Poisson semigroup and the Gruet formula for the heat kernels on spaces of constant curvature

TL;DR

This paper develops new methods to derive explicit formulas for Poisson and heat kernels on spaces of constant curvature, including Euclidean, spherical, and hyperbolic spaces, with new formulas and elementary derivations.

Contribution

It introduces new elementary methods to derive explicit Poisson and heat kernel formulas on constant curvature spaces, including new Gruet formulas for Euclidean and spherical spaces.

Findings

Derived new Gruet formulas for Euclidean and spherical heat kernels.

Provided an elementary derivation of the hyperbolic heat kernel formula.

Unified approach to heat kernel formulas across different constant curvature spaces.

Abstract

This paper is concerned with the Poisson and heat equations on spaces of constant curvature. More explicitly we provide new methods for obtaining old and new explicit formulas for the Poisson and heat semigroups on the Euclidean, spherical and hyperbolic spaces $\R^n$, $\S^n$ and $\H^n$ . We obtain the Gruet formula for the heat kernels in Euclidean and spherical spaces $\R^n$ and $\S^n$, which are new and we provide a new elementary method to derive the classical Gruet formula Gruet\cite{Gruet} for the kernel of the heat semigroup on the hyperbolic space $\H^n$.