Moment bounds and geometric ergodicity of diffusions with random switching and unbounded transition rates

A diffusion with random switching is a Markov process that consists of a stochastic differential equation part X_t and a continuous Markov jump process part Y_t. Such systems have a wide range of applications, where the transition rates of Y_t may not be bounded or Lipschitz. A new analytical framework is developed to understand the stability and ergodicity of these processes and allows for genuinely unbounded transition rates. Assuming the averaged dynamics is dissipative, the first part of this paper explicitly demonstrates how to construct a polynomial Lyapunov function and furthermore moment bounds. When the transition rates have multiple scales, this construction comes interestingly as a dual process of the averaging of fast transitions. The coefficients of the Lyapunov function can be seen as the potential dissipation of each regime in different scales, and a comparison principle comes naturally under this interpretation. On the basis of these results, the second part of this paper establishes geometric ergodicity for the joint processes. This can be achieved in two scenarios. If there is a commonly accessible regime that satisfies the minorization condition, the geometric convergence to the ergodic measure takes place in the total variation distance. If there is contraction on average, the geometric convergence takes place in a proper Wasserstein distance and is proved through an application of the asymptotic coupling framework.