Bessel Potentials, Hitting Distributions and Green Functions

2006/12/07 by T. Byczkowski, M. Ryznar, Michał Ryznar +5 · 1 citation
Mathematics · #60J65 #Advanced Mathematical Physics Problems #Differential Equations and Boundary Problems #FOS: Mathematics #Numerical methods in inverse problems #Probability (math.PR) #math.PR #msc:60J65

paper · pdf · doi:10.48550/arxiv.math/0612176

arxiv created 2006/12/07 · openalex publication_date 2006/12/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The purpose of this paper is to find explicit formulas for basic objects pertaining the local potential theory of the operator (I-Δ)^α/2, 0<α<2. The potential theory of this operator is based on Bessel potentials J_α=(I-Δ)^-α/2. We compute the \it harmonic measure of the half-space and write a concise form of the corresponding \it Green function for the operator (I-Δ)^α/2. To achieve this we analyze the so-called \it relativistic α-stable process on \R^d space, killed when exiting the half-space. In terms of this process we are dealing here with the 1-\it potential theory or, equivalently, potential theory of Schrödinger operator based on the generator of the process with Kato's potential q=-1.

Cited by

• Revisiting the blow-up criterion and the maximal existence time for solutions of the parabolic-elliptic Keller-Segel system in 2d-Euclidean space

Related

• Bessel potentials and Green functions on pseudo-Euclidean spaces
• On the eigenfunctions of Sturm--Liouville operators with potentials --- distributions
• Fractional powers of Bessel operator and its numerical calculation
• Threshold between short and long-range potentials for non-local Schrödinger operators
• Explicit formulas for the Bessel models: odd general spin groups