Chaos hypothesis for a system interacting through shared resources

We study the chaos hypothesis for a wide class of pure-jump multitype interacting systems. The interaction may be strong, there is no symmetry assumption, and the system is not necessarily Markovian. We use interaction graphs and coupling and study in a precise way how a chain reaction is constituted by a series of direct interactions. We obtain the chaos hypothesis in variation norm with speed of convergence and deduce from it convergence of general empirical measures. We couple the interaction graph to a Boltzmann tree and show that the variation norm between the processes constructed on each goes to zero. This proves propagation of chaos in total variation with speed of convergence when the Boltzmann trees converge. Under light symmetry assumptions, we characterize the limit law by a nonlinear martingale problem.