Topics in large-volume behavior of interacting particle systems: emergent rhythms and propagation of chaos

This thesis concerns the large-volume behavior of systems of stochastic interacting particles. The first two chapters deal with emergent self-sustained rhythmic behaviors and investigate some possible mechanisms at their origin, whereas the third one is dedicated to the conditional propagation of chaos. In Chapter 1 we investigate the emergence of collective periodic behaviors from the combination of specific interaction networks and Brownian noise. We consider a toy model of two populations of frustratedly interacting diffusions. We show that, in the thermodynamic limit, a periodic law might arise, in some parameter regimes, for an intermediate noise intensity. Despite the frustrated interaction network, no rhythmic behavior is present in the absence of noise. Thus, this phenomenon goes under the name of noise-induced periodicity. In Chapter 2 we investigate the emergence of collective oscillations in a mean-field contact process where the dynamics of the interaction terms is subject to dissipation. We show that, in the thermodynamic limit, the system is attracted towards a stable equilibrium point, but persistent, noise-induced, oscillations might arise for the correspondent fluctuation process in an appropriate regime of the parameters. Therefore, self-sustained rhythmic behaviors survive as finite-size effects when the number of individuals in the system is large but finite. In Chapter 3 we consider a system of SDEs driven by Poisson random measures and whose coefficients depend on the empirical measure of the system. In such finite system, particles are subject to simultaneous jumps whose distribution lies in the normal domain of attraction of a strictly stable law. We prove strong existence and uniqueness of the limit McKean-Vlasov system, which is an infinite-exchangeable system of SDEs driven by a common stable process and whose coefficients depend on the conditional law of any of its coordinates, given such common process. We then show the convergence of the finite to the limit system, also providing explicit error bounds for finite time marginals.