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Diffusion processes on fractal fields: heat kernel estimates and large deviations
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  • Published: 15 July 2003

Diffusion processes on fractal fields: heat kernel estimates and large deviations

  • B.M. Hambly1 &
  • T. Kumagai2 

Probability Theory and Related Fields volume 127, pages 305–352 (2003)Cite this article

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Abstract

A fractal field is a collection of fractals with, in general, different Hausdorff dimensions, embedded in ℝ2. We will construct diffusion processes on such fields which behave as Brownian motion in ℝ2 outside the fractals and as the appropriate fractal diffusion within each fractal component of the field. We will discuss the properties of the diffusion process in the case where the fractal components tile ℝ2. By working in a suitable shortest path metric we will establish heat kernel bounds and large deviation estimates which determine the trajectories followed by the diffusion over short times.

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford, OX1 3LB, UK

    B.M. Hambly

  2. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    T. Kumagai

Authors
  1. B.M. Hambly
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  2. T. Kumagai
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Correspondence to B.M. Hambly.

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Hambly, B., Kumagai, T. Diffusion processes on fractal fields: heat kernel estimates and large deviations. Probab. Theory Relat. Fields 127, 305–352 (2003). https://doi.org/10.1007/s00440-003-0284-0

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  • Received: 23 April 2002

  • Revised: 01 May 2003

  • Published: 15 July 2003

  • Issue Date: November 2003

  • DOI: https://doi.org/10.1007/s00440-003-0284-0

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Keywords

  • Brownian Motion
  • Short Path
  • Diffusion Process
  • Heat Kernel
  • Hausdorff Dimension
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