A Markov process for an infinite age-structured population

TL;DR

This paper constructs and analyzes a Markov process modeling an infinite age-structured population in a general habitat, proving existence, uniqueness, and ergodic properties of the process and its stationary state.

Contribution

It provides an explicit construction of a Markov process for an infinite age-structured population with detailed analysis of its properties, including uniqueness and ergodicity.

Findings

Unique solution to the martingale problem for the process

Existence of a unique stationary state

Process is ergodic under certain conditions

Abstract

For an infinite system of particles arriving in and departing from a habitat $X$ -- a locally compact Polish space with a positive Radon measure $\chi$ -- a Markov process is constructed in an explicit way. Along with its location $x\in X$, each particle is characterized by age $\alpha\geq 0$ -- time since arriving. As the state space one takes the set of marked configurations $\widehat{\Gamma}$, equipped with a metric that makes it a complete and separable metric space. The stochastic evolution of the system is described by a Kolmogorov operator $L$, expressed through the measure $\chi$ and a departure rate $m(x,\alpha)\geq 0$, and acting on bounded continuous functions $F:\widehat{\Gamma}\to \mathds{R}$. For this operator, we pose the martingale problem and show that it has a unique solution, explicitly constructed in the paper. We also prove that the corresponding process has a unique stationary state and is temporarily egrodic if the rate of departure is separated away from zero.