Hausdorff dimension of range and graph for general Markov processes

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.