June  2018, 38(6): 2731-2762. doi: 10.3934/dcds.2018115

Asymptotic properties of various stochastic Cucker-Smale dynamics

Institut de Mathématiques de Toulouse, 118, route de Narbonne, F-31062 Toulouse Cedex 9, France

Received  March 2017 Revised  December 2017 Published  April 2018

Starting from the stochastic Cucker-Smale model introduced in [14], we look into its asymptotic behaviours for different kinds of interaction. First in term of ergodicity, when $t$ goes to infinity, seeking invariant probability measures and using Lyapunov functionals. Second, when the number $N$ of particles becomes large, leading to results about propagation of chaos.

Citation: Laure Pédèches. Asymptotic properties of various stochastic Cucker-Smale dynamics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2731-2762. doi: 10.3934/dcds.2018115
References:
[1]

M. AguehR. Illner and A. Richardson, Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type, Kinet. Relat. Models, 4 (2011), 1-16. doi: 10.3934/krm.2011.4.1. Google Scholar

[2]

S. AhnY. Ha and A. Richardson, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51 (2011), 103301, 17 pp. Google Scholar

[3]

D. Aldous, Exchangeability and related topics, Lectures Notes in Math., 1117 (1983), 1-198. Google Scholar

[4]

D. BakryP. Cattiaux and A. Guillin, Rate of convergence for ergodic Markov processes: Lyapunov versus Poincaré, J. Funct. Anal., 254 (2008), 727-759. doi: 10.1016/j.jfa.2007.11.002. Google Scholar

[5] P. Billingsley, Convergence of Probability Measures, Wiley, 1968. Google Scholar
[6]

F. BolleyJ. Cañizo and J. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces and swarming, Math. Models Methods Appl. Sci., 21 (2011), 2179-2210. doi: 10.1142/S0218202511005702. Google Scholar

[7]

P. CattiauxD. Chafaï and A. Guillin, Central limit theorems for additive functionals of ergodic Markov diffusions processes, ALEA Lat. Am. J. Probab. Math. Stat., 9 (2012), 337-382. Google Scholar

[8]

P. Cattiaux, F. Delebecque and L. Pédèches, Stochastic Cucker-Smale models: Old and new, Preprint.Google Scholar

[9]

F. Cucker and J.-G. Dong, Avoiding collisions in flocks, IEEE Trans. Automat. Control, 55 (2010), 1238-1243. doi: 10.1109/TAC.2010.2042355. Google Scholar

[10]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296. doi: 10.1016/j.matpur.2007.12.002. Google Scholar

[11]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-861. doi: 10.1109/TAC.2007.895842. Google Scholar

[12]

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

[13]

D. DownS. Meyn and R. Tweedie, Exponential and uniform ergodicity of Markov processes, Ann. Probab., 23 (1995), 1671-1691. doi: 10.1214/aop/1176987798. Google Scholar

[14]

S.-Y. HaK. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469. doi: 10.4310/CMS.2009.v7.n2.a9. Google Scholar

[15]

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

[16]

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

[17]

K. Itô and M. Nisio, On stationary solutions of a stochastic differential equation, J. Math. Kyoto Univ., 4 (1964), 1-75. doi: 10.1215/kjm/1250524705. Google Scholar

[18]

S. Méléard, Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models, Lectures Notes in Math., 1627 (1995), 42-95. Google Scholar

[19] S. Meyn and R. Tweedie, Markov Chains and Stochastic Stability, Springer-Verlag, 1993. Google Scholar
[20]

L. Pédèches, Exponential ergodicity for a class of non-Markovian stochastic processes, Preprint.Google Scholar

[21]

M. Scheutzow, Qualitative behaviour of stochastic delay equations with a bounded memory, Stochastics, 12 (1984), 41-80. doi: 10.1080/17442508408833294. Google Scholar

[22]

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

[23]

A. Sznitman, Topics in propagation of chaos, Lectures Notes in Math., 1464 (1991), 165-251. Google Scholar

[24]

T. TonN. Linh and A. Yagi, Flocking and non-flocking behavior in a stochastic Cucker-Smale system, Anal. Appl. (Singap.), 12 (2014), 63-73. doi: 10.1142/S0219530513500255. Google Scholar

show all references

References:
[1]

M. AguehR. Illner and A. Richardson, Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type, Kinet. Relat. Models, 4 (2011), 1-16. doi: 10.3934/krm.2011.4.1. Google Scholar

[2]

S. AhnY. Ha and A. Richardson, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51 (2011), 103301, 17 pp. Google Scholar

[3]

D. Aldous, Exchangeability and related topics, Lectures Notes in Math., 1117 (1983), 1-198. Google Scholar

[4]

D. BakryP. Cattiaux and A. Guillin, Rate of convergence for ergodic Markov processes: Lyapunov versus Poincaré, J. Funct. Anal., 254 (2008), 727-759. doi: 10.1016/j.jfa.2007.11.002. Google Scholar

[5] P. Billingsley, Convergence of Probability Measures, Wiley, 1968. Google Scholar
[6]

F. BolleyJ. Cañizo and J. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces and swarming, Math. Models Methods Appl. Sci., 21 (2011), 2179-2210. doi: 10.1142/S0218202511005702. Google Scholar

[7]

P. CattiauxD. Chafaï and A. Guillin, Central limit theorems for additive functionals of ergodic Markov diffusions processes, ALEA Lat. Am. J. Probab. Math. Stat., 9 (2012), 337-382. Google Scholar

[8]

P. Cattiaux, F. Delebecque and L. Pédèches, Stochastic Cucker-Smale models: Old and new, Preprint.Google Scholar

[9]

F. Cucker and J.-G. Dong, Avoiding collisions in flocks, IEEE Trans. Automat. Control, 55 (2010), 1238-1243. doi: 10.1109/TAC.2010.2042355. Google Scholar

[10]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296. doi: 10.1016/j.matpur.2007.12.002. Google Scholar

[11]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-861. doi: 10.1109/TAC.2007.895842. Google Scholar

[12]

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

[13]

D. DownS. Meyn and R. Tweedie, Exponential and uniform ergodicity of Markov processes, Ann. Probab., 23 (1995), 1671-1691. doi: 10.1214/aop/1176987798. Google Scholar

[14]

S.-Y. HaK. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469. doi: 10.4310/CMS.2009.v7.n2.a9. Google Scholar

[15]

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

[16]

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

[17]

K. Itô and M. Nisio, On stationary solutions of a stochastic differential equation, J. Math. Kyoto Univ., 4 (1964), 1-75. doi: 10.1215/kjm/1250524705. Google Scholar

[18]

S. Méléard, Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models, Lectures Notes in Math., 1627 (1995), 42-95. Google Scholar

[19] S. Meyn and R. Tweedie, Markov Chains and Stochastic Stability, Springer-Verlag, 1993. Google Scholar
[20]

L. Pédèches, Exponential ergodicity for a class of non-Markovian stochastic processes, Preprint.Google Scholar

[21]

M. Scheutzow, Qualitative behaviour of stochastic delay equations with a bounded memory, Stochastics, 12 (1984), 41-80. doi: 10.1080/17442508408833294. Google Scholar

[22]

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

[23]

A. Sznitman, Topics in propagation of chaos, Lectures Notes in Math., 1464 (1991), 165-251. Google Scholar

[24]

T. TonN. Linh and A. Yagi, Flocking and non-flocking behavior in a stochastic Cucker-Smale system, Anal. Appl. (Singap.), 12 (2014), 63-73. doi: 10.1142/S0219530513500255. Google Scholar

Figure 1.  Case $\gamma = 0.2$, with $D = 0.1$, $\lambda = 5$, $\beta = 2$ and $K_{\gamma} = 8.3$
Figure 2.  Case $\gamma = 2$, with $D = 0.1$, $\lambda = 5$, $\beta = 2$ and $K_{\gamma} = 17$
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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 & Applied Analysis, 2018, 17 (2) : 505-538. doi: 10.3934/cpaa.2018028

[8]

Young-Pil Choi, Samir Salem. Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition. Kinetic & Related Models, 2019, 12 (3) : 573-592. doi: 10.3934/krm.2019023

[9]

Young-Pil Choi, Seung-Yeal Ha, Jeongho Kim. Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication. Networks & Heterogeneous Media, 2018, 13 (3) : 379-407. doi: 10.3934/nhm.2018017

[10]

Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034

[11]

Marco Caponigro, Massimo Fornasier, Benedetto Piccoli, Emmanuel Trélat. Sparse stabilization and optimal control of the Cucker-Smale model. Mathematical Control & Related Fields, 2013, 3 (4) : 447-466. doi: 10.3934/mcrf.2013.3.447

[12]

Young-Pil Choi, Jan Haskovec. Cucker-Smale model with normalized communication weights and time delay. Kinetic & Related Models, 2017, 10 (4) : 1011-1033. doi: 10.3934/krm.2017040

[13]

Seung-Yeal Ha, Dongnam Ko, Yinglong Zhang. Remarks on the critical coupling strength for the Cucker-Smale model with unit speed. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2763-2793. doi: 10.3934/dcds.2018116

[14]

Ioannis Markou. Collision-avoiding in the singular Cucker-Smale model with nonlinear velocity couplings. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5245-5260. doi: 10.3934/dcds.2018232

[15]

Agnieszka B. Malinowska, Tatiana Odzijewicz. Optimal control of the discrete-time fractional-order Cucker-Smale model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 347-357. doi: 10.3934/dcdsb.2018023

[16]

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

[17]

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 & Related Models, 2019, 12 (5) : 1045-1067. doi: 10.3934/krm.2019039

[18]

Jiu-Gang Dong, Seung-Yeal Ha, Doheon Kim. Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5569-5596. doi: 10.3934/dcdsb.2019072

[19]

Young-Pil Choi, Cristina Pignotti. Emergent behavior of Cucker-Smale model with normalized weights and distributed time delays. Networks & Heterogeneous Media, 2019, 14 (4) : 789-804. doi: 10.3934/nhm.2019032

[20]

Benedetto Piccoli, Francesco Rossi. Measure dynamics with Probability Vector Fields and sources. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6207-6230. doi: 10.3934/dcds.2019270

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (69)
  • HTML views (140)
  • Cited by (0)

Other articles
by authors

[Back to Top]