Hartman-Watson distribution and hyperbolic-like Heat kernels

TL;DR

This paper establishes connections between different formulas for heat kernels on hyperbolic spaces and harmonic AN groups, using probabilistic methods involving Brownian motion and exponential functionals, leading to new integral representations.

Contribution

It introduces new integral representations for heat kernels on hyperbolic and harmonic AN groups by relating existing formulas through probabilistic techniques.

Findings

Derived a new integral representation for the heat kernel of the Maass Laplacian.

Connected Gruet and Millson formulas via Yor's joint distribution result.

Provided integral formulas that do not depend on the parity of the group's center dimension.

Abstract

We relate Gruet formula for the heat kernel on real hyperbolic spaces to the commonly used one derived from Millson induction. The bridge between both formulas is settled by Yor result on the joint distribution of a Brownian motion and of its exponential functional at fixed time. This result allows further to relate Gruet formula with real parameter to the heat kernel of the hyperbolic Jacobi operator and to derive a new integral representation for the heat kernel of the Maass Laplacian. When applied to harmonic AN groups, Yor result yields also new a integral representation of their corresponding heat kernels which does not distinguish the parity of the dimension of the center of the Lie group N.