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

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.

Minor changes and improvements; journal ref added. arXiv admin note: text overlap with arXiv:1502.06783

References in corpus (6)

• A microscopic probabilistic description of a locally regulated population and macroscopic approximations
• Spatial birth and death processes as solutions of stochastic equations
• Age-structured Trait Substitution Sequence Process and Canonical Equation
• Long term behaviour of locally interacting birth-and-death processes
• Asymptotic shape and the speed of propagation of continuous-time continuous-space birth processes
• Maximal irreducibility measure for spatial birth-and-death processes