April  2020, 13(2): 211-247. doi: 10.3934/krm.2020008

Stochastic Cucker-Smale flocking dynamics of jump-type

1. 

School of Mathematics and Natural Sciences, University of Wuppertal, Gauẞstraẞe 20, 42119 Wuppertal, Germany

2. 

Department of Mathematics, Bielefeld University, Universitätsstraẞe 25, 33615 Bielefeld, Germany

* Corresponding author: Martin Friesen

Received  June 2018 Revised  July 2019 Published  January 2020

We present a stochastic version of the Cucker-Smale flocking dynamics described by a system of $ N $ interacting particles. The velocity aligment of particles is purely discontinuous with unbounded and, in general, non-Lipschitz continuous interaction rates. Performing the mean-field limit as $ N \to \infty $ we identify the limiting process with a solution to a nonlinear martingale problem associated with a McKean-Vlasov stochastic equation with jumps. Moreover, we show uniqueness and stability for the kinetic equation by estimating its solutions in the total variation and Wasserstein distance. Finally, we prove uniqueness in law for the McKean-Vlasov equation, i.e. we establish propagation of chaos.

Citation: Martin Friesen, Oleksandr Kutoviy. Stochastic Cucker-Smale flocking dynamics of jump-type. Kinetic and Related Models, 2020, 13 (2) : 211-247. doi: 10.3934/krm.2020008
References:
[1]

S. M. 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]

S. AlbeverioB. Rüdiger and P. Sundar, The Enskog process, J. Stat. Phys, 167 (2017), 90-122.  doi: 10.1007/s10955-017-1743-9.

[3]

A. V. BobylevI. M. Gamba and V. A. Panferov, Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions, J. Statist. Phys, 116 (2004), 1651-1682.  doi: 10.1023/B:JOSS.0000041751.11664.ea.

[4]

J.-Y. Chemin, Fluides Parfaits Incompressibles, Astérisque, 1995.

[5]

C.-C. ChenS.-Y. Ha and X. Zhang, The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements, Commun. Pure Appl. Anal, 17 (2018), 505-538.  doi: 10.3934/cpaa.2018028.

[6]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[7]

F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math, 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.

[8]

S. Ethier and T. G. Kurtz, Markov Processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiles & Sons, Inc., New York, 1986, Characterization and convergence. doi: 10.1002/9780470316658.

[9]

N. Fournier, Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition, Ann. Appl. Probab, 25 (2015), 860-897.  doi: 10.1214/14-AAP1012.

[10]

N. Fournier and S. Mischler, Rate of convergence of the Nanbu particle system for hard potentials and Maxwell molecules, Ann. Probab, 44 (2016), 589-627.  doi: 10.1214/14-AOP983.

[11]

N. Fournier and C. Mouhot, On the well-posedness of the spatially homogeneous Boltzmann equation with a moderate angular singularity, Comm. Math. Phys, 289 (2009), 803-824.  doi: 10.1007/s00220-009-0807-3.

[12]

M. Friesen, B. Rüdiger and P. Sundar, The Enskog process for hard and soft potentials,, NoDEA Nonlinear Differential Equations Appl., 26 (2019), Art. 20, 42pp. doi: 10.1007/s00030-019-0566-6.

[13]

M. Friesen, B. Rüdiger and P. Sundar, On uniqueness for the Enskog process for hard and soft potentials, to appear.

[14]

S.-Y. HaJ. JeongS. E. NohQ. Xiao and X. Zhang, Emergent dynamics of Cucker-Smale flocking particles in a random environment, J. Differential Equations, 262 (2017), 2554-2591.  doi: 10.1016/j.jde.2016.11.017.

[15]

S.-Y. Ha and D. Levy, Particle, kinetic and fluid models for phototaxis, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 77-108.  doi: 10.3934/dcdsb.2009.12.77.

[16]

S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci, 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.

[17]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.

[18]

J. Horowitz and R. L. Karandikar, Martingale problems associated with the Boltzmann equation, in Seminar on Stochastic Processes, 1989 (San Diego, CA, 1989), vol. 18 of Progr. Probab., Birkhäuser Boston, Boston, MA, 1990, 75–122.

[19]

P.-E. Jabin and Z. Wang, Quantitative estimates of propagation of chaos for stochastic systems with $W^{-1, \infty}$ kernels, Invent. Math, 214 (2018), 523-591.  doi: 10.1007/s00222-018-0808-y.

[20]

J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, vol. 288 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2nd edition, Springer-Verlag, Berlin, 2003, URL http://dx.doi.org/10.1007/978-3-662-05265-5. doi: 10.1007/978-3-662-05265-5.

[21]

T. G. Kurtz, Equivalence of stochastic equations and martingale problems,, in Stochastic Analysis 2010, Springer Heidelberg, 2011, 113–130. doi: 10.1007/978-3-642-15358-7_6.

[22]

S. Mischler and C. Mouhot, Kac's program in kinetic theory, Invent. Math, 193 (2013), 1-147.  doi: 10.1007/s00222-012-0422-3.

[23]

P. B. Mucha and J. Peszek, The Cucker-Smale equation: Singular communication weight, measure-valued solutions and weak-atomic uniqueness, Arch. Ration. Mech. Anal, 227 (2018), 273-308.  doi: 10.1007/s00205-017-1160-x.

[24]

B. PiccoliF. Rossi and E. Trélat, Control to flocking of the kinetic Cucker-Smale model, SIAM J. Math. Anal, 47 (2015), 4685-4719.  doi: 10.1137/140996501.

[25]

P. E. Protter, Stochastic Integration and Differential Equations, vol. 21 of Stochastic Modelling and Applied Probability, Springer-Verlag, Berlin, 2005, Second edition. Version 2.1, Corrected third printing. doi: 10.1007/978-3-662-10061-5.

[26]

S. Serfaty and M. Duerinckx, Mean field limit for coulomb-type flows, arXiv: 1803.08345 [math.AP].

[27]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math, 68 (2007/08), 694-719.  doi: 10.1137/060673254.

[28]

A.-S. Sznitman, Topics in propagation of chaos,, in École d'Été de Probabilités de Saint-Flour XIX-1989, vol. 1464 of Lecture Notes in Math., Springer, Berlin, 1991, 165–251. doi: 10.1007/BFb0085169.

[29]

H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrsch. Verw. Gebiete, 46 (1978/79), 67-105.  doi: 10.1007/BF00535689.

[30]

C. Villani, Optimal Transport, vol. 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

show all references

References:
[1]

S. M. 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]

S. AlbeverioB. Rüdiger and P. Sundar, The Enskog process, J. Stat. Phys, 167 (2017), 90-122.  doi: 10.1007/s10955-017-1743-9.

[3]

A. V. BobylevI. M. Gamba and V. A. Panferov, Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions, J. Statist. Phys, 116 (2004), 1651-1682.  doi: 10.1023/B:JOSS.0000041751.11664.ea.

[4]

J.-Y. Chemin, Fluides Parfaits Incompressibles, Astérisque, 1995.

[5]

C.-C. ChenS.-Y. Ha and X. Zhang, The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements, Commun. Pure Appl. Anal, 17 (2018), 505-538.  doi: 10.3934/cpaa.2018028.

[6]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[7]

F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math, 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.

[8]

S. Ethier and T. G. Kurtz, Markov Processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiles & Sons, Inc., New York, 1986, Characterization and convergence. doi: 10.1002/9780470316658.

[9]

N. Fournier, Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition, Ann. Appl. Probab, 25 (2015), 860-897.  doi: 10.1214/14-AAP1012.

[10]

N. Fournier and S. Mischler, Rate of convergence of the Nanbu particle system for hard potentials and Maxwell molecules, Ann. Probab, 44 (2016), 589-627.  doi: 10.1214/14-AOP983.

[11]

N. Fournier and C. Mouhot, On the well-posedness of the spatially homogeneous Boltzmann equation with a moderate angular singularity, Comm. Math. Phys, 289 (2009), 803-824.  doi: 10.1007/s00220-009-0807-3.

[12]

M. Friesen, B. Rüdiger and P. Sundar, The Enskog process for hard and soft potentials,, NoDEA Nonlinear Differential Equations Appl., 26 (2019), Art. 20, 42pp. doi: 10.1007/s00030-019-0566-6.

[13]

M. Friesen, B. Rüdiger and P. Sundar, On uniqueness for the Enskog process for hard and soft potentials, to appear.

[14]

S.-Y. HaJ. JeongS. E. NohQ. Xiao and X. Zhang, Emergent dynamics of Cucker-Smale flocking particles in a random environment, J. Differential Equations, 262 (2017), 2554-2591.  doi: 10.1016/j.jde.2016.11.017.

[15]

S.-Y. Ha and D. Levy, Particle, kinetic and fluid models for phototaxis, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 77-108.  doi: 10.3934/dcdsb.2009.12.77.

[16]

S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci, 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.

[17]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.

[18]

J. Horowitz and R. L. Karandikar, Martingale problems associated with the Boltzmann equation, in Seminar on Stochastic Processes, 1989 (San Diego, CA, 1989), vol. 18 of Progr. Probab., Birkhäuser Boston, Boston, MA, 1990, 75–122.

[19]

P.-E. Jabin and Z. Wang, Quantitative estimates of propagation of chaos for stochastic systems with $W^{-1, \infty}$ kernels, Invent. Math, 214 (2018), 523-591.  doi: 10.1007/s00222-018-0808-y.

[20]

J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, vol. 288 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2nd edition, Springer-Verlag, Berlin, 2003, URL http://dx.doi.org/10.1007/978-3-662-05265-5. doi: 10.1007/978-3-662-05265-5.

[21]

T. G. Kurtz, Equivalence of stochastic equations and martingale problems,, in Stochastic Analysis 2010, Springer Heidelberg, 2011, 113–130. doi: 10.1007/978-3-642-15358-7_6.

[22]

S. Mischler and C. Mouhot, Kac's program in kinetic theory, Invent. Math, 193 (2013), 1-147.  doi: 10.1007/s00222-012-0422-3.

[23]

P. B. Mucha and J. Peszek, The Cucker-Smale equation: Singular communication weight, measure-valued solutions and weak-atomic uniqueness, Arch. Ration. Mech. Anal, 227 (2018), 273-308.  doi: 10.1007/s00205-017-1160-x.

[24]

B. PiccoliF. Rossi and E. Trélat, Control to flocking of the kinetic Cucker-Smale model, SIAM J. Math. Anal, 47 (2015), 4685-4719.  doi: 10.1137/140996501.

[25]

P. E. Protter, Stochastic Integration and Differential Equations, vol. 21 of Stochastic Modelling and Applied Probability, Springer-Verlag, Berlin, 2005, Second edition. Version 2.1, Corrected third printing. doi: 10.1007/978-3-662-10061-5.

[26]

S. Serfaty and M. Duerinckx, Mean field limit for coulomb-type flows, arXiv: 1803.08345 [math.AP].

[27]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math, 68 (2007/08), 694-719.  doi: 10.1137/060673254.

[28]

A.-S. Sznitman, Topics in propagation of chaos,, in École d'Été de Probabilités de Saint-Flour XIX-1989, vol. 1464 of Lecture Notes in Math., Springer, Berlin, 1991, 165–251. doi: 10.1007/BFb0085169.

[29]

H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrsch. Verw. Gebiete, 46 (1978/79), 67-105.  doi: 10.1007/BF00535689.

[30]

C. Villani, Optimal Transport, vol. 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

Figure 1.  Change of velocities with $ \eta = 3/4 $
[1]

Xing Huang, Feng-Yu Wang. Mckean-Vlasov sdes with drifts discontinuous under wasserstein distance. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1667-1679. doi: 10.3934/dcds.2020336

[2]

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

[3]

Lining Ru, Xiaoping Xue. Flocking of Cucker-Smale model with intrinsic dynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6817-6835. doi: 10.3934/dcdsb.2019168

[4]

Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic and Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381

[5]

Roberto Natalini, Thierry Paul. On the mean field limit for Cucker-Smale models. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2873-2889. doi: 10.3934/dcdsb.2021164

[6]

Seung-Yeal Ha, Jeongho Kim, Peter Pickl, Xiongtao Zhang. A probabilistic approach for the mean-field limit to the Cucker-Smale model with a singular communication. Kinetic and Related Models, 2019, 12 (5) : 1045-1067. doi: 10.3934/krm.2019039

[7]

Seung-Yeal Ha, Jeongho Kim, Xiongtao Zhang. Uniform stability of the Cucker-Smale model and its application to the Mean-Field limit. Kinetic and Related Models, 2018, 11 (5) : 1157-1181. doi: 10.3934/krm.2018045

[8]

Hyunjin Ahn, Seung-Yeal Ha, Jeongho Kim. Uniform stability of the relativistic Cucker-Smale model and its application to a mean-field limit. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4209-4237. doi: 10.3934/cpaa.2021156

[9]

Seung-Yeal Ha, Doheon Kim, Weiyuan Zou. Slow flocking dynamics of the Cucker-Smale ensemble with a chemotactic movement in a temperature field. Kinetic and Related Models, 2020, 13 (4) : 759-793. doi: 10.3934/krm.2020026

[10]

Hyeong-Ohk Bae, Young-Pil Choi, Seung-Yeal Ha, Moon-Jin Kang. Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4419-4458. doi: 10.3934/dcds.2014.34.4419

[11]

Huyên Pham. Linear quadratic optimal control of conditional McKean-Vlasov equation with random coefficients and applications. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 7-. doi: 10.1186/s41546-016-0008-x

[12]

Laure Pédèches. Asymptotic properties of various stochastic Cucker-Smale dynamics. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2731-2762. doi: 10.3934/dcds.2018115

[13]

Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic and Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1

[14]

Seung-Yeal Ha, Jinwook Jung, Peter Kuchling. Emergence of anomalous flocking in the fractional Cucker-Smale model. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5465-5489. doi: 10.3934/dcds.2019223

[15]

Chun-Hsien Li, Suh-Yuh Yang. A new discrete Cucker-Smale flocking model under hierarchical leadership. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2587-2599. doi: 10.3934/dcdsb.2016062

[16]

Seung-Yeal Ha, Dongnam Ko, Yinglong Zhang, Xiongtao Zhang. Emergent dynamics in the interactions of Cucker-Smale ensembles. Kinetic and Related Models, 2017, 10 (3) : 689-723. doi: 10.3934/krm.2017028

[17]

Yu-Jhe Huang, Zhong-Fu Huang, Jonq Juang, Yu-Hao Liang. Flocking of non-identical Cucker-Smale models on general coupling network. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 1111-1127. doi: 10.3934/dcdsb.2020155

[18]

Jan Haskovec, Ioannis Markou. Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime. Kinetic and Related Models, 2020, 13 (4) : 795-813. doi: 10.3934/krm.2020027

[19]

Zhisu Liu, Yicheng Liu, Xiang Li. Flocking and line-shaped spatial configuration to delayed Cucker-Smale models. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3693-3716. doi: 10.3934/dcdsb.2020253

[20]

Chiun-Chuan Chen, Seung-Yeal Ha, Xiongtao Zhang. The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements. Communications on Pure and Applied Analysis, 2018, 17 (2) : 505-538. doi: 10.3934/cpaa.2018028

2020 Impact Factor: 1.432

Metrics

  • PDF downloads (333)
  • HTML views (139)
  • Cited by (1)

Other articles
by authors

[Back to Top]