A Markov process for an infinite age-structured population

Abstract

A Markov process is constructed in an explicit way for an infinite system of entities arriving in and departing from a habitat X, which is a locally compact Polish space with a positive Radon measure χ. Along with its location x ∈ X, each particle is characterized by age α ≥ 0 – time since arriving. As the state space one takes the set of marked configurations Γ̂, 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 χ and a departure rate m(x, α) ≥ 0, and acting on bounded continuous functions F : Γ̂→ 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.