Ergodicity and exponential [beta]-mixing bounds for multidimensional diffusions with jumps

Let X be a multidimensional diffusion with jumps. We provide sets of conditions under which: X fulfils the ergodic theorem for any initial distribution; and X is exponentially [beta]-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) for a suitable unbounded function g. 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.