The Annals of Probability

Measure-valued branching processes associated with random walks on $p$-adics

Sergio Albeverio and Xuelei Zhao

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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.

Article information

Ann. Probab., Volume 28, Number 4 (2000), 1680-1710.

First available in Project Euclid: 18 April 2002

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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

Random walks $p$-adic spaces measure-valued branching processes nonlinear evolution equations absolute continuity self-similarity local extinction


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.

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