Potential theory of one-dimensional geometric stable processes

Abstract

The purpose of this paper is to find optimal estimates for the Green function and the Poisson kernel for a half-line and intervals of the geometric stable process with parameter $\alpha \in (0,2]$. This process has an infinitesimal generator of the form $-\log (1+(-\Delta {)}^{\alpha /2})$. As an application we prove the scale invariant Harnack inequality as well as the boundary Harnack principle.

28 pages