On the emergence of random initial conditions in fluid limits

TL;DR

This paper investigates how population processes with small initial populations and large carrying capacities converge to fluid limits with random initial conditions, revealing a stochastic aspect in the initial state at large times.

Contribution

It demonstrates that, at large times, the process converges to a fluid limit with a random initial condition linked to a martingale limit, extending classical results.

Findings

Convergence to fluid limits with random initial conditions at large times.

The random initial condition relates to the martingale limit of a linear birth-death process.

Provides a new perspective on the initial state in population process limits.

Abstract

The paper presents a phenomenon occurring in population processes that start near zero and have large carrying capacity. By the classical result of Kurtz~(1970), such processes, normalized by the carrying capacity, converge on finite intervals to the solutions of ordinary differential equations, also known as the fluid limit. When the initial population is small relative to carrying capacity, this limit is trivial. Here we show that, viewed at suitably chosen times increasing to infinity, the process converges to the fluid limit, governed by the same dynamics, but with a random initial condition. This random initial condition is related to the martingale limit of an associated linear birth and death process.