Dirichlet Heat Kernel for the Laplacian in a Ball

  • Jacek Małecki
  • Grzegorz SerafinEmail author
Open Access


We provide sharp two-sided estimates on the Dirichlet heat kernel k1(t, x, y) for the Laplacian in a ball. The result accurately describes the exponential behaviour of the kernel for small times and significantly improves the qualitatively sharp results known so far. As a consequence we obtain the full description of the kernel k1(t, x, y) in terms of its global two-sided sharp estimates. Such precise estimates were possible to obtain due to the enrichment of analytical methods with probabilistic tools.


Laplacian Ball Heat kernel Estimates Exit time 

Mathematics Subject Classification (2010)

35K08 60J65 


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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Faculty of Pure and Applied MathematicsWrocław University of Science and TechnologyWrocławPoland

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