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The Heat Equation on the Spaces of Positive Definite Matrices

Published online by Cambridge University Press:  20 November 2018

P. Sawyer
P. Sawyer*
Affiliation:
Department of Mathematics University of Ottawa Ottawa, Ontario K1N6N5
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Abstract

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The main topic of this paper is the study of the fundamental solution of the heat equation for the symmetric spaces of positive definite matrices, Pos(n,R).

Our first step is to develop a “False Abel Inverse Transform” which transforms functions of compact support on an euclidean space into integrable functions on the symmetric space. The transform is shown to satisfy the relation is the usual Laplacian with a constant drift).

Using this transform, we find explicit formulas for the heat kernel in the cases n = 2 and n = 3. These formulas allow us to give the asymptotic development for the heat kernel as t tends to infinity. Finally, we give an upper and lower bound of the same type for the heat kernel.

Keywords

58G30(53C35,58G11)
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 3 , 01 June 1992 , pp. 624 - 651
Copyright
Copyright © Canadian Mathematical Society 1992

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