February  2020, 40(2): 683-707. doi: 10.3934/dcds.2020057

On mean field systems with multi-classes

1. 

Department of Applied Mathematics, Faculty of Applied Science, Ho Chi Minh City University of Technology, Vietnam National University Ho Chi Minh City, 268 Ly Thuong Kiet, District 10, Ho Chi Minh City, Vietnam

2. 

Department of Mathematics, University of Puerto Rico, Rio Piedras campus, San Juan, PR 00925, Puerto Rico

3. 

Department of Mathematics, Mechanics and Informatics, Hanoi National University, 334 Nguyen Trai, Thanh Xuan, Hanoi Vietnam

* Corresponding author: Dung Tien Nguyen

Dedicated to Professor Gang George Yin on the occasion of his 65th birthday.

Received  May 2018 Revised  August 2019 Published  November 2019

This work focuses on stochastic systems of weakly interacting particles containing different populations represented by multi-classes. The dynamics of each particle depends not only on the empirical measure of the whole population but also on those of different populations. The limits of such systems as the number of particles tends to infinity are investigated. We establish the existence, uniqueness, and basic properties of solutions to the limiting McKean-Vlasov equations of these systems and then obtain the rate of convergence of the sequences of empirical measures associated with the systems to their limits in terms of the $ p^{\text{th}} $ Monge-Wasserstein distance.

Citation: Dung Tien Nguyen, Son Luu Nguyen, Nguyen Huu Du. On mean field systems with multi-classes. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 683-707. doi: 10.3934/dcds.2020057
References:
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A. BudhirajaP. Dupuis and M. Fischer, Large deviation properties of weakly interacting processes via weak convergence methods, Ann. Probab., 40 (2012), 74-102.  doi: 10.1214/10-AOP616.  Google Scholar

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A. Budhiraja and R. Wu, Some fluctuation results for weakly interacting multi-type particle systems, Stochastic Processes and their Applications, 126 (2016), 2253-2296.  doi: 10.1016/j.spa.2016.01.010.  Google Scholar

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R. CarmonaF. Delarue and D. Lacker, Mean field games with common noise, Ann. Probab., 44 (2016), 3740-3803.  doi: 10.1214/15-AOP1060.  Google Scholar

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P. ContucciI. Gallo and G. Menconi, Phase transitions in social sciences: Two-populations mean field theory, International Journal of Modern Physics B, 22 (2008), 2199-2212.   Google Scholar

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[24]

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[26]

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[27]

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[28]

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[29]

A. S. Sznitman, Topics in propagation of chaos, Ecole d'Eté de Probabilités de Saint-Flour XIX - 1989, Ed. by P. L. Hennequin. Berlin, Heidelberg: Springer Berlin Heidelberg, 1464 (1991), 165–251. doi: 10.1007/BFb0085169.  Google Scholar

[30]

C. Villani, Optimal transport: Old and new, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar

[31]

F. Xi and G. Yin, Asymptotic properties of a mean-field model with a continuous-state-dependent switching process, Journal of Applied Probability, 46 (2009), 221-243.  doi: 10.1239/jap/1238592126.  Google Scholar

show all references

References:
[1]

L. AndreisP. D. Pra and M. Fischer, McKean-Vlasov limit for interacting systems with simultaneous jumps, Stochastic Analysis and Applications, 36 (2018), 960-995.  doi: 10.1080/07362994.2018.1486202.  Google Scholar

[2]

J. Baladron, D. Fasoli, O. Faugeras and J. Touboul, Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons, The Journal of Mathematical Neuroscience, 2 (2012), Art. 10, 50 pp. doi: 10.1186/2190-8567-2-10.  Google Scholar

[3]

A. Bensoussan, J. Frehse and P. Yam, Mean Field Games and Mean Field Type Control Theory, Springer Briefs in Mathematics, Springer New York, 2013. doi: 10.1007/978-1-4614-8508-7.  Google Scholar

[4]

F. Bolley, Separability and completeness for the Wasserstein distance, Séminaire de Probabilités XLI (eds. C. Donati-Martin, M. Émery, A. Rouault, and C. Stricker), Berlin, Heidelberg: Springer Berlin Heidelberg, 1934 (2008), 371–377. doi: 10.1007/978-3-540-77913-1_17.  Google Scholar

[5]

V. S. Borkar and K. S. Kumar, McKean-Vlasov limit in portfolio optimization, Stochastic Analysis and Applications, 28 (2010), 884-906.  doi: 10.1080/07362994.2010.482836.  Google Scholar

[6]

A. BudhirajaP. Dupuis and M. Fischer, Large deviation properties of weakly interacting processes via weak convergence methods, Ann. Probab., 40 (2012), 74-102.  doi: 10.1214/10-AOP616.  Google Scholar

[7]

A. Budhiraja and R. Wu, Some fluctuation results for weakly interacting multi-type particle systems, Stochastic Processes and their Applications, 126 (2016), 2253-2296.  doi: 10.1016/j.spa.2016.01.010.  Google Scholar

[8]

R. CarmonaF. Delarue and D. Lacker, Mean field games with common noise, Ann. Probab., 44 (2016), 3740-3803.  doi: 10.1214/15-AOP1060.  Google Scholar

[9]

F. Collet, Macroscopic limit of a bipartite Curie-Weiss model: A dynamical approach, Journal of Statistical Physics, 157 (2014), 1301-1319.  doi: 10.1007/s10955-014-1105-9.  Google Scholar

[10]

P. ContucciI. Gallo and G. Menconi, Phase transitions in social sciences: Two-populations mean field theory, International Journal of Modern Physics B, 22 (2008), 2199-2212.   Google Scholar

[11]

D. A. Dawson, Critical dynamics and fluctuations for a mean-field model of cooperative behavior, Journal of Statistical Physics, 31 (1983), 29-85.  doi: 10.1007/BF01010922.  Google Scholar

[12]

D. A. Dawson and J. Vaillancourt, Stochastic McKean-Vlasov equations, Nonlinear Differential Equations and Applications NoDEA, 2 (1995), 199-229.  doi: 10.1007/BF01295311.  Google Scholar

[13]

D. A. Dawson and J. Gärtner, Large deviations from the Mckean-Vlasov limit for weakly interacting diffusions, Stochastics, 20 (1987), 247-308.  doi: 10.1080/17442508708833446.  Google Scholar

[14]

N. Fournier and A. Guillin, On the rate of convergence in Wasserstein distance of the empirical measure, Probability Theory and Related Fields, 162 (2015), 707-738.  doi: 10.1007/s00440-014-0583-7.  Google Scholar

[15]

T. A. HoangS. L. Nguyen and G. Yin, Near optimality and near equilibrium for controlled systems with wideband noise for hybrid systems, Dynamics of Continuous Discrete and Impulsive Systems, Series A: Mathematical Analysis, 23 (2016), 163-194.   Google Scholar

[16]

T. A. Hoang and G. Yin, Properties for a class of multi-type mean-field models, Communications in Information and Systems, 15 (2015), 489-519.  doi: 10.4310/CIS.2015.v15.n4.a4.  Google Scholar

[17]

M. Huang, Large-population LQG games involving a major player: The Nash certainty equivalence principle, SIAM Journal on Control and Optimization, 48 (2010), 3318-3353.  doi: 10.1137/080735370.  Google Scholar

[18]

M. HuangP. E. Caines and R. P. Malhamé, Social optima in mean field LQG control: Centralized and decentralized strategies, IEEE Trans. Automat. Control, 57 (2012), 1736-1751.  doi: 10.1109/TAC.2012.2183439.  Google Scholar

[19]

M. HuangR. P. Malhamé and P. E. Caines, Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Communications in Information & Systems, 6 (2006), 221-252.  doi: 10.4310/CIS.2006.v6.n3.a5.  Google Scholar

[20]

V. Kolokoltsov and M. Troeva, On the mean field games with common noise and the McKean-Vlasov SPDEs, Stochastic Analysis and Applications, 37 (2019), 522-549.  doi: 10.1080/07362994.2019.1592690.  Google Scholar

[21]

T. G. Kurtz and J. Xiong, Particle representations for a class of nonlinear SPDEs, Stochastic Processes and their Applications, 83 (1999), 103-126.  doi: 10.1016/S0304-4149(99)00024-1.  Google Scholar

[22]

J. M. Lasry and P. L. Lions, Jeux à champ moyen. I - Le cas stationnaire, Comptes Rendus Mathematique, 343 (2006), 619-625.  doi: 10.1016/j.crma.2006.09.019.  Google Scholar

[23]

S. Méléard, Convergence of the fluctuations for interacting diffusions with jumps associated with boltzmann equations, Stochastics and Stochastic Reports, 63 (1998), 195-225.  doi: 10.1080/17442509808834148.  Google Scholar

[24]

S. L. Nguyen and M. Huang, Linear-quadratic-Gaussian mixed games with continuum-parametrized minor players, SIAM Journal on Control and Optimization, 50 (2012), 2907-2937.  doi: 10.1137/110841217.  Google Scholar

[25]

S. L. Nguyen, G. Yin and T. A. Hoang, On laws of large numbers for systems with mean-field interactions and Markovian switching, Stochastic Processes and their Applications, In press (2019), Available from: https://doi.org/10.1016/j.spa.2019.02.014. Google Scholar

[26]

M. Nourian and P. E. Caines, $\epsilon$-Nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents, SIAM Journal on Control and Optimization, 51 (2013), 3302-3331.  doi: 10.1137/120889496.  Google Scholar

[27]

K. Oelschlager, A martingale approach to the law of large numbers for weakly interacting stochastic processes, The Annals of Probability, 12 (1984), 458-479.  doi: 10.1214/aop/1176993301.  Google Scholar

[28]

T. Shiga and H. Tanaka, Central limit theorem for a system of Markovian particles with mean field interactions, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 69 (1985), 439–459. doi: 10.1007/BF00532743.  Google Scholar

[29]

A. S. Sznitman, Topics in propagation of chaos, Ecole d'Eté de Probabilités de Saint-Flour XIX - 1989, Ed. by P. L. Hennequin. Berlin, Heidelberg: Springer Berlin Heidelberg, 1464 (1991), 165–251. doi: 10.1007/BFb0085169.  Google Scholar

[30]

C. Villani, Optimal transport: Old and new, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar

[31]

F. Xi and G. Yin, Asymptotic properties of a mean-field model with a continuous-state-dependent switching process, Journal of Applied Probability, 46 (2009), 221-243.  doi: 10.1239/jap/1238592126.  Google Scholar

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