Bessel bridge representation for heat kernel in hyperbolic space

TL;DR

This paper introduces a novel Bessel bridge representation for the heat kernel on hyperbolic space, providing new insights and formulas that differ from classical results, with potential applications to related geometric and stochastic processes.

Contribution

It derives a new Bessel bridge representation for the heat kernel on hyperbolic space, extending the methodology to other symmetric spaces and processes.

Findings

Recovers the known heat kernel in 3D hyperbolic space

Provides a different bridge representation from McKean and Gruet's formulas

Applicable to Cartan-Hadamard spaces and hyperbolic Bessel processes

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 representations for heat kernel on a Cartan-Hadamard radially symmetric space and for the transition density of hyperbolic Bessel process.