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THE HEAT KERNEL ON NONCOMPACT SYMMETRIC SPACES Jean--Philippe Anker & Patrick Ostellari
 

Summary: THE HEAT KERNEL ON NONCOMPACT SYMMETRIC SPACES
Jean--Philippe Anker & Patrick Ostellari
In memory of F. I. KarpeleviŸc (1927--2000)
The heat kernel plays a central role in mathematics. It occurs in several fields :
analysis, geometry and -- last but not least -- probability theory. In this survey, we
shall focus on its analytic aspects, specifically sharp bounds, in the particular setting of
Riemannian symmetric spaces of noncompact type. It is a natural tribute to KarpeleviŸc,
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 first 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 briefly 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.
Thus G is a semisimple Lie group (real, connected, noncompact, with finite center) or
more generally a reductive Lie group in the Harish­Chandra class and K is a maximal

  

Source: Anker, Jean-Philippe - Laboratoire de Mathématiques et Applications, Physique Mathématique, Université d'Orléans

 

Collections: Mathematics