Bessel bridge representation for the heat kernel in hyperbolic space

Abstract

This article shows a Bessel bridge representation for the transition density of Brownian motion on the Poincare space. This transition density is also referred to as the heat kernel on the hyperbolic space in differential geometry literature. The representation recovers the well-known closed form expression for the heat kernel on hyperbolic space in dimension three. However, the newly derived bridge representation is different from the McKean kernel in dimension two and from the Gruet's formula in higher dimensions. The methodology is also applicable to the derivation of an analogous Bessel bridge representation for the heat kernel on a Cartan-Hadamard radially symmetric space and for the transition density of the hyperbolic Bessel process.