Potential theory of one-dimensional geometric stable processes
  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. 
Tomasz Grzywny and Micha{\l} Ryznar
math.PR