Stochastic mean-field limit: Non-lipschitz forces and swarming

François Bolley, José A. CaÑizo, José A. Carrillo

Research output: Contribution to journalArticleResearchpeer-review

143 Citations (Scopus)


We consider general stochastic systems of interacting particles with noise which are relevant as models for the collective behavior of animals, and rigorously prove that in the mean-field limit the system is close to the solution of a kinetic PDE. Our aim is to include models widely studied in the literature such as the CuckerSmale model, adding noise to the behavior of individuals. The difficulty, as compared to the classical case of globally Lipschitz potentials, is that in several models the interaction potential between particles is only locally Lipschitz, the local Lipschitz constant growing to infinity with the size of the region considered. With this in mind, we present an extension of the classical theory for globally Lipschitz interactions, which works for only locally Lipschitz ones. © 2011 World Scientific Publishing Company.
Original languageEnglish
Pages (from-to)2179-2210
JournalMathematical Models and Methods in Applied Sciences
Issue number11
Publication statusPublished - 1 Nov 2011


  • collective behavior
  • CuckerSmale
  • diffusion
  • Mean-field limit


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