In this paper, we consider the Cucker-Smale ocking particles which are subject to the same velocity-dependent noise, which exhibits a phase change phenomenon occurs bringing the system from a "non flocking" to a "flocking" state as the strength of noises decreases. We rigorously show the stochastic mean-field limit from the many-particle Cucker-Smale system with multiplicative noises to the Vlasov-type stochastic partial differential equation as the number of particles goes to infinity. More precisely, we provide a quantitative error estimate between solutions to the stochastic particle system and measure-valued solutions to the expected limiting stochastic partial differential equation by using the Wasserstein distance. For the limiting equation, we construct global-in-time measure-valued solutions and study the stability and large-time behavior showing the convergence of velocities to their mean exponentially fast almost surely.
Bibliographical noteFunding Information:
Y-PC is supported by NRF grant (no. 2017R1C1B2012918 and 2017R1A4A1014735) and POSCO Science Fellowship of POSCO TJ Park Foundation. SS is supported by the Fondation des Sciences Mathematiques de Paris, and Paris Sciences-Lettres Universite. The authors are also grateful to the reviewers for improving this article through their comments.
© American Institute of Mathematical Sciences.
All Science Journal Classification (ASJC) codes
- Numerical Analysis
- Modelling and Simulation