Two-time-scale stochastic functional differential equations: Inclusion of infinite delay and coupled segment processes

Abstract

This paper focuses on two-time-scale stochastic functional differential equations (SFDEs). It features in inclusion of infinite delay and coupling of slow and fast components. The coupling is through the segment processes of the slow and fast processes. The main difficulties include infinite delay and the coupling of segment processes involving fast and slow motions. Concentrating on weak convergence, the tightness of the segment process is established on a space of continuous functions. In addition, the Hölder continuity and boundedness for the segment process of the slow component, uniform boundedness for the segment process of a fixed-x SFDE, exponential ergodicity, and continuous dependence on parameters are obtained to carry out the desired asymptotic analysis, and also as byproducts, which are interesting in their own right. Then using the martingale problem formulation, an average principle is established by a direct averaging, which involves detailed computations and subtle estimates. Finally, two classes of special SFDEs, stochastic integro-differential equations and stochastic delay differential equations with two-time scales are investigated.