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

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Kyushu University 21st Century COE Program Development of Dynamic Mathematics with High Functionality九州大学21世紀COEプログラム「機能数理学の構築と展開」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 β-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.