The Fractal Character of Localizable Measure-Valued Processes, II. Localizable Processes and Backward Trees

Abstract

Regard a large population of infinitesimal particles (i.e. a measure) in the case, when the particles evolve (i.e. move, branch, die) independently of each other. Those evolutions we will call localizable. In the present part of this paper we answer at first the question about the structure of localizable evolutions, which take place in one step (independent clustering of measures). In localizable evolutions one can follow the path of any particle (and their predecessors) surviving up to some fixed time. The totality of these paths is the backward tere corresponding to this time point. In the case of continuous time localizable Markov evolutions the random backward tree measure is constructed via finite-dimensional distributions using the extension theorem from part I.