The heat kernel on noncompact symmetric spaces, Lie groups and symmetric spaces

The heat kernel plays a central role in mathematics. It occurs in several elds: analysis, geometry and { last but not least { probability theory. In this survey, we shall focus on its analytic aspects, specically sharp bounds, in the particular setting of Riemannian symmetric spaces of noncompact type. It is a natural tribute to Karpelevic, whose pioneer work [Ka] inspired further study of the geometry of theses spaces and of the analysis of the Laplacian thereon. This survey is based on lectures delivered by the rst author in May 2002 at IHP in Paris during the Special Quarter Heat kernels, random walks & analysis on manifolds & graphs. Both authors would like to thank the organizers for their great job, as well as Martine Babillot, Gilles Carron, Sasha Grigor'yan and Jean{Pierre Otal for stimulating discussions. 1. Preliminaries We shall brie y review some basics about noncompact Riemannian symmetric spaces X = G=K and we shall otherwise refer to standard texbooks ([GV]; [H1], [H2], [H3]; [Kn]) for their structure and harmonic analysis thereon