Hausdorff Dimension of Range and Graph for General Markov Processes

We establish the Hausdorff dimension of the graph of general Markov processes on \mathbbR^d based on 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 \mathbbR^d, it holds almost surely that \textdim_\mathcalH\mathrmGrX(0, 1)= \mathbb1_\\alpha<1\+(2-1/\alpha)\mathbb1_\\alpha\ge1, d=1\+(d\wedge \alpha)\mathbb1_\\alpha\ge1, d\ge2\. We also systematically prove the corresponding results about the Hausdorff dimension of the range of the processes.