Evolution of states of an infinite fission-death system

TL;DR

This paper models the evolution of an infinite system of interacting entities with traits, incorporating state-dependent death and fission, and constructs the evolution of their probability measures over time.

Contribution

It introduces a new mathematical framework for the evolution of infinite fission-death systems with state-dependent interactions and constructs the evolution of sub-Poissonian measures.

Findings

Successfully constructed the evolution of states over time

Extended the class of measures to sub-Poissonian measures

Provided a rigorous mathematical foundation for such systems

Abstract

The evolution of an infinite system of interacting point entities with traits $x\in \mathds{R}^d$ is studied. The elementary acts of the evolution are state-dependent death of an entity with rate that includes a competition term and independent fission in the course of which an entity gives birth to two new entities and simultaneously disappears. The states of the system are probability measures on the corresponding configuration space and the main result of the paper is the construction of the evolution $\mu_0\to \mu_t$, $t>0$, of states in the class of sub-Poissonian measures.