Two-time-scale stochastic partial differential equations driven by $\alpha$-stable noises: Averaging principles

TL;DR

This paper develops averaging principles for two-time-scale stochastic partial differential equations driven by non-square integrable -stable noises, including Markov switching and jump processes, using a semigroup approach for stronger convergence results.

Contribution

It introduces averaging principles for SPDEs driven by -stable noises with regime switching and jumps, expanding applicability and overcoming non-square integrability challenges.

Findings

Established pth moment convergence for systems with -stable noise.

Extended averaging principles to systems with Markov switching and jump processes.

Used semigroup approach for stronger convergence results.

Abstract

This paper focuses on stochastic partial differential equations (SPDEs) under two-time-scale formulation. Distinct from the work in the existing literature, the systems are driven by $\alpha$-stable processes with $\alpha \in(1,2)$. In addition, the SPDEs are either modulated by a continuous-time Markov chain with a finite state space or have an addition fast jump component. The inclusion of the Markov chain is for the needs of treating random environment, whereas the addition of the fast jump process enables the consideration of discontinuity in the sample paths of the fast processes. Assuming either a fast changing Markov switching or an additional fast-varying jump process, this work aims to obtain the averaging principles for such systems. There are several distinct difficulties. First, the noise is not square integrable. Second, in our setup, for the underlying SPDE, there is only a unique mild solution and as a result, there is only mild It\^{o}'s formula that can be used. Moreover, another new aspect is the addition of the fast regime switching and the addition of the fast varying jump processes in the formulation, which enlarges the applicability of the underlying systems. To overcome these difficulties, a semigroup approach is taken. Under suitable conditions, it is proved that the $p$th moment convergence takes place with $p\in(1,\alpha )$, which is stronger than the usual weak convergence approaches.