  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.
