Potential theory of subordinate Brownian motions with Gaussian components

TL;DR

This paper investigates the potential theory of subordinate Brownian motions with Gaussian components, establishing boundary Harnack principles, Green function estimates, and Martin boundary identification in smooth domains.

Contribution

It provides new boundary Harnack principles and sharp Green function estimates for subordinate Brownian motions with Gaussian components in $C^{1,1}$ domains.

Findings

Boundary Harnack principle with explicit decay rate

Sharp two-sided Green function estimates

Identification of Martin boundary with Euclidean boundary

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.