Reflecting diffusions and hyperbolic Brownian motions in multidimensional spheres

We consider diffusion processes moving inside spheres aS, a"e (d) and reflecting orthogonally on their surfaces. We present stochastic differential equations governing the reflecting diffusions and explicitly derive their kernels and distributions. Reflection is obtained by means of the inversion with respect to the sphere . The particular cases of Ornstein-Uhlenbeck process and Brownian motion are examined in detail. The hyperbolic Brownian motion on the Poincar, half-space a"i (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, disc D within spheres concentric with D.