The Kolmogorov equation in the stochastic fragmentation theory and branching processes with infinite collection of particle types

Abstract

The stochastic model for the description of the so-called fragmentation process in frameworks of Kolmogorov approach is proposed. This model is represented as the branching process with continuum set (0,∞) of particle types. Each type r∈(0,∞) corresponds to the set of fragments having the size r. It is proved that the branching condition of this process represents the basic equation of the Kolmogorov theory.