Two-sided Green function estimates for killed subordinate Brownian motions

Abstract

A subordinate Brownian motion is a Lévy process which can be obtained by replacing the time of the Brownian motion by an independent subordinator. The infinitesimal generator of a subordinate Brownian motion is $-\varphi (-\Delta )$, where $\varphi$ is the Laplace exponent of the subordinator. In this paper, we consider a large class of subordinate Brownian motions without diffusion component and with $\varphi$ comparable to a regularly varying function at infinity. This class of processes includes symmetric stable processes, relativistic stable processes, sums of independent symmetric stable processes, sums of independent relativistic stable processes, and much more. We give sharp two-sided estimates on the Green functions of these subordinate Brownian motions in any bounded $\kappa$-fat open set $D$. When $D$ is a bounded $C^{1,1}$ open set, we establish an explicit form of the estimates in terms of the distance to the boundary. As a consequence of such sharp Green function estimates, we obtain a boundary Harnack principle in $C^{1,1}$ open sets with explicit rate of decay.

33 pages

References in corpus (3)

• Spectral analysis of subordinate Brownian motions in half-line
• Boundary Harnack inequalities for regional fractional Laplacian
• Potential Theory of Subordinate Brownian Motions Revisited

Cited by in corpus (22)

• Boundary Harnack inequality for Markov processes with jumps
• Dirichlet Heat Kernel Estimates for Rotationally Symmetric Lévy processes
• Suprema of Lévy processes
• Spectral analysis of subordinate Brownian motions in half-line
• Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes
• First passage times for subordinate Brownian motions
• One-dimensional quasi-relativistic particle in the box
• Barriers, exit time and survival probability for unimodal Lévy processes
• Global uniform boundary Harnack principle with explicit decay rate and its application
• Scale invariant boundary Harnack principle at infinity for Feller processes
• Estimates of Dirichlet heat kernels for unimodal Lévy processes with low intensity of small jumps
• Limit Theorems for Some Critical Superprocesses
• Minimal thinness for subordinate Brownian motion in half space
• Censored symmetric Lévy processes
• Minimal thinness with respect to symmetric Lévy processes
• Estimates of Dirichlet heat kernels for subordinate Brownian motions
• Limit theorems for a class of critical superprocesses with stable branching
• Sharp two-sided Green function estimates for Dirichlet forms degenerate at the boundary
• First exit times from a bounded interval for Lévy processes
• Potential theory of subordinate killed Brownian motion
• Central Limit Theorems for Supercritical Branching Markov Processes
• On the trace of unimodal Lévy processes on Lipschitz domains