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Invariant measures for a stochastic Fokker-Planck equation
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On a Fokker-Planck equation for wealth distribution
| 1. | Department of Mathematics, University of Pavia, Pavia, Italy |
| 2. | Department of Mathematics, University of Pavia, and IMATI-CNR, Pavia, Italy |
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 [
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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [6] |
A. Chakraborti,
Distributions of money in models of market economy, Int. J. Modern Phys. C, 13 (2002), 1315-1321.
doi: 10.1142/S0129183102003905. |
| [7] |
A. Chakraborti and B. K. Chakrabarti,
Statistical mechanics of money: Effects of saving propensity, Eur. Phys. J. B, 17 (2000), 167-170.
|
| [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. |
| [9] |
H. Chernoff,
A note on an inequality involving the normal distribution, Ann. Probab., 9 (1981), 533-535.
doi: 10.1214/aop/1176994428. |
| [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. |
| [11] |
B. Düring, D. Matthes and G. Toscani, Kinetic Equations modelling Wealth Redistribution: A comparison of approaches, Phys. Rev. E, 78 (2008), 056103, 12pp. |
| [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.
|
| [13] |
W. Feller,
Two singular diffusion problems, Ann. Math., 54 (1951), 173-182.
doi: 10.2307/1969318. |
| [14] |
W. Feller,
An Introduction to Probability Theory and Its Applications, Vol. Ⅰ. John Wiley & Sons Inc., New York, 1968. |
| [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. |
| [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. |
| [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. |
| [18] |
C. A. Klaassen,
On an inequality of Chernoff, Ann. Probability, 13 (1985), 966-974.
doi: 10.1214/aop/1176992917. |
| [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. |
| [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. |
| [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. |
| [22] |
L. Pareschi and G. Toscani, Interacting Multiagent Systems. Kinetic Equations & Monte Carlo Methods, Oxford University Press, Oxford, 2013.
|
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [6] |
A. Chakraborti,
Distributions of money in models of market economy, Int. J. Modern Phys. C, 13 (2002), 1315-1321.
doi: 10.1142/S0129183102003905. |
| [7] |
A. Chakraborti and B. K. Chakrabarti,
Statistical mechanics of money: Effects of saving propensity, Eur. Phys. J. B, 17 (2000), 167-170.
|
| [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. |
| [9] |
H. Chernoff,
A note on an inequality involving the normal distribution, Ann. Probab., 9 (1981), 533-535.
doi: 10.1214/aop/1176994428. |
| [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. |
| [11] |
B. Düring, D. Matthes and G. Toscani, Kinetic Equations modelling Wealth Redistribution: A comparison of approaches, Phys. Rev. E, 78 (2008), 056103, 12pp. |
| [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.
|
| [13] |
W. Feller,
Two singular diffusion problems, Ann. Math., 54 (1951), 173-182.
doi: 10.2307/1969318. |
| [14] |
W. Feller,
An Introduction to Probability Theory and Its Applications, Vol. Ⅰ. John Wiley & Sons Inc., New York, 1968. |
| [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. |
| [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. |
| [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. |
| [18] |
C. A. Klaassen,
On an inequality of Chernoff, Ann. Probability, 13 (1985), 966-974.
doi: 10.1214/aop/1176992917. |
| [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. |
| [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. |
| [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. |
| [22] |
L. Pareschi and G. Toscani, Interacting Multiagent Systems. Kinetic Equations & Monte Carlo Methods, Oxford University Press, Oxford, 2013.
|
| [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. |
| [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. |
| [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. |
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