Evans defined a notion of what it means for a set $B$ to be polar for a process indexed by a tree. The main result herein is that a tree picked from a Galton-Watson measure whose offspring distribution has mean $m$ and finite variance will almost surely have precisely the same polar sets as a deterministic tree of the same growth rate. This implies that deterministic and nondeterministic trees behave identically in a variety of probability models. Mapping subsets of Euclidean space to trees and polar sets to capacity criteria, it follows that certain random Cantor sets are capacity-equivalent to each other and to deterministic Cantor sets. An extension to branching processes in varying environment is also obtained.
Publié le : 1995-07-14
Classification:
Galton-Watson,
branching,
tree,
polar sets,
percolation,
capacity,
random Cantor sets,
60J80,
60J45,
60D05,
60G60
@article{1176988175,
author = {Pemantle, Robin and Peres, Yuval},
title = {Galton-Watson Trees with the Same Mean Have the Same Polar Sets},
journal = {Ann. Probab.},
volume = {23},
number = {3},
year = {1995},
pages = { 1102-1124},
language = {en},
url = {http://dml.mathdoc.fr/item/1176988175}
}
Pemantle, Robin; Peres, Yuval. Galton-Watson Trees with the Same Mean Have the Same Polar Sets. Ann. Probab., Tome 23 (1995) no. 3, pp. 1102-1124. http://gdmltest.u-ga.fr/item/1176988175/