Averaging principle for two-time-scale stochastic functional differential equations with past-dependent switching

This paper aimed to establish the averaging principle for the two-timescale stochastic functional differential equations with past-dependent switching. Initially, the existence and uniqueness of solutions, as well as moment estimates, were obtained by using classical interlacing techniques. Furthermore, the interaction between the fast and slow processes was derived based on the properties of the Poisson random measure. Subsequently, employing the coupling method and the integration by parts formula for the generator, the exponential ergodicity of the frozen Markov chain and the Lipschitz continuity of its invariant measures were proved. In addition to the challenges posed by the dependence on history, the Lipschitz condition under uniform norms that the generator satisfies also introduced computational and proof difficulties. Therefore, more refined estimates were provided for the segment process. Based on these results, together with weak convergence and martingale methods, the averaging principle for the original system was established. Finally, two examples were provided to illustrate the differences between the results presented here and the classical results.