Green function estimates for subordinate Brownian motions : stable and beyond

TL;DR

This paper establishes sharp Green function estimates and boundary Harnack inequalities for subordinate Brownian motions with specific Laplace exponents, extending results to geometric stable and relativistic processes.

Contribution

It provides explicit Green function estimates and boundary Harnack inequalities for a broad class of subordinate Brownian motions with mild conditions on the Laplace exponent.

Findings

Proved scale invariant boundary Harnack inequality for subordinate Brownian motions.

Derived explicit two-sided Green function estimates in bounded $C^{1,1}$ domains.

Extended results to geometric stable and relativistic stable processes.

Abstract

A subordinate Brownian motion $X$ is a L\'evy process which can be obtained by replacing the time of the Brownian motion by an independent subordinator. In this paper, when the Laplace exponent $\phi$ of the corresponding subordinator satisfies some mild conditions, we first prove the scale invariant boundary Harnack inequality for $X$ on arbitrary open sets. Then we give an explicit form of sharp two-sided estimates on the Green functions of these subordinate Brownian motions in any bounded $C^{1,1}$ open set. As a consequence, we prove the boundary Harnack inequality for $X$ on any $C^{1,1}$ open set with explicit decay rate. Unlike {KSV2, KSV4}, our results cover geometric stable processes and relativistic geometric stable process, i.e. the cases when the subordinator has the Laplace exponent $$\phi(\lambda)=\log(1+\lambda^{\alpha/2}) (0<\alpha\leq 2, d > \alpha)$$ and $$\phi(\lambda)=\log(1+(\lambda+m^{\alpha/2})^{2/\alpha}-m) (0<\alpha<2,\, m>0, d >2) .$$