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.
Publié le : 1997-07-14
Classification:
Brownian motion,
random fractal,
Dirichlet form,
spectral dimension,
transition density estimate,
general branching process,
60J60,
60J25,
60J65
@article{1024404506,
author = {Hambly, B. M.},
title = {Brownian motion on a random recursive Sierpinski gasket},
journal = {Ann. Probab.},
volume = {25},
number = {4},
year = {1997},
pages = { 1059-1102},
language = {en},
url = {http://dml.mathdoc.fr/item/1024404506}
}
Hambly, B. M. Brownian motion on a random recursive Sierpinski gasket. Ann. Probab., Tome 25 (1997) no. 4, pp. 1059-1102. http://gdmltest.u-ga.fr/item/1024404506/