Averaging principles for two-time-scale neutral stochastic delay partial differential equations driven by fractional Brownian motions

AbstractWe prove the validity of averaging principles for two-time-scale neutral stochastic delay partial differential equations (NSDPDEs) driven by fractional Brownian motions (fBms) under two-time-scale formulation. Firstly, in the sense of mean-square convergence, we obtain not only the averaging principles for NSDPDEs involving two-time-scale Markov switching with a single weakly recurrent class but also for the case of two-time-scale Markov switching with multiple weakly irreducible classes. Secondly, averaging principles for NSDPDEs driven by fBms with random delay modulated by two-time-scale Markovian switching are established. We proved that there is a limit process in which the fast changing noise is averaged out. The limit process is substantially simpler than that of the original full fast–slow system.Keywords: Averaging principlesneutral stochastic delay partial differential equationsrandom delayfractional Brownian motionstwo-time-scale Markov switching2010 Mathematics Subject Classifications: Primary: 60G22Secondary: 60H15 Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingPei's work was partially supported by National Natural Science Foundation of China (NSFC) [grant number 12172285], NSFC-Chongqing [grant number cstc2021jcyj-msxmX0296], Shaanxi Fundamental Science Research Project for Mathematics and Physics [grant number 22JSQ027], Fundamental Research Funds for the Central Universities, Young Talent Fund of the University Association for Science and Technology in Shaanxi, China. Xu's work was partially supported by NSFC [grant number 12072264], and NSFC Key International (Regional) Joint Research Program [grant number 12120101002].