Boundary Harnack principle for the absolute value of a one-dimensional subordinate Brownian motion killed at $0$

TL;DR

This paper establishes the boundary Harnack principle and related properties for the absolute value of a one-dimensional subordinate Brownian motion killed at zero, under certain scaling conditions.

Contribution

It proves the boundary Harnack principle and Green function estimates for this process, extending classical results to subordinate Brownian motions with specific scaling behaviors.

Findings

Boundary Harnack principle established for the process.

Green function estimates comparable to killed subordinate Brownian motion.

Properties of the harmonic compensated resolvent kernel analyzed.

Abstract

We prove the Harnack inequality and boundary Harnack principle for the absolute value of a one-dimensional recurrent subordinate Brownian motion killed upon hitting $0$, when $0$ is regular for itself and the Laplace exponent of the subordinator satisfies certain global scaling conditions. Using the conditional gauge theorem for symmetric Hunt processes we prove that the Green function of this process killed outside of some interval $(a,b)$ is comparable to the Green function of the corresponding killed subordinate Brownian motion. We also consider several properties of the compensated resolvent kernel $h$, which is harmonic for our process on $(0,1)$.