Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition

Young-Pil Choi; Samir Salem

doi:10.3934/krm.2019023

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2019, Volume 12, Issue 3: 573-592. Doi: 10.3934/krm.2019023

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Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition

Young-Pil Choi^1, , and
Samir Salem^2,

1.
Department of Mathematics and Institute of Applied Mathematics, Inha University, Incheon 402–751, Korea
2.
CEREMADE UMR CNRS 7534, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny 75775, Paris Cedex 16, France

^* Corresponding author: Young-Pil Choi
^* Corresponding author: Young-Pil Choi

Received: April 2018

Revised: September 2018

Published: June 2019

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Abstract

In this paper, we consider the Cucker-Smale flocking 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.

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Mathematics Subject Classification: Primary: 60H10, 60H15; Secondary: 82C22, 92D25.

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References

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