Mean-field limit of particle systems with absorption

Abstract

In this work, we consider one-dimensional particles interacting in mean-field type through a bounded kernel. In addition, when particles hit some barrier (say zero), they are removed from the system. This absorption of particles is instantaneously felt by the others, as, contrary to the usual mean-field setting, particles interact only with other non-absorbed particles. This makes the interaction singular as it happens through hitting times of the given barrier. In addition, the diffusion coefficient of each particle is non uniformly elliptic. We show that the particle system admits a weak solution. Through Partial Girsanov transforms we are able to relate our particles with independent stopped Brownian motions, and prove tightness and convergence to a mean-fied limit stochastic differential equation when the number of particles tends to infinity. Further, we study the limit and establish the existence and uniqueness of the classical solution to the corresponding nonlinear Fokker-Planck equation under some continuity assumption on the interacting kernel. This yields the strong well-posedness of the mean-field limit SDE and confirms that our convergence result is indeed a propagation of chaos result.