Random Walks and Diffusions on Fractals

  • Sheldon Goldstein

Abstract

We investigate the asymptotic motion of a random walker, which at time n is at X(n), on certain “fractal lattices”. For the “Sierpinski lattice” in dimension d we show that as → ∞, the process Y(t) ≡ X([(d+3) t])/2 converges in distribution (so that, in particular, |X(n)| ~ nγ, where γ = (ln 2)/ln(d + 3)) to a diffusion on the Sierpinski gasket, a Cantor set of Lebesgue measure zero. The analysis is based on a simple “renormalization group” type argument, involving self-similarity and “decimation invariance”.

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References

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    S. Kusuoka, A diffusion process on a fractal, preprint.Google Scholar
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    P. Billingsley, Convergence of Probability Measures, Wiley, New York (1968).MATHGoogle Scholar
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    R.A. Guyer, Phys. Rev. A 29, 2751 (1984).MathSciNetCrossRefGoogle Scholar
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    R. Rammal and G. Toulouse, J. Physique-Lettres 44, L-13 (1983).Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1987

Authors and Affiliations

  • Sheldon Goldstein
    • 1
  1. 1.Department of MathematicsRutgers UniversityNew BrunswickUSA

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