Abstract
We consider diffusion processes \( {{\left( {{{{\underline{\mathrm{X}}}}_d}(t)} \right)}_{{t\geqslant 0}}} \) moving inside spheres \( S_R^d \) ⊂ ℝd 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 \( S_R^d \). The particular cases of Ornstein–Uhlenbeck process and Brownian motion are examined in detail.
The hyperbolic Brownian motion on the Poincaré half-space 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 Poincaré disc D within spheres concentric with D.
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Aryasova, O., De Gregorio, A. & Orsingher, E. Reflecting diffusions and hyperbolic Brownian motions in multidimensional spheres. Lith Math J 53, 241–263 (2013). https://doi.org/10.1007/s10986-013-9206-8
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DOI: https://doi.org/10.1007/s10986-013-9206-8