Asymmetric attractive zero-range processes with particle destruction at the origin

Clément Erignoux (1, 2, 3, 4) , Marielle Simon (1, 2, 4) , Linjie Zhao (1)

1 Paradyse - Systèmes de particules et systèmes dynamiques
2 PSPM - Probabilités, statistique, physique mathématique
3 DRACULA - Modélisation multi-échelle des dynamiques cellulaires : application à l'hématopoïese
4 ICJ - Institut Camille Jordan

We investigate the macroscopic behavior of asymmetric attractive zero-range processes on $\mathbb{Z}$ where particles are destroyed at the origin at a rate of order $N^\beta$, where $\beta \in \mathbb{R}$ and $N\in\mathbb{N}$ is the scaling parameter. We prove that the hydrodynamic limit of this particle system is described by the unique entropy solution of a hyperbolic conservation law, supplemented by a boundary condition depending on the range of $\beta$. Namely, if $\beta \geqslant 0$, then the boundary condition prescribes the particle current through the origin, whereas if $\beta<0$, the destruction of particles at the origin has no macroscopic effect on the system and no boundary condition is imposed at the hydrodynamic limit.

Domaines: Probabilités [math.PR]

Clément Erignoux, Marielle Simon, Linjie Zhao. Asymmetric attractive zero-range processes with particle destruction at the origin. Stochastic Processes and their Applications, 2023, 159, pp.1-33. ⟨10.1016/j.spa.2023.01.015⟩. ⟨hal-03431141⟩