The heat kernel and its estimates. (English) Zbl 1201.58025

Kotani, Motoko (ed.) et al., Probabilistic approach to geometry. Proceedings of the 1st international conference, Kyoto, Japan, 28th July – 8th August, 2008. Tokyo: Mathematical Society of Japan (MSJ) (ISBN 978-4-931469-58-7/hbk). Advanced Studies in Pure Mathematics 57, 405-436 (2010).
The paper provides a nice overview on the probabilistic approach to study the heat kernel. This subject attracts considerable attention in analysis, probability and geometry. The author begins with a survey of upper bounds and two-sided bounds of the diffusive heat kernel. Then, these fundamental results are used to study manifolds with finitely many ends. Finally, heat kernels with Dirichlet boundary conditions are considered.
For the entire collection see [Zbl 1190.60003].


58J65 Diffusion processes and stochastic analysis on manifolds
60J65 Brownian motion
35K05 Heat equation