Bernoulli

  • Bernoulli
  • Volume 23, Number 1 (2017), 645-669.

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

Jianhai Bao, George Yin, and Chenggui Yuan

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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 Itô’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.

Article information

Source
Bernoulli, Volume 23, Number 1 (2017), 645-669.

Dates
Received: July 2013
Revised: March 2014
First available in Project Euclid: 27 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.bj/1475001370

Digital Object Identifier
doi:10.3150/14-BEJ677

Mathematical Reviews number (MathSciNet)
MR3556788

Zentralblatt MATH identifier
1360.60118

Keywords
$\alpha $-stable process averaging principle invariant measure stochastic partial differential equation strong convergence

Citation

Bao, Jianhai; Yin, George; Yuan, Chenggui. Two-time-scale stochastic partial differential equations driven by $\alpha $-stable noises: Averaging principles. Bernoulli 23 (2017), no. 1, 645--669. doi:10.3150/14-BEJ677. https://projecteuclid.org/euclid.bj/1475001370


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