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

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 $\H^n$.