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

TL;DR

This paper introduces a general stochastic calculus-based method to compute hitting distributions for Ornstein-Uhlenbeck type diffusions, providing explicit Poisson kernel representations for specific geometric domains.

Contribution

It develops a novel approach using skew product representation to derive hitting distributions, surpassing previous Feynman-Kac based methods.

Findings

Explicit Poisson kernel formulas for half-spaces and balls

Application to hyperbolic Brownian motion and Ornstein-Uhlenbeck processes

Enhanced computational technique for hitting distributions

Abstract

The purpose of the paper is to provide a general method for computing hitting distributions of some regular subsets D for Ornstein-Uhlenbeck type operators of the form 1/2\Delta + 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 the paper is based on stochastic calculus and on skew product representation of multidimensional Brownian motion and yields more complete results as those based on Feynmann-Kac technique.