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 and Continuous Dynamical Systems, 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.

[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. 

[3]

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

[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.

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

[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. 

[8]

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

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

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

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

[21]

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

[22]

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

[23]

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

[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.

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.

[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. 

[3]

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

[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.

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

[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. 

[8]

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

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

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

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

[21]

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

[22]

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

[23]

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

[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.

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 $K_{\gamma} = 17$
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