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 L\'evy 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 $C^{1,1}$ 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 $C^{1,1}$ open set $D$ and identify the Martin boundary of $D$ with respect to the subordinate Brownian motion with the Euclidean boundary.