Averaging principles for functional stochastic partial differential equations driven by a fractional Brownian motion modulated by two-time-scale Markovian switching processes

Motivated by applications of hybrid systems, this work considers functional stochastic partial differential equations (FSPDEs) driven by a fractional Brownian motion (fBm) modulated by a two-time-scale Markov chain with a finite state space. Our aim is to obtain an averaging principle for such systems with fast–slow Markov switching processes. Under suitable conditions, it is proved that there is a limit process in which the fast changing “noise” is averaged out and the limit is an average with respect to the stationary measure of the fast-varying processes. The limit process, being substantially simpler than that of the original system, can be used to reduce the computational complexity. There are several difficulties in our problems. First, because of the use of fBm, the techniques of martingale problem formulation can no longer be used. Second, there is no strong solution available and the underlying FSPDEs admit only a unique mild solution. Moreover, although the regime-switching enlarges the applicability of the underlying systems, to treat such systems is more difficult. To overcome the difficulties, fixed point theorem together with the use of stopping time argument, and a semigroup approach are used.