Averaging principles for SPDEs driven by fractional Brownian motions with random delays modulated by two-time-scale Markov switching processes

This work considers stochastic partial differential equations (SPDEs) driven by fractional Brownian motions (fBm) with random delays modulated by two-time scale Markov switching processes leading to a two-time scale formulation. The two-time scale Markov chains have a fast-varying component and a slowly evolving component. Our aim is to obtain an averaging principle for such systems. Under suitable conditions, it is proved that there is a limit process in which the fast changing “noise” is averaged out. The slow component has a limit that is an average with respect to the stationary distribution of the fast component. The limit process is substantially simpler than the original system leading to reduction of the computational complexity.