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

Pierre del Moral (1) , Julian Tugaut (2)

1 UNSW - University of New South Wales [Sydney]
2 ICJ - Institut Camille Jordan

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. However, it has been proven that the lack of convexity of the external force implies that there is no uniform propagation of chaos if the diffusion coefficient is small enough. 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, see \cite{BGG1,BGG2}. 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.

Mots clés: Nonlinear diffusions; Propagation of chaos; Feynman-Kac; McKean-Vlasov models; Functional inequality

Domaines: Probabilités [math.PR]

Pierre del Moral, Julian Tugaut. Uniform propagation of chaos and creation of chaos for a class of nonlinear diffusions. 2013. ⟨hal-00798813v4⟩