Uniform propagation of chaos and creation of chaos for a class of nonlinear diffusions

Abstract

We are interested in nonlinear diffusions in which the own law intervenes in the drift. This kind of diffusions corresponds to the hydrodynamical limit of some particle system. One also talks about propagation of chaos. It is well known, for McKean-Vlasov diffusions, that such a propagation of chaos holds on finite-time interval. We here aim to establish a uniform propagation of chaos even if the external force is not convex, with a diffusion coefficient sufficiently large. The idea consists in combining the propagation of chaos on a finite-time interval with a functional inequality, already used by Bolley, Gentil and Guillin. Here, we also deal with a case in which the system at time t = 0 is not chaotic and we show under easily checked assumptions that the system becomes chaotic as the number of particles goes to infinity together with the time. This yields the first result of this type for mean field particle diffusion models as far as we know.

Keywords:

• Nonlinear diffusions
• propagation of chaos
• Feynman-Kac
• McKean-Vlasov models
• functional inequality

Acknowledgments

I would like to thank Arnaud Guillin, François Bolley and Ivan Gentil for the email that they sent me on Wednesday 18th of April 2012. (J.T.)

Disclosure statement

No potential conflict of interest was reported by the authors.