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Stochastic Cucker-Smale flocking dynamics of jump-type

  • * Corresponding author: Martin Friesen

    * Corresponding author: Martin Friesen 
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  • 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.

    Mathematics Subject Classification: Primary: 35Q83, 60F05; Secondary: 60K35.


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

  • [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.
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