Hitting half-spaces or spheres by Ornstein–Uhlenbeck type diffusions

Tomasz Byczkowski ¹   ; Jakub Chorowski ¹   ; Piotr Graczyk ²   ; Jacek Małecki ¹

¹ Institute of Mathematics and Computer Sciences Wrocław University of Technology 50-370 Wrocław, Poland
² Département de Mathématiques Université d'Angers 49045 Angers Cedex 01, France

Colloquium Mathematicum, Tome 129 (2012) no. 2, pp. 145-171

Résumé

The purpose of the paper is to provide a general method for computing the hitting distributions of some regular subsets $D$ for Ornstein–Uhlenbeck type operators of the form ${1\over 2}\varDelta + F\cdot \nabla $, with $F$ bounded and orthogonal to the boundary of $D$. As an important application we obtain integral representations of the Poisson kernel for a half-space and balls for hyperbolic Brownian motion and for the classical Ornstein–Uhlenbeck process. The method developed in this paper is based on stochastic calculus and on the skew product representation of multidimensional Brownian motion and yields more complete results than those based on the Feynman–Kac technique.

Détail

DOI : 10.4064/cm129-2-1

Keywords: purpose paper provide general method computing hitting distributions regular subsets ornstein uhlenbeck type operators form vardelta cdot nabla bounded orthogonal boundary important application obtain integral representations poisson kernel half space balls hyperbolic brownian motion classical ornstein uhlenbeck process method developed paper based stochastic calculus skew product representation multidimensional brownian motion yields complete results those based feynman kac technique