Probability Theory and Related Fields

, Volume 124, Issue 1, pp 1–25 | Cite as

Self-repelling walk on the Sierpiński gasket

  • B.M. Hambly
  • Kumiko Hattori
  • Tetsuya Hattori


 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.


Brownian Motion Continuum Limit Sample Path Iterate Logarithm 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • B.M. Hambly
    • 1
  • Kumiko Hattori
    • 2
  • Tetsuya Hattori
    • 3
  1. 1.Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, UK. e-mail:
  2. 2.Department of Mathematical Sciences, Shinshu University, Asahi, Matsumoto, 390-8621, Japan. e-mail:
  3. 3.Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan. e-mail:

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