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April  2018, 11(2): 337-355. doi: 10.3934/krm.2018016

On a Fokker-Planck equation for wealth distribution

Marco Torregrossa 1,, and  Giuseppe Toscani 2,

1. 

Department of Mathematics, University of Pavia, Pavia, Italy
2. 

Department of Mathematics, University of Pavia, and IMATI-CNR, Pavia, Italy

* Corresponding authorr: Marco Torregrossa

Received  January 2017 Revised  May 2017 Published  January 2018

Fund Project: This work has been written within the activities of the National Group of Mathematical Physics (GNFM) of INdAM (National Institute of High Mathematics), and partially supported by the MIUR-PRIN Grant 2015PA5MP7 "Calculus of Variations".

We study here a Fokker-Planck equation with variable coefficient of diffusion and boundary conditions which appears in the study of the wealth distribution in a multi-agent society [2, 10, 22]. In particular, we analyze the large-time behavior of the solution, by showing that convergence to the steady state can be obtained in various norms at different rates.

Keywords: Wealth distribution, kinetic theory, Fokker-Planck equations, large-time behavior.
Mathematics Subject Classification: Primary: 91D10; Secondary: 35Q84, 82B21, 94A17.
Citation: Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic & Related Models, 2018, 11 (2) : 337-355. doi: 10.3934/krm.2018016
References:
[1]

A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations, 26 (2001), 43-100.  doi: 10.1081/PDE-100002246.  Google Scholar

[2]

J. F. Bouchaud and M. Mézard, Wealth condensation in a simple model of economy, Physica A, 282 (2000), 536-545.  doi: 10.1016/S0378-4371(00)00205-3.  Google Scholar

[3]

M. J. Cáceres and G. Toscani, Kinetic approach to long time behavior of linearized fast diffusion equations, J. Statist. Phys., 128 (2007), 883-925.  doi: 10.1007/s10955-007-9329-6.  Google Scholar

[4]

J. A. Carrillo and G. Toscani, Contractive probability metrics and asymptotic behavior of dissipative kinetic equations, Riv. Mat. Univ. Parma (7), 6 (2007), 75-198.   Google Scholar

[5]

J. A. Carrillo, S. Cordier and G. Toscani, Over-populated tails for conservative-in-the-mean inelastic Maxwell models, Discr. Cont. Dynamical Syst. A, 24 (2009), 59-81.  doi: 10.3934/dcds.2009.24.59.  Google Scholar

[6]

A. Chakraborti, Distributions of money in models of market economy, Int. J. Modern Phys. C, 13 (2002), 1315-1321.  doi: 10.1142/S0129183102003905.  Google Scholar

[7]

A. Chakraborti and B. K. Chakrabarti, Statistical mechanics of money: Effects of saving propensity, Eur. Phys. J. B, 17 (2000), 167-170.   Google Scholar

[8]

A. Chatterjee, B. K. Chakrabarti and R. B. Stinchcombe, Master equation for a kinetic model of trading market and its analytic solution, Phys. Rev. E, 72 (2005), 026126.  doi: 10.1103/PhysRevE.72.026126.  Google Scholar

[9]

H. Chernoff, A note on an inequality involving the normal distribution, Ann. Probab., 9 (1981), 533-535.  doi: 10.1214/aop/1176994428.  Google Scholar

[10]

S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy, J. Statist. Phys., 120 (2005), 253-277.  doi: 10.1007/s10955-005-5456-0.  Google Scholar

[11]

B. Düring, D. Matthes and G. Toscani, Kinetic Equations modelling Wealth Redistribution: A comparison of approaches, Phys. Rev. E, 78 (2008), 056103, 12pp.  Google Scholar

[12]

B. Düring, D. Matthes and G. Toscani, A Boltzmann-type approach to the formation of wealth distribution curves, (Notes of the Porto Ercole School, June 2008), Riv. Mat. Univ. Parma, 1 (2009), 199-261.   Google Scholar

[13]

W. Feller, Two singular diffusion problems, Ann. Math., 54 (1951), 173-182.  doi: 10.2307/1969318.  Google Scholar

[14]

W. Feller, An Introduction to Probability Theory and Its Applications, Vol. Ⅰ. John Wiley & Sons Inc., New York, 1968.  Google Scholar

[15]

G. Furioli, A. Pulvirenti, E. Terraneo and G. Toscani, Fokker-Planck equations in the modelling of socio-economic phenomena, Math. Mod. Meth. Appl. Scie., 27 (2017), 115-158.  doi: 10.1142/S0218202517400048.  Google Scholar

[16]

G. Gabetta, G. Toscani and B. Wennberg, Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation, J. Statist. Phys., 81 (1995), 901-934.  doi: 10.1007/BF02179298.  Google Scholar

[17]

O. Johnson and A. Barron, Fisher information inequalities and the central limit theorem, Probab. Theory Related Fields, 129 (2004), 391-409.  doi: 10.1007/s00440-004-0344-0.  Google Scholar

[18]

C. A. Klaassen, On an inequality of Chernoff, Ann. Probability, 13 (1985), 966-974.  doi: 10.1214/aop/1176992917.  Google Scholar

[19]

C. Le Bris and P. L. Lions, Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients, Comm. Partial Differential Equations, 33 (2008), 1272-1317.  doi: 10.1080/03605300801970952.  Google Scholar

[20]

D. Matthes, A. Juengel and G. Toscani, Convex Sobolev inequalities derived from entropy dissipation, Arch. Rat. Mech. Anal., 199 (2011), 563-596.  doi: 10.1007/s00205-010-0331-9.  Google Scholar

[21]

D. Matthes and G. Toscani, On steady distributions of kinetic models of conservative economies, J. Statist. Phys., 130 (2008), 1087-1117.  doi: 10.1007/s10955-007-9462-2.  Google Scholar

[22] L. Pareschi and G. Toscani, Interacting Multiagent Systems. Kinetic Equations & Monte Carlo Methods, Oxford University Press, Oxford, 2013.   Google Scholar
[23]

V. Pareto, Cours d'Économie Politique, Tome Premier, Rouge Éd., Lausanne 1896; Tome second, Pichon Éd., Paris, 1897. doi: 10.3917/droz.paret.1964.01.  Google Scholar

[24]

G. Toscani, Entropy dissipation and the rate of convergence to equilibrium for the Fokker-Planck equation, Quart. Appl. Math., 57 (1999), 521-541.  doi: 10.1090/qam/1704435.  Google Scholar

[25]

G. Toscani and C. Villani, Probability Metrics and Uniqueness of the Solution to the Boltzmann Equation for a Maxwell Gas, J. Statist. Phys., 94 (1999), 619-637.  doi: 10.1023/A:1004589506756.  Google Scholar

show all references

References:
[1]

A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations, 26 (2001), 43-100.  doi: 10.1081/PDE-100002246.  Google Scholar

[2]

J. F. Bouchaud and M. Mézard, Wealth condensation in a simple model of economy, Physica A, 282 (2000), 536-545.  doi: 10.1016/S0378-4371(00)00205-3.  Google Scholar

[3]

M. J. Cáceres and G. Toscani, Kinetic approach to long time behavior of linearized fast diffusion equations, J. Statist. Phys., 128 (2007), 883-925.  doi: 10.1007/s10955-007-9329-6.  Google Scholar

[4]

J. A. Carrillo and G. Toscani, Contractive probability metrics and asymptotic behavior of dissipative kinetic equations, Riv. Mat. Univ. Parma (7), 6 (2007), 75-198.   Google Scholar

[5]

J. A. Carrillo, S. Cordier and G. Toscani, Over-populated tails for conservative-in-the-mean inelastic Maxwell models, Discr. Cont. Dynamical Syst. A, 24 (2009), 59-81.  doi: 10.3934/dcds.2009.24.59.  Google Scholar

[6]

A. Chakraborti, Distributions of money in models of market economy, Int. J. Modern Phys. C, 13 (2002), 1315-1321.  doi: 10.1142/S0129183102003905.  Google Scholar

[7]

A. Chakraborti and B. K. Chakrabarti, Statistical mechanics of money: Effects of saving propensity, Eur. Phys. J. B, 17 (2000), 167-170.   Google Scholar

[8]

A. Chatterjee, B. K. Chakrabarti and R. B. Stinchcombe, Master equation for a kinetic model of trading market and its analytic solution, Phys. Rev. E, 72 (2005), 026126.  doi: 10.1103/PhysRevE.72.026126.  Google Scholar

[9]

H. Chernoff, A note on an inequality involving the normal distribution, Ann. Probab., 9 (1981), 533-535.  doi: 10.1214/aop/1176994428.  Google Scholar

[10]

S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy, J. Statist. Phys., 120 (2005), 253-277.  doi: 10.1007/s10955-005-5456-0.  Google Scholar

[11]

B. Düring, D. Matthes and G. Toscani, Kinetic Equations modelling Wealth Redistribution: A comparison of approaches, Phys. Rev. E, 78 (2008), 056103, 12pp.  Google Scholar

[12]

B. Düring, D. Matthes and G. Toscani, A Boltzmann-type approach to the formation of wealth distribution curves, (Notes of the Porto Ercole School, June 2008), Riv. Mat. Univ. Parma, 1 (2009), 199-261.   Google Scholar

[13]

W. Feller, Two singular diffusion problems, Ann. Math., 54 (1951), 173-182.  doi: 10.2307/1969318.  Google Scholar

[14]

W. Feller, An Introduction to Probability Theory and Its Applications, Vol. Ⅰ. John Wiley & Sons Inc., New York, 1968.  Google Scholar

[15]

G. Furioli, A. Pulvirenti, E. Terraneo and G. Toscani, Fokker-Planck equations in the modelling of socio-economic phenomena, Math. Mod. Meth. Appl. Scie., 27 (2017), 115-158.  doi: 10.1142/S0218202517400048.  Google Scholar

[16]

G. Gabetta, G. Toscani and B. Wennberg, Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation, J. Statist. Phys., 81 (1995), 901-934.  doi: 10.1007/BF02179298.  Google Scholar

[17]

O. Johnson and A. Barron, Fisher information inequalities and the central limit theorem, Probab. Theory Related Fields, 129 (2004), 391-409.  doi: 10.1007/s00440-004-0344-0.  Google Scholar

[18]

C. A. Klaassen, On an inequality of Chernoff, Ann. Probability, 13 (1985), 966-974.  doi: 10.1214/aop/1176992917.  Google Scholar

[19]

C. Le Bris and P. L. Lions, Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients, Comm. Partial Differential Equations, 33 (2008), 1272-1317.  doi: 10.1080/03605300801970952.  Google Scholar

[20]

D. Matthes, A. Juengel and G. Toscani, Convex Sobolev inequalities derived from entropy dissipation, Arch. Rat. Mech. Anal., 199 (2011), 563-596.  doi: 10.1007/s00205-010-0331-9.  Google Scholar

[21]

D. Matthes and G. Toscani, On steady distributions of kinetic models of conservative economies, J. Statist. Phys., 130 (2008), 1087-1117.  doi: 10.1007/s10955-007-9462-2.  Google Scholar

[22] L. Pareschi and G. Toscani, Interacting Multiagent Systems. Kinetic Equations & Monte Carlo Methods, Oxford University Press, Oxford, 2013.   Google Scholar
[23]

V. Pareto, Cours d'Économie Politique, Tome Premier, Rouge Éd., Lausanne 1896; Tome second, Pichon Éd., Paris, 1897. doi: 10.3917/droz.paret.1964.01.  Google Scholar

[24]

G. Toscani, Entropy dissipation and the rate of convergence to equilibrium for the Fokker-Planck equation, Quart. Appl. Math., 57 (1999), 521-541.  doi: 10.1090/qam/1704435.  Google Scholar

[25]

G. Toscani and C. Villani, Probability Metrics and Uniqueness of the Solution to the Boltzmann Equation for a Maxwell Gas, J. Statist. Phys., 94 (1999), 619-637.  doi: 10.1023/A:1004589506756.  Google Scholar

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