一般马氏过程象集和图集的 Hausdorff 维数

Hausdorff dimension of range and graph for general Markov processes

  • 摘要: 在小时间内停留或离开小球的概率估计下, 我们建立了~\R^d~上一般马氏过程图集的~Hausdorff~维数. 特别地, 我们的结果表明在~\R^d~上对称扩散过程~(\alpha=2)~或对称~\alpha-stable~型过程~(\alpha\in(0,2))~中, 几乎处处有\dim_\mathcalH\mathrmGrX(0,1)= \Ii_\\alpha<1\+(2-1/\alpha)\Ii_\\alpha\ge1,d=1\+(d\wedge \alpha)\Ii_\\alpha\ge1,d\ge2\. 我们也系统地证明了马氏过程象集的~Hausdorff~维数.

     

    Abstract: We establish the Hausdorff dimension for the graph of general Markov processes on \R^d, under some probability estimates of the processes staying or leaving small balls in small time. In particular, our results indicate that, for symmetric diffusion processes (with \alpha=2) or symmetric \alpha-stable-like processes (with \alpha\in (0,2)) on \R^d, it holds almost surely that \dim_\mathcalH\mathrmGrX(0,1)= \Ii_\\alpha<1\+(2-1/\alpha)\Ii_\\alpha\ge1,d=1\+(d\wedge \alpha)\Ii_\\alpha\ge1,d\ge2\. We also systematically prove the corresponding results about the Hausdorff dimension for the range of the processes.

     

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