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On-diagonal oscillation of the heat kernels on post-critically finite self-similar fractals
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  • Published: 23 March 2012

On-diagonal oscillation of the heat kernels on post-critically finite self-similar fractals

  • Naotaka Kajino1 

Probability Theory and Related Fields volume 156, pages 51–74 (2013)Cite this article

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Abstract

For the canonical heat kernels p t (x, y) associated with Dirichlet forms on post-critically finite self-similar fractals, e.g. the transition densities (heat kernels) of Brownian motion on affine nested fractals, the non-existence of the limit \({\lim_{t\downarrow 0}t^{d_{s}/2}p_{t}(x,x)}\) is established for a “generic” (in particular, almost every) point x, where d s denotes the spectral dimension. Furthermore the same is proved for any point x in the case of the d-dimensional standard Sierpinski gasket with d ≥ 2 and the N-polygasket with N ≥ 3 odd, e.g. the pentagasket (N = 5) and the heptagasket (N = 7).

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Authors and Affiliations

  1. Department of Mathematics, University of Bielefeld, Postfach 10 01 31, 33501, Bielefeld, Germany

    Naotaka Kajino

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  1. Naotaka Kajino
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Correspondence to Naotaka Kajino.

Additional information

The author was supported by the Japan Society for the Promotion of Science (JSPS Research Fellow PD (20 · 6088)).

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Cite this article

Kajino, N. On-diagonal oscillation of the heat kernels on post-critically finite self-similar fractals. Probab. Theory Relat. Fields 156, 51–74 (2013). https://doi.org/10.1007/s00440-012-0420-9

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  • Received: 12 June 2011

  • Revised: 30 January 2012

  • Accepted: 29 February 2012

  • Published: 23 March 2012

  • Issue Date: June 2013

  • DOI: https://doi.org/10.1007/s00440-012-0420-9

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Keywords

  • Post-critically finite self-similar fractals
  • Affine nested fractals
  • Dirichlet form
  • Heat kernel
  • Oscillation
  • Short time asymptotics

Mathematics Subject Classification

  • Primary 28A80
  • 60J35
  • Secondary 31C25
  • 58C40
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