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Self-repelling walk on the Sierpiński gasket
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  • Published: September 2002

Self-repelling walk on the Sierpiński gasket

  • B.M. Hambly1,
  • Kumiko Hattori2 &
  • Tetsuya Hattori3 

Probability Theory and Related Fields volume 124, pages 1–25 (2002)Cite this article

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  • 8 Citations

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Abstract.

 We construct a one-parameter family of self-repelling processes on the Sierpiński gasket, by taking continuum limits of self-repelling walks on the pre-Sierpiński gaskets. We prove that our model interpolates between the Brownian motion and the self-avoiding process on the Sierpiński gasket. Namely, we prove that the process is continuous in the parameter in the sense of convergence in law, and that the order of Hölder continuity of the sample paths is also continuous in the parameter. We also establish a law of the iterated logarithm for the self-repelling process. Finally we show that this approach yields a new class of one-dimensional self-repelling processes.

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, UK. e-mail: hambly@maths.ox.ac.uk, , , , , , GB

    B.M. Hambly

  2. Department of Mathematical Sciences, Shinshu University, Asahi, Matsumoto, 390-8621, Japan. e-mail: hattori@gipac.shinshu-u.ac.jp, , , , , , JP

    Kumiko Hattori

  3. Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan. e-mail: hattori@math.nagoya-u.ac.jp, , , , , , JP

    Tetsuya Hattori

Authors
  1. B.M. Hambly
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  2. Kumiko Hattori
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  3. Tetsuya Hattori
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Additional information

Received: 8 March 2001 / Revised version: 30 May 2001 / Published online: 22 August 2002

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Hambly, B., Hattori, K. & Hattori, T. Self-repelling walk on the Sierpiński gasket. Probab Theory Relat Fields 124, 1–25 (2002). https://doi.org/10.1007/s004400100192

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  • Issue Date: September 2002

  • DOI: https://doi.org/10.1007/s004400100192

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Keywords

  • Brownian Motion
  • Continuum Limit
  • Sample Path
  • Iterate Logarithm
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