Ergodicity and exponential ββ-mixing bounds for multidimensional diffusions with jumps

Let XX be a multidimensional diffusion with jumps. We provide sets of conditions under which: XX fulfils the ergodic theorem for any initial distribution; and XX is exponentially ββ-mixing. Utilizing the Foster–Lyapunov drift criteria developed by Meyn and Tweedie, we extend several existing results concerning diffusions. We also obtain the boundedness of moments of g(Xt)g(Xt) for a suitable unbounded function gg. Our results can cover a wide variety of diffusions with jumps by selecting suitable test functions, and serve as fundamental tools for statistical analyses concerning the processes.