Exponential ergodicity for stochastic functional differential equations with Markovian switching

This paper investigates exponential ergodicity for several kinds of stochastic functional differential equations (SFDEs) with Markovian switching. Firstly, we derive exponential ergodicity for SFDEs with Markovian switching using the Krylov-Bogoliubov theorem in the space C equipped with the uniform topology. Then, we obtain exponential ergodicity for neutral SFDEs with Markovian switching by making use of M-matrix theory and a generalized Razumikhin-type argument. Finally, we focus on SFDEs driven by Lévy processes with Markovian switching and prove exponential ergodicity through the Kurtz criterion of tightness for the space D endowed with the Skorohod topology. A concrete example is also given to illustrate our main results.