Potential theory of subordinate Brownian motions with Gaussian components

Panki Kim; Renming Song; Zoran Vondracek

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arXiv · 1106.5858

Potential theory of subordinate Brownian motions with Gaussian components

Panki Kim·Renming Song·Zoran Vondracek

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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 $C^{1,1}$ 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.

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BibTeX RIS

Panki Kim, Renming Song, Zoran Vondracek. 2012-04-05. Potential theory of subordinate Brownian motions with Gaussian components. https://arxiv.org/abs/1106.5858

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