Ergodicity and strong limit results for two-time-scale functional stochastic differential equations

This article focuses on a class of two-time-scale functional stochastic differential equations, where the phase space of the segment processes is infinite-dimensional. The systems under consideration have a fast-varying component and a slowly varying one. First, the ergodicity of the fast-varying component is obtained. Then inspired by the Khasminskii’s approach, an averaging principle, in the sense of convergence in the pth moment uniformly in time within a finite time interval, is developed.