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Heat content and Brownian motion for some regions with a fractal boundary
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  • Published: December 1994

Heat content and Brownian motion for some regions with a fractal boundary

  • M. van den Berg1 

Probability Theory and Related Fields volume 100, pages 439–456 (1994)Cite this article

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Summary

LetD be an open, bounded set in euclidean space ℝm (m=2, 3, ...) with boundary ∂D. SupposeD has temperature 0 at timet=0, while ∂D is kept at temperature 1 for allt>0. We use brownian motion to obtain estimates for the solution of corresponding heat equation and to obtain results for the asymptotic behaviour ofE D (t), the amount of heat inD at timet, ast→0+. For the triadic von Koch snowflakeK our results imply that

$$c^{ - 1} t^{1{\mathbf{ }} - {\mathbf{ }}(\log {\mathbf{ }}2)/\log {\mathbf{ }}3} \leqq {\mathbf{ }}E_K (t) \leqq ct^{1{\mathbf{ }} - {\mathbf{ }}(\log {\mathbf{ }}2)/\log {\mathbf{ }}3} ,{\mathbf{ }}0 \leqq t \leqq c^{ - 1} ,$$

for some constantc>1.

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Authors and Affiliations

  1. Department of Mathematics, Heriot-Watt University, Riccarton, EH14 4AS, Edinburgh, UK

    M. van den Berg

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  1. M. van den Berg
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van den Berg, M. Heat content and Brownian motion for some regions with a fractal boundary. Probab. Th. Rel. Fields 100, 439–456 (1994). https://doi.org/10.1007/BF01268989

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  • Received: 30 November 1993

  • Revised: 28 December 1994

  • Issue Date: December 1994

  • DOI: https://doi.org/10.1007/BF01268989

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Mathematics Subject Classification

  • 60J45
  • 60J60
  • 60J65
  • 35K05
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