Hausdorff dimensions of inverse images and collision time sets for symmetric Markov processes

In this paper, we establish the Hausdorff dimensions of inverse images and collision time sets for a large class of symmetric Markov processes on metric measure spaces. We apply the approach in the works by Hawkes and Jain–Pruitt, and make full use of heat kernel estimates. In particular, the results efficiently apply to symmetric diffusion processes, symmetric stable-like processes, and symmetric diffusion processes with jumps in $d$-sets.