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July 1997 Brownian motion on a random recursive Sierpinski gasket
B. M. Hambly
Ann. Probab. 25(3): 1059-1102 (July 1997). DOI: 10.1214/aop/1024404506


We introduce a random recursive fractal based on the Sierpinski gasket and construct a diffusion upon the fractal via a Dirichlet form. This form and its symmetrizing measure are determined by the electrical resistance of the fractal. The effective resistance provides a metric with which to discuss the properties of the fractal and the diffusion. The main result is to obtain uniform upper and lower bounds for the transition density of the Brownian motion on the fractal in terms of this metric. The bounds are not tight as there are logarithmic corrections due to the randomness in the environment, and the behavior of the shortest paths in the effective resistance metric is not well understood. The results are deduced from the study of a suitable general branching process.


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B. M. Hambly. "Brownian motion on a random recursive Sierpinski gasket." Ann. Probab. 25 (3) 1059 - 1102, July 1997.


Published: July 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0895.60081
MathSciNet: MR1457612
Digital Object Identifier: 10.1214/aop/1024404506

Primary: 60J25 , 60J60 , 60J65

Keywords: Brownian motion , Dirichlet form , general branching process , Random fractal , Spectral dimension , transition density estimate

Rights: Copyright © 1997 Institute of Mathematical Statistics


Vol.25 • No. 3 • July 1997
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