The coagulation-fragmentation hierarchy with homogeneous rates and underlying stochastic dynamics

TL;DR

This paper introduces a hierarchical system of equations modeling the dynamics of infinite clusters undergoing coagulation and fragmentation, linking stochastic interval partition processes with solutions to the coagulation-fragmentation equation.

Contribution

It establishes a weak solution framework using correlation measures for stochastic split-merge processes and provides conditions for reversibility and asymptotic behavior.

Findings

Correlation measures solve the hierarchical system

Reversibility conditions for the stochastic processes

Asymptotic solutions to the coagulation-fragmentation equation

Abstract

A hierarchical system of equations is introduced to describe dynamics of `sizes' of infinite clusters which coagulate and fragmentate with homogeneous rates of certain form. We prove that this system of equations is solved weakly by correlation measures for stochastic dynamics of interval partitions evolving according to some split-merge transformations. Regarding those processes, a sufficient condition for a distribution to be reversible is given. Also, an asymptotic result for properly rescaled processes is shown to obtain a solution to a nonlinear equation called the coagulation-fragmentation equation.