The Annals of Probability
- Ann. Probab.
- Volume 28, Number 4 (2000), 1680-1710.
Measure-valued branching processes associated with random walks on $p$-adics
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.
Ann. Probab., Volume 28, Number 4 (2000), 1680-1710.
First available in Project Euclid: 18 April 2002
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60G57: Random measures 60C65
Secondary: 11E95: $p$-adic theory 47H20: Semigroups of nonlinear operators [See also 37L05, 47J35, 54H15, 58D07] 60G30: Continuity and singularity of induced measures 60F99
Albeverio, Sergio; Zhao, Xuelei. Measure-valued branching processes associated with random walks on $p$-adics. Ann. Probab. 28 (2000), no. 4, 1680--1710. doi:10.1214/aop/1019160503. https://projecteuclid.org/euclid.aop/1019160503