Ergodicity for regime-switching neutral stochastic functional differential equations with infinite delay

TL;DR

This paper establishes exponential ergodicity for a class of regime-switching neutral stochastic functional differential equations with infinite delay, using coupling, Lyapunov functions, and M-matrix theory.

Contribution

It extends ergodicity results to RNSFDEs with infinite delay and countably infinite switching states, employing novel analytical techniques.

Findings

Proved well-posedness of NSFDEs without switching under dissipativity.

Derived exponential ergodicity in Wasserstein distance for finite state space.

Extended ergodicity results to infinite state space using finite partition and Lyapunov methods.

Abstract

This work focuses on a class of regime-switching neutral stochastic functional differential equations (RNSFDEs) with infinite delay, in which the switching component can possess finite or countably infinite many states. To ensure the well-posedness of the underlying process, we first investigate the well-posedness for NSFDEs without Markovian switching under dissipativity conditions, and obtain the desired result by Skorohod's representation. By utilizing the moment estimate of exponential functionals of the switching component, we derive the exponential ergodicity in Wasserstein distance for RNSFDEs with a finite state space using the coupling method. To address the difficulty posed by the infinite state space, we obtain the same exponential ergodicity by applying the finite partition method along with Lyapunov functions and M-matrix theory.