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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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