Geometric Ergodicity for diffusions with random regime switching and unbounded rates

Diffusions with random switching are stochastic processes that consist of a diffusion process Xt and a continuous Markov jump process Yt. Both components interact with each other as Yt controls the dynamical regimes of Xt and Xt effects the transition rate of Yt. This type of systems includes a wide range of models used in various fields, and many applications require the transition rates of Yt to be genuinely unbounded due to energy principles. This paper produces concrete criterions for diffusions with random switching to be stable and geometrically ergodic.