Measure-valued branching random walks (superprocesses) on $p$-adics
are introduced and investigated. The uniqueness and existence of solutions to
associated linear and nonlinear heat-type (parabolic) equations are proved,
provided some condition on the parameter of the random walks is satisfied. The
solutions of these equations are shown to be locally constant if their initial
values are. Moreover, the heat-type equations can be identified with a system
of ordinary differential equations. Conditions for the measure-valued branching
stable random walks to possess the property of quasi-self-similarity are given,
as well as a sufficient and necessary condition for these processes to be
locally extinct. The latter result is consistent with the Euclidean case in the
sense that the critical value for measure-valued branching stable processes to
be locally extinct is the Hausdorff dimension of the image of the underlying
processes divided by the dimension of the state space.
Publié le : 2000-10-14
Classification:
Random walks,
$p$-adic spaces,
measure-valued branching processes,
nonlinear evolution equations,
absolute continuity,
self-similarity,
local extinction,
60G57,
60C65,
11E95,
47H20,
60G30,
60F99
@article{1019160503,
author = {Albeverio, Sergio and Zhao, Xuelei},
title = {Measure-valued branching processes associated with random walks on
$p$-adics},
journal = {Ann. Probab.},
volume = {28},
number = {1},
year = {2000},
pages = { 1680-1710},
language = {en},
url = {http://dml.mathdoc.fr/item/1019160503}
}
Albeverio, Sergio; Zhao, Xuelei. Measure-valued branching processes associated with random walks on
$p$-adics. Ann. Probab., Tome 28 (2000) no. 1, pp. 1680-1710. http://gdmltest.u-ga.fr/item/1019160503/