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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 |
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.
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. |
show all references
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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