Potential Theory of Geometric Stable Processes

Abstract

In this paper we study the potential theory of symmetric geometric stable processes by realizing them as subordinate Brownian motions with geometric stable subordinators. More precisely, we establish the asymptotic behaviors of the Green function and the Levy density of symmetric geometric stable processes. The asymptotics of these functions near zero exhibit features that are very different from the ones for stable processes. The Green function behaves near zero as 1/(|x| d log 2|x|), while the Levy density behaves like 1/|x| d . We also study the asymptotic behaviors of the Green function and Levy density of subordinate Brownian motions with iterated geometric stable subordinators. As an application, we establish estimates on the capacity of small balls for these processes, as well as mean exit time estimates from small balls and a Harnack inequality for these processes.