Hitting half-spaces by Bessel-Brownian diffusions

Details Hitting half-spaces by Bessel-Brownian diffusions Hitting half-spaces by Bessel-Brownian diffusions

2009-04-11

arxiv.org/abs/0904.1803v1

Potential Anal. vol. 33 no. 1 (2010), 47-83

Mathematics

Probability

Scientific paper

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.

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