied for a class of multiscale stochastic partial differential equations driven by $\alpha$-stable process, where $\alpha\in(1,2)$. Using the technique of Poisson equation, the orders of strong and weak convergence are given $1-1/\alpha$ and $1-r$ for any $r\in (0,1)$ respectively. The main results extend Wiener noise considered by Br\'{e}hier in [6] and Ge et al. in [17] to $\alpha$-stable process, and the finite dimensional case considered by Sun et al. in [39] to the infinite dimensional case.