Hitting half-spaces by Bessel-Brownian diffusions

TL;DR

This paper derives explicit formulas for the joint distribution of hitting times and locations for half-spaces by a diffusion combining Bessel processes and Brownian motion, with applications to Poisson kernels and stable Lévy processes.

Contribution

It provides new explicit formulas for hitting distributions involving Bessel-Brownian diffusions, extending previous results and applying to stable Lévy processes.

Findings

Formulas for joint distributions of hitting times and places.

Explicit Poisson kernel formulas for certain operators.

Applications to stable Lévy process hitting distributions.

Abstract

The purpose of the paper is to find explicit formulas describing the joint distributions of the first hitting time and place for half-spaces of codimension one for a diffusion in $\R^{n+1}$, composed of one-dimensional Bessel process and independent n-dimensional Brownian motion. The most important argument is carried out for the two-dimensional situation. We show that this amounts to computation of distributions of various integral functionals with respect to a two-dimensional process with independent Bessel components. As a result, we provide a formula for the Poisson kernel of a half-space or of a strip for the operator $(I-\Delta)^{\alpha/2}$, $0<\alpha<2$. In the case of a half-space, this result was recently found, by different methods, in [6]. As an application of our method we also compute various formulas for first hitting places for the isotropic stable L\'evy process.