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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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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DOI: https://doi.org/10.1007/s004400100192
Keywords
- Brownian Motion
- Continuum Limit
- Sample Path
- Iterate Logarithm