Hitting spheres on hyperbolic spaces
Valentina Cammarota, Enzo Orsingher

TL;DR
This paper analyzes the behavior of hyperbolic Brownian motion, deriving hitting probabilities, exit distributions, and Poisson kernels in hyperbolic spaces and on spheres, with asymptotic and small domain approximations.
Contribution
It provides explicit formulas for hitting distributions and exit probabilities in hyperbolic spaces and spheres, extending classical Euclidean results to hyperbolic geometry.
Findings
Derived boundary hitting probabilities for hyperbolic Brownian motion.
Obtained Poisson kernels for hyperbolic and spherical domains.
Analyzed asymptotic behavior and small domain limits.
Abstract
For a hyperbolic Brownian motion on the Poincar\'e half-plane , starting from a point of hyperbolic coordinates inside a hyperbolic disc of radius , we obtain the probability of hitting the boundary at the point . For we derive the asymptotic Cauchy hitting distribution on and for small values of and we obtain the classical Euclidean Poisson kernel. The exit probabilities from a hyperbolic annulus in of radii and are derived and the transient behaviour of hyperbolic Brownian motion is considered. Similar probabilities are calculated also for a Brownian motion on the surface of the three dimensional sphere. For the hyperbolic half-space we obtain…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
