Mean field limit of non-smooth systems and differential inclusions

Abstract

In this paper, we study deterministic limits of Markov processes made of several interacting objects. While most classical results assume that the limiting dynamics has Lipschitz properties, we show that these conditions are not necessary to prove convergence to a deterministic system. We show that under mild assumptions, the stochastic system converges to the set of solutions of a differential inclusion and we provide simple way to compute the limiting inclusion. When this differential inclusion satisfies a one-sided Lipschitz condition, there exists a unique solution of this differential inclusion and we show convergence in probability with explicit bounds. This extends the applicability of mean field techniques to systems exhibiting threshold dynamics such as queuing systems with boundary conditions or controlled dynamics. This is illustrated by applying our results to several types of systems: fluid limits of priority queues, best response dynamics in games, push-pull queues with a large number of sources and a large number of servers and self-adapting computing systems.