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A diffusion defined on a fractal state space

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  • Krebs, William B.

Abstract

In the plane, we define a fractal known as the Vicsek snowflake in terms of a family of affine contractions in 2. We show that the Vicsek snowflake is a nested fractal in the sense of Lindstrøm (1990). We define random walks on the Vicsek snowflake and explicitly find an invariant probability for random walk. From this invariant probability, we construct a Brownian motion on the Vicsek snowflake. We show that this Brownian motion is the unique diffusion limit under weak convergence of rescaled random walks with any probability parameter. We show that Brownian motion on the Vicsek snowflake has a scaling property reminiscent of Brownian motion in 1. Using a coupling argument, we show that our Brownian motion has transition densities with respect to Hausdorff measure on the snowflake.

Suggested Citation

  • Krebs, William B., 1991. "A diffusion defined on a fractal state space," Stochastic Processes and their Applications, Elsevier, vol. 37(2), pages 199-212, April.
  • Handle: RePEc:eee:spapps:v:37:y:1991:i:2:p:199-212
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    Cited by:

    1. Hambly, B. M. & Metz, V., 1998. "The homogenization problem for the Vicsek set," Stochastic Processes and their Applications, Elsevier, vol. 76(2), pages 167-190, August.
    2. Radchenko, Vadym & Zähle, Martina, 2012. "Heat equation with a general stochastic measure on nested fractals," Statistics & Probability Letters, Elsevier, vol. 82(3), pages 699-704.

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    Keywords

    diffusions fractals;

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