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

Mohamed Vall Ould Moustapha

TL;DR

The paper addresses explicit formulas for the Poisson and heat semigroups on spaces of constant curvature by leveraging subordination and Laplace transform relations between the semigroups. It derives Gruet-type representations for heat kernels in the Euclidean and spherical settings and provides a new elementary derivation of the Gruet formula on hyperbolic space, unifying the treatment across $\mathbb{R}^n$, $\mathbb{S}^n$, and $\mathbb{H}^n$. The results yield detailed recurrence relations across dimensions and analytic representations (contour/integral forms) for the kernels, clarifying the role of the Laplace-Beltrami operator and Brownian motion on these manifolds. These explicit kernels facilitate potential-theoretic and probabilistic analyses on spaces of constant curvature and offer practical tools for applications in mathematical physics and geometric analysis.

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 $\R^n$, $§^n$ and $\H^n$ . We obtain the Gruet formula for the heat kernels in Euclidean and spherical spaces $\R^n$ and $§^n$, 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 $\H^n$.

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

TL;DR

The paper addresses explicit formulas for the Poisson and heat semigroups on spaces of constant curvature by leveraging subordination and Laplace transform relations between the semigroups. It derives Gruet-type representations for heat kernels in the Euclidean and spherical settings and provides a new elementary derivation of the Gruet formula on hyperbolic space, unifying the treatment across , , and . The results yield detailed recurrence relations across dimensions and analytic representations (contour/integral forms) for the kernels, clarifying the role of the Laplace-Beltrami operator and Brownian motion on these manifolds. These explicit kernels facilitate potential-theoretic and probabilistic analyses on spaces of constant curvature and offer practical tools for applications in mathematical physics and geometric analysis.

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 , and . We obtain the Gruet formula for the heat kernels in Euclidean and spherical spaces and , 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 .
Paper Structure (4 sections, 11 theorems, 18 equations)

This paper contains 4 sections, 11 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Heat and Poisson semigroups on the Euclidean space $I\!\!R^n$
  3. Heat and Poisson semigroups on the spherical space
  4. Heat and Poisson semigroups on the hyperbolic spaces $I\!\!H^n$

Key Result

Proposition 1.1

Let $e^{t A_n}$ and $e^{-y\sqrt{-{A}_n}}$ be the heat and the Poisson semigroup operators on spaces of constant curvature, the following formulas hold: i) $e^{-y \sqrt{-A_n}}=\frac{y}{\sqrt{\pi}}L_u\left[e^{A_n/4 u}u^{-1/2}\right](y^2)$, ii) $e^{t A_n}=(4 t)^{-1/2}L_{y^2}^{-1}\left[\frac{\sqrt{\pi}e

Theorems & Definitions (21)

  • Proposition 1.1
  • proof
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Theorem 2.1
  • proof
  • ...and 11 more