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

Abstract

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.