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

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