Weak convergence and stability of functional diffusion systems with singularly perturbed regime switching

This paper focuses on a class of functional diffusion systems with singularly perturbed regime switching, where the modulating Markov chain has a large state space and undergoes weak and strong interactions. By using the martingale method and weak convergence, this paper shows that the underlying system will weakly converge to a limit system, which is simpler than the original system. For a class of integro-differential diffusion system with singularly perturbed regime switching, as a class of special functional diffusion system, this paper demonstrates that if the limit system is moment exponentially stable, the original system with singular perturbation is also moment exponentially stable under suitable conditions. This result is interesting since the limit system is always simpler. Finally, an example is given to illustrate this result.