Empirical Measures and Vlasov Hierarchies

François Golse (1) , Clément Mouhot (2) , Valeria Ricci (3)

1 CMLS - Centre de Mathématiques Laurent Schwartz
2 DPMMS - Department of Pure Mathematics and Mathematical Statistics
3 Dipartimento di Matematica e Informatica

The present note reviews some aspects of the mean field limit for Vlasov type equations with Lipschitz continuous interaction kernel. We discuss in particular the connection between the approach involving the N-particle empirical measure and the formulation based on the BBGKY hierarchy. This leads to a more direct proof of the quantitative estimates on the propagation of chaos obtained on a more general class of interacting systems in [S. Mischler, C. Mouhot, B. Wennberg, arXiv:1101.4727]. Our main result is a stability estimate on the BBGKY hierarchy uniform in the number of particles, which implies a stability estimate in the sense of the Monge-Kantorovich distance with exponent 1 on the infinite mean field hierarchy. This last result amplifies Spohn's uniqueness theorem [H. Spohn, Math. Meth. Appl. Sci. 3 (1981), 445-455].

Keywords: Vlasov type equation; Mean field limit; Empirical measure; BBGKY hierarchy; Monge-Kantorovich distance

Domains: Mathematics [math]; Analysis of PDEs [math.AP]; Physics [physics]; Mathematical Physics [math-ph]; Mathematics [math]; Mathematical Physics [math-ph]

François Golse, Clément Mouhot, Valeria Ricci. Empirical Measures and Vlasov Hierarchies. Kinetic and Related Models , 2013, 6 (2013), pp.919-943. ⟨10.3934/krm.2013.6.919⟩. ⟨hal-00856485v2⟩