Reflecting diffusions and hyperbolic Brownian motions in multidimensional spheres

TL;DR

This paper studies reflecting diffusion processes inside multidimensional spheres and hyperbolic spaces, deriving explicit formulas for their distributions and examining special cases like Brownian motion and Ornstein-Uhlenbeck processes.

Contribution

It introduces explicit stochastic differential equations and kernels for reflecting diffusions in spheres and hyperbolic spaces, including their hyperbolic counterparts.

Findings

Explicit kernels and distributions for reflecting diffusions in spheres.

Analysis of reflecting hyperbolic Brownian motion in various models.

Detailed examination of special cases like Ornstein-Uhlenbeck and standard Brownian motion.

Abstract

Diffusion processes $(\underline{\bf X}_d(t))_{t\geq 0}$ moving inside spheres $S_R^d \subset\mathbb{R}^d$ and reflecting orthogonally on their surfaces $\partial S_R^d$ are considered. The stochastic differential equations governing the reflecting diffusions are presented and their kernels and distributions explicitly derived. Reflection is obtained by means of the inversion with respect to the sphere $S_R^d$. The particular cases of Ornstein-Uhlenbeck process and Brownian motion are examined in detail. The hyperbolic Brownian motion on the Poincar\`e half-space $\mathbb{H}_d$ is examined in the last part of the paper and its reflecting counterpart within hyperbolic spheres is studied. Finally a section is devoted to reflecting hyperbolic Brownian motion in the Poincar\`e disc $D$ within spheres concentric with $D$.