Kesten (1969) gives necessary and sufficient conditions for single points to be polar for Levy processes. In this article we investigate when single points are polar for processes obtained by adding functions to stable processes.
T. S. Mountford. "Time Inhomogeneous Markov Processes and the Polarity of Single Points." Ann. Probab. 17 (2) 573 - 585, April, 1989. https://doi.org/10.1214/aop/1176991416