Advanced Studies in Pure Mathematics

The heat kernel and its estimates

Laurent Saloff-Coste

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After a short survey of some of the reasons that make the heat kernel an important object of study, we review a number of basic heat kernel estimates. We then describe recent results concerning (a) the heat kernel on certain manifolds with ends, and (b) the heat kernel with the Neumann or Dirichlet boundary condition in inner uniform Euclidean domains.

Article information

Probabilistic Approach to Geometry, M. Kotani, M. Hino and T. Kumagai, eds. (Tokyo: Mathematical Society of Japan, 2010), 405-436

Received: 20 November 2008
Revised: 3 March 2009
First available in Project Euclid: 24 November 2018

Permanent link to this document euclid.aspm/1543086330

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60] 60J65: Brownian motion [See also 58J65] 35K05: Heat equation

Heat kernel Brownian motion uniform domains


Saloff-Coste, Laurent. The heat kernel and its estimates. Probabilistic Approach to Geometry, 405--436, Mathematical Society of Japan, Tokyo, Japan, 2010. doi:10.2969/aspm/05710405.

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