Spatial birth-and-death processes with a finite number of particles

TL;DR

This paper constructs and analyzes spatial birth-and-death processes with finite particles, proving key properties and exploring the behavior of an aggregation model, including extinction probabilities and growth rates.

Contribution

It establishes existence, uniqueness, and Markov properties for these processes under linear growth conditions and links them rigorously to their generators.

Findings

Proved existence and uniqueness of solutions.

Established martingale properties and Markov characteristics.

Estimated extinction probabilities and growth rates.

Abstract

Spatial birth-and-death processes with time dependent rates are obtained as solutions to certain stochastic equations. The existence, uniqueness, uniqueness in law and the strong Markov property of unique solutions are proven when the integral of the birth rate over $\mathbb{R} ^ \mathrm{d}$ grows not faster than linearly with the number of particles of the system. Martingale properties of the constructed process provide a rigorous connection to the heuristic generator. We also study pathwise behavior of an aggregation model. The probability of extinction and the growth rate of the number of particles conditioning on non-extinction are estimated.