Mean-field limit for the stochastic Vicsek model

François Bolley (1) , José Cañizo (2) , José Antonio Carrillo (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 UAB - Departament de Matemàtiques [Barcelona]

We consider the continuous version of the Vicsek model with noise, proposed as a model for collective behavior of individuals with a fixed speed. We rigorously derive the kinetic mean-field partial differential equation satisfied when the number $N$ of particles tends to infinity, quantifying the convergence of the law of one particle to the solution of the PDE. For this we adapt a classical coupling argument to the present case in which both the particle system and the PDE are defined on a surface rather than on the whole space $\rr^d$. As part of the study we give existence and uniqueness results for both the particle system and the PDE.

Domaines: Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]

François Bolley, José Cañizo, José Antonio Carrillo. Mean-field limit for the stochastic Vicsek model. Applied Mathematics Letters, 2012, 25 (3), pp.339-343. ⟨hal-00563915⟩