Vietnam Journal of Mathematics

, Volume 45, Issue 1–2, pp 51–75 | Cite as

Consensus Convergence with Stochastic Effects

  • Josselin Garnier
  • George Papanicolaou
  • Tzu-Wei YangEmail author


We consider a stochastic, continuous state and time opinion model where each agent’s opinion locally interacts with other agents’ opinions in the system, and there is also exogenous randomness. The interaction tends to create clusters of common opinion. By using linear stability analysis of the associated nonlinear Fokker–Planck equation that governs the empirical density of opinions in the limit of infinitely many agents, we can estimate the number of clusters, the time to cluster formation, and the critical strength of randomness so as to have cluster formation. We also discuss the cluster dynamics after their formation, the width and the effective diffusivity of the clusters. Finally, the long-term behavior of clusters is explored numerically. Extensive numerical simulations confirm our analytical findings.


Flocking Opinion dynamics Mean field Interacting random processes 

Mathematics Subject Classification (2010)

92D25 35Q84 60K35 


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Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016

Authors and Affiliations

  • Josselin Garnier
    • 1
  • George Papanicolaou
    • 2
  • Tzu-Wei Yang
    • 3
    Email author
  1. 1.Laboratoire de Probabilités et Modèles AléatoiresLaboratoire Jacques-Louis Lions, Université Paris DiderotParisFrance
  2. 2.Department of MathematicsStanford UniversityStanfordUSA
  3. 3.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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