Potential theory of subordinate Brownian motions with Gaussian components

Abstract

In this paper we study a subordinate Brownian motion with a Gaussian component and a rather general discontinuous part. The assumption on the subordinator is that its Laplace exponent is a complete Bernstein function with a Lévy density satisfying a certain growth condition near zero. The main result is a boundary Harnack principle with explicit boundary decay rate for non-negative harmonic functions of the process in C1,1 open sets. As a consequence of the boundary Harnack principle, we establish sharp two-sided estimates on the Green function of the subordinate Brownian motion in any bounded C1,1 open set D and identify the Martin boundary of D with respect to the subordinate Brownian motion with the Euclidean boundary.