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 and Continuous Dynamical Systems, 2020, 40 (2) : 683-707. doi: 10.3934/dcds.2020057
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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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. 

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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.

show all references

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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. 

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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.

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