Stochastic ordering, attractiveness and couplings in non-conservative particle systems

Abstract

We analyse the stochastic comparison of interacting particle systems allowing for multiple arrivals, departures and non-conservative jumps of individuals between sites. That is, if $k$ individuals leave site $x$ for site $y$, a possibly different number $l$ arrive at destination. This setting includes new models, when compared to the conservative case, such as metapopulation models with deaths during migrations. It implies a sharp increase of technical complexity, given the numerous changes to consider. Known results are significantly generalised, even in the conservative case, as no particular form of the transition rates is assumed. We obtain necessary and sufficient conditions on the rates for the stochastic comparison of the processes and prove their equivalence with the existence of an order-preserving Markovian coupling. As a corollary, we get necessary and sufficient conditions for the attractiveness of the processes. A salient feature of our approach lies in the presentation of the coupling in terms of solutions to network flow problems. We illustrate the applicability of our results to a flexible family of population models described as interacting particle systems, with a range of parameters controlling births, deaths, catastrophes or migrations. We provide explicit conditions on the parameters for the stochastic comparison and attractiveness of the models, showing their usefulness in studying their limit behaviour. Additionally, we give three examples of constructing the coupling.