# American Institute of Mathematical Sciences

June  2019, 12(3): 573-592. doi: 10.3934/krm.2019023

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

 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

Received  April 2018 Revised  September 2018 Published  February 2019

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.

Citation: Young-Pil Choi, Samir Salem. Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition. Kinetic and Related Models, 2019, 12 (3) : 573-592. doi: 10.3934/krm.2019023
##### References:
 [1] S. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51 (2010), 103301, 17pp. doi: 10.1063/1.3496895. [2] F. Bolley, J. A. Canizo and J. A. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces and swarming, Math. Models Methods Appl. Sci., 21 (2011), 2179-2210.  doi: 10.1142/S0218202511005702. [3] J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, Collective Dynamics from Bacteria to Crowds: An Excursion Through Modeling, Analysis and Simulation (eds. A. Muntean, F. Toschi), Springer-Verlag Wien, 553 (2014), 1–46. doi: 10.1007/978-3-7091-1785-9_1. [4] J. A. Carrillo, Y.-P. Choi, M. Hauray and S. Salem, Mean-field limit for collective behavior models with sharp sensitivity regions, J. Eur. Math. Soc., 21 (2019), 121-161.  doi: 10.4171/JEMS/832. [5] J. A. Carrillo, Y.-P. Choi and S. Pérez, A review on attractive-repulsive hydrodynamics for consensus in collective behavior, Active Particles Vol.I: Advances in Theory, Models, Applications (eds. N. Bellomo, P. Degond, and E. Tadmor), Birkhäuser Basel, (2017), 259–298. [6] P. Cattiaux, F. Delebecque and L. Pédèches, Stochastic Cucker-Smale models: Old and new, Ann. Appl. Probab., 28 (2018), 3239-3286.  doi: 10.1214/18-AAP1400. [7] Y.-P. Choi, S.-Y. Ha, and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, Active Particles Vol.I: Advances in Theory, Models, Applications (eds. N. Bellomo, P. Degond, and E. Tadmor), Birkhäuser Basel, (2017), 299–331. [8] M. Coghi and F. Flandoli, Propagation of chaos for interacting particles subject to environmental noise, Ann. Appl. Probab., 26 (2016), 1407-1442.  doi: 10.1214/15-AAP1120. [9] F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842. [10] R. Dobrushin, Vlasov equations, Funct. Anal. Appl., 13 (1979), 115-123. [11] R. Durrett, Stochastic Calculus: A Practical Introduction, Vol. 6, CRC press, 1996. [12] S.-Y. Ha, J. Jeong, S. E. Noh, Q. Xiao and X. Zhang, Emergent dynamics of Cucker-Smale flocking particles in a random environment, J. Differ. Equat., 262 (2017), 2554-2591.  doi: 10.1016/j.jde.2016.11.017. [13] S.-Y. Ha, K. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Comm. Math. Sci., 7 (2009), 453-469.  doi: 10.4310/CMS.2009.v7.n2.a9. [14] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981. [15] E. Pardoux and A. Rascanu, Stochastic Differential Equations, Backward SDEs, Partial Differential Equations, Springer International Publishing, 2014. doi: 10.1007/978-3-319-05714-9. [16] L. Pédèches, Asymptotic properties of various stochastic Cucker-Smale dynamics, Discrete Contin. Dyn. Syst., 38 (2018), 2731-2762.  doi: 10.3934/dcds.2018115. [17] D. Revus and M. Yor, Continuous Martingales and Brownian Motion, Springer-Verlag Berlin Heidelberg, 1999. doi: 10.1007/978-3-662-06400-9. [18] A.-S. Sznitman, Topics in propagation of chaos, École d'Été de Probabilités de Saint-Flour XIX-1989, Springer, Berlin, 1464 (1991), 165–251. doi: 10.1007/BFb0085169. [19] T. V. Ton, N. T. H. Linh and A. Yagi, Flocking and non-flocking behavior in a stochastic Cucker-Smale system, Anal. Appl., 12 (2014), 63-73.  doi: 10.1142/S0219530513500255.

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##### References:
 [1] S. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51 (2010), 103301, 17pp. doi: 10.1063/1.3496895. [2] F. Bolley, J. A. Canizo and J. A. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces and swarming, Math. Models Methods Appl. Sci., 21 (2011), 2179-2210.  doi: 10.1142/S0218202511005702. [3] J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, Collective Dynamics from Bacteria to Crowds: An Excursion Through Modeling, Analysis and Simulation (eds. A. Muntean, F. Toschi), Springer-Verlag Wien, 553 (2014), 1–46. doi: 10.1007/978-3-7091-1785-9_1. [4] J. A. Carrillo, Y.-P. Choi, M. Hauray and S. Salem, Mean-field limit for collective behavior models with sharp sensitivity regions, J. Eur. Math. Soc., 21 (2019), 121-161.  doi: 10.4171/JEMS/832. [5] J. A. Carrillo, Y.-P. Choi and S. Pérez, A review on attractive-repulsive hydrodynamics for consensus in collective behavior, Active Particles Vol.I: Advances in Theory, Models, Applications (eds. N. Bellomo, P. Degond, and E. Tadmor), Birkhäuser Basel, (2017), 259–298. [6] P. Cattiaux, F. Delebecque and L. Pédèches, Stochastic Cucker-Smale models: Old and new, Ann. Appl. Probab., 28 (2018), 3239-3286.  doi: 10.1214/18-AAP1400. [7] Y.-P. Choi, S.-Y. Ha, and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, Active Particles Vol.I: Advances in Theory, Models, Applications (eds. N. Bellomo, P. Degond, and E. Tadmor), Birkhäuser Basel, (2017), 299–331. [8] M. Coghi and F. Flandoli, Propagation of chaos for interacting particles subject to environmental noise, Ann. Appl. Probab., 26 (2016), 1407-1442.  doi: 10.1214/15-AAP1120. [9] F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842. [10] R. Dobrushin, Vlasov equations, Funct. Anal. Appl., 13 (1979), 115-123. [11] R. Durrett, Stochastic Calculus: A Practical Introduction, Vol. 6, CRC press, 1996. [12] S.-Y. Ha, J. Jeong, S. E. Noh, Q. Xiao and X. Zhang, Emergent dynamics of Cucker-Smale flocking particles in a random environment, J. Differ. Equat., 262 (2017), 2554-2591.  doi: 10.1016/j.jde.2016.11.017. [13] S.-Y. Ha, K. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Comm. Math. Sci., 7 (2009), 453-469.  doi: 10.4310/CMS.2009.v7.n2.a9. [14] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981. [15] E. Pardoux and A. Rascanu, Stochastic Differential Equations, Backward SDEs, Partial Differential Equations, Springer International Publishing, 2014. doi: 10.1007/978-3-319-05714-9. [16] L. Pédèches, Asymptotic properties of various stochastic Cucker-Smale dynamics, Discrete Contin. Dyn. Syst., 38 (2018), 2731-2762.  doi: 10.3934/dcds.2018115. [17] D. Revus and M. Yor, Continuous Martingales and Brownian Motion, Springer-Verlag Berlin Heidelberg, 1999. doi: 10.1007/978-3-662-06400-9. [18] A.-S. Sznitman, Topics in propagation of chaos, École d'Été de Probabilités de Saint-Flour XIX-1989, Springer, Berlin, 1464 (1991), 165–251. doi: 10.1007/BFb0085169. [19] T. V. Ton, N. T. H. Linh and A. Yagi, Flocking and non-flocking behavior in a stochastic Cucker-Smale system, Anal. Appl., 12 (2014), 63-73.  doi: 10.1142/S0219530513500255.
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