Reflecting diffusions and hyperbolic Brownian motions in multidimensional spheres

Diffusion processes $({\underset{‾}{\mathbf{X}}}_d(t){)}_{t\ge 0}$ moving inside spheres $S_R^d\subset {\mathbb{R}}^d$ and reflecting orthogonally on their surfaces $\mathrm{\partial }{S}_R^d$ 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 $D$.