Note on basic features of large time behaviour of heat kernels

Matthias Keller 1 , Daniel Lenz 2 , Hendrik Vogt 3 , and Radosław Wojciechowski 4
  • 1 Mathematisches Institut, Friedrich Schiller Universität Jena, 03477 Jena, Germany
  • 2 Mathematisches Institut, Friedrich Schiller Universität Jena, 03477 Jena, Germany
  • 3 Institut für Mathematik, Technische Universität Hamburg-Harburg, 21073 Hamburg, Germany
  • 4 York College of the City University of New York, Jamaica, NY 11451, USA

Abstract

Large time behaviour of heat semigroups (and, more generally, of positive selfadjoint semigroups) is studied. Convergence of the semigroup to the ground state and of averaged logarithms of kernels to the ground state energy is shown in the general framework of positivity improving selfadjoint semigroups. This framework encompasses all irreducible semigroups coming from Dirichlet forms as well as suitable perturbations thereof. It includes, in particular, Laplacians on connected manifolds, metric graphs and discrete graphs.

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