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Modern Stochastics: Theory and Applications


Spatial birth-and-death processes with a finite number of particles
Volume 9, Issue 3 (2022), pp. 279–312
Pub. online: 19 April 2022      Type: Research Article      Open accessOpen Access
Received
27 September 2021
Revised
14 March 2022
Accepted
14 March 2022
Published
19 April 2022

Abstract

The aim of this work is to establish essential properties of spatial birth-and-death processes with general birth and death rates on ${\mathbb{R}^{\mathrm{d}}}$. 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.
The pathwise behavior of an aggregation model is also studied. The probability of extinction and the growth rate of the number of particles under condition of nonextinction are estimated.

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© 2022 The Author(s). Published by VTeX
Open access article under the CC BY license.

Keywords
Spatial birth-and-death processes stochastic equation graphical representation interacting particles

MSC2010
60K35 60J25

Funding
VB was supported by the Department of Computer Science at the University of Verona. VB acknowledges a partial support of the DFG through the SFB 701 (Bielefeld University) and the IRTG (IGK) 1132 “Stochastics and Real World Models”.

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