June  2013, 6(2): 245-268. doi: 10.3934/krm.2013.6.245

Large deviations for the solution of a Kac-type kinetic equation

1. 

Dipartimento di Matematica, Università degli Studi di Pavia, Via Ferrata 1, 27100, Pavia, Italy

2. 

Dipartimento di Matematica, Politecnico di Milano, P.zza Leonardo da Vinci 32, 20133, Milanod, Italy

Received  October 2012 Revised  November 2012 Published  February 2013

The aim of this paper is to study large deviations for the self-similar solution of a Kac-type kinetic equation. Under the assumption that the initial condition belongs to the domain of normal attraction of a stable law of index $\alpha < 2$ and under suitable assumptions on the collisional kernel, precise asymptotic behavior of the large deviations probability is given.
Citation: Federico Bassetti, Lucia Ladelli. Large deviations for the solution of a Kac-type kinetic equation. Kinetic & Related Models, 2013, 6 (2) : 245-268. doi: 10.3934/krm.2013.6.245
References:
[1]

K. B. Athreya and P. E. Ney, "Branching Processes,", Reprint of the 1972 original, (1972).   Google Scholar

[2]

F. Bassetti and L. Ladelli, Self similar solutions in one-dimensional kinetic models: A probabilistic view,, Ann. Appl. Prob., 22 (2012), 1928.   Google Scholar

[3]

F. Bassetti, L. Ladelli and D. Matthes, Central limit theorem for a class of one-dimensional kinetic equations,, Probab. Theory Related Fields, 150 (2011), 77.  doi: 10.1007/s00440-010-0269-8.  Google Scholar

[4]

F. Bassetti, L. Ladelli and E. Regazzini, Probabilistic study of the speed of approach to equilibrium for an inelastic Kac model,, J. Stat. Phys., 133 (2008), 683.  doi: 10.1007/s10955-008-9630-z.  Google Scholar

[5]

F. Bassetti, L. Ladelli and G. Toscani, Kinetic models with randomly perturbed binary collisions,, J. Stat. Phys., 142 (2011), 686.  doi: 10.1007/s10955-011-0136-8.  Google Scholar

[6]

F. Bassetti and E. Perversi, Speed of convergence to equilibrium in Wasserstein metrics for Kac's like kinetic equations,, Electron. J. Probab., 18 (2013), 1.   Google Scholar

[7]

B. Basu, B. K. Chackabarti, S. R. Chackavart and K. Gangopadhyay, eds., "Econophysics & Economics of Games, Social Choices and Quantitative Techniques,", Springer-Verlag, (2010).   Google Scholar

[8]

D. Ben-Avraham, E. Ben-Naim, K. Lindenberg and A. Rosas, Self-similarity in random collision processes,, Phys. Rev. E, 68 (2003).   Google Scholar

[9]

A. V. Bobylev and C. Cercignani, Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions,, J. Statist. Phys., 110 (2003), 333.  doi: 10.1023/A:1021031031038.  Google Scholar

[10]

A. V. Bobylev, C. Cercignani and I. M. Gamba, Generalized kinetic Maxwell type models of granular gases,, in, 1937 (2008), 23.  doi: 10.1007/978-3-540-78277-3_2.  Google Scholar

[11]

A. V. Bobylev, C. Cercignani and I. M. Gamba, On the self-similar asymptotics for generalized nonlinear kinetic Maxwell models,, Comm. Math. Phys., 291 (2009), 599.  doi: 10.1007/s00220-009-0876-3.  Google Scholar

[12]

A. Dembo and O. Zeitouni, "Large Deviations Techniques and Applications,", Second edition, (1998).   Google Scholar

[13]

E. Dolera, E. Gabetta and E. Regazzini, Reaching the best possible rate of convergence to equilibrium for solutions of Kac's equation via central limit theorem,, Ann. Appl. Probab., 19 (2009), 186.  doi: 10.1214/08-AAP538.  Google Scholar

[14]

E. Dolera and E. Regazzini, The role of the central limit theorem in discovering sharp rates of convergence to equilibrium for the solution of the Kac equation,, Ann. Appl. Probab., 20 (2010), 430.  doi: 10.1214/09-AAP623.  Google Scholar

[15]

D. Duffie, G. Giroux and G. Manso, Information percolation,, American Economic Journal: Microeconomics, 2 (2010), 100.   Google Scholar

[16]

B. Fristedt and L. Gray, "A Modern Approach to Probability Theory,", Probability and its Applications, (1997).   Google Scholar

[17]

E. Gabetta and E. Regazzini, Central limit theorem for the solution of the Kac equation,, Ann. Appl. Probab., 18 (2008), 2320.  doi: 10.1214/08-AAP524.  Google Scholar

[18]

E. Gabetta and E. Regazzini, Central limit theorem for the solution of the Kac equation: Speed of approach to equilibrium in weak metrics,, Probab. Theory Related Fields, 146 (2010), 451.  doi: 10.1007/s00440-008-0196-0.  Google Scholar

[19]

C. C. Heyde, On large deviation problems for sums of random variables which are not attracted to the normal law,, Ann. Math. Statist., 38 (1967), 1575.   Google Scholar

[20]

C. C. Heyde, A contribution to the theory of large deviations for sums of independent random variables,, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 7 (1967), 303.   Google Scholar

[21]

C. C. Heyde, On large deviation probabilities in the case of attraction to a non-normal stable law,, Sankhyā Ser. A, 30 (1968), 253.   Google Scholar

[22]

I. A. Ibragimov and Yu. V. Linnik, "Independent and Stationary Sequences of Random Variables,", Wolters-Noordhoff Publishing, (1971).   Google Scholar

[23]

M. Kac, Foundations of kinetic theory,, in, (1956), 1954.   Google Scholar

[24]

Z. Kielek, An application of convolution iterates to evolution equation in Banach space,, Univ. Iagel. Acta Math., 27 (1988), 247.   Google Scholar

[25]

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

[26]

H. P. McKean, Jr., Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas,, Arch. Rational Mech. Anal., 21 (1966), 343.   Google Scholar

[27]

L. Pareschi and G. Toscani, Self-similarity and power-like tails in nonconservative kinetic models,, J. Statist. Phys., 124 (2006), 747.  doi: 10.1007/s10955-006-9025-y.  Google Scholar

[28]

M. Patriarca, E. Heinsalu and A. Chakraborti, Basic kinetic wealth-exchange models: Common features and open problems,, Eur. Phys. J. B, 73 (2010), 145.   Google Scholar

[29]

A. Pulvirenti and G. Toscani, Asymptotic properties of the inelastic Kac model,, J. Statist. Phys., 114 (2004), 1453.  doi: 10.1023/B:JOSS.0000013964.98706.00.  Google Scholar

[30]

V. Vinogradov, "Refined Large Deviation Limit Theorems,", Pitman Research Notes in Mathematics Series, 315 (1994).   Google Scholar

[31]

V. M. Yakovenko, Statistical mechanics approach to econophysics,, in, ().   Google Scholar

show all references

References:
[1]

K. B. Athreya and P. E. Ney, "Branching Processes,", Reprint of the 1972 original, (1972).   Google Scholar

[2]

F. Bassetti and L. Ladelli, Self similar solutions in one-dimensional kinetic models: A probabilistic view,, Ann. Appl. Prob., 22 (2012), 1928.   Google Scholar

[3]

F. Bassetti, L. Ladelli and D. Matthes, Central limit theorem for a class of one-dimensional kinetic equations,, Probab. Theory Related Fields, 150 (2011), 77.  doi: 10.1007/s00440-010-0269-8.  Google Scholar

[4]

F. Bassetti, L. Ladelli and E. Regazzini, Probabilistic study of the speed of approach to equilibrium for an inelastic Kac model,, J. Stat. Phys., 133 (2008), 683.  doi: 10.1007/s10955-008-9630-z.  Google Scholar

[5]

F. Bassetti, L. Ladelli and G. Toscani, Kinetic models with randomly perturbed binary collisions,, J. Stat. Phys., 142 (2011), 686.  doi: 10.1007/s10955-011-0136-8.  Google Scholar

[6]

F. Bassetti and E. Perversi, Speed of convergence to equilibrium in Wasserstein metrics for Kac's like kinetic equations,, Electron. J. Probab., 18 (2013), 1.   Google Scholar

[7]

B. Basu, B. K. Chackabarti, S. R. Chackavart and K. Gangopadhyay, eds., "Econophysics & Economics of Games, Social Choices and Quantitative Techniques,", Springer-Verlag, (2010).   Google Scholar

[8]

D. Ben-Avraham, E. Ben-Naim, K. Lindenberg and A. Rosas, Self-similarity in random collision processes,, Phys. Rev. E, 68 (2003).   Google Scholar

[9]

A. V. Bobylev and C. Cercignani, Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions,, J. Statist. Phys., 110 (2003), 333.  doi: 10.1023/A:1021031031038.  Google Scholar

[10]

A. V. Bobylev, C. Cercignani and I. M. Gamba, Generalized kinetic Maxwell type models of granular gases,, in, 1937 (2008), 23.  doi: 10.1007/978-3-540-78277-3_2.  Google Scholar

[11]

A. V. Bobylev, C. Cercignani and I. M. Gamba, On the self-similar asymptotics for generalized nonlinear kinetic Maxwell models,, Comm. Math. Phys., 291 (2009), 599.  doi: 10.1007/s00220-009-0876-3.  Google Scholar

[12]

A. Dembo and O. Zeitouni, "Large Deviations Techniques and Applications,", Second edition, (1998).   Google Scholar

[13]

E. Dolera, E. Gabetta and E. Regazzini, Reaching the best possible rate of convergence to equilibrium for solutions of Kac's equation via central limit theorem,, Ann. Appl. Probab., 19 (2009), 186.  doi: 10.1214/08-AAP538.  Google Scholar

[14]

E. Dolera and E. Regazzini, The role of the central limit theorem in discovering sharp rates of convergence to equilibrium for the solution of the Kac equation,, Ann. Appl. Probab., 20 (2010), 430.  doi: 10.1214/09-AAP623.  Google Scholar

[15]

D. Duffie, G. Giroux and G. Manso, Information percolation,, American Economic Journal: Microeconomics, 2 (2010), 100.   Google Scholar

[16]

B. Fristedt and L. Gray, "A Modern Approach to Probability Theory,", Probability and its Applications, (1997).   Google Scholar

[17]

E. Gabetta and E. Regazzini, Central limit theorem for the solution of the Kac equation,, Ann. Appl. Probab., 18 (2008), 2320.  doi: 10.1214/08-AAP524.  Google Scholar

[18]

E. Gabetta and E. Regazzini, Central limit theorem for the solution of the Kac equation: Speed of approach to equilibrium in weak metrics,, Probab. Theory Related Fields, 146 (2010), 451.  doi: 10.1007/s00440-008-0196-0.  Google Scholar

[19]

C. C. Heyde, On large deviation problems for sums of random variables which are not attracted to the normal law,, Ann. Math. Statist., 38 (1967), 1575.   Google Scholar

[20]

C. C. Heyde, A contribution to the theory of large deviations for sums of independent random variables,, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 7 (1967), 303.   Google Scholar

[21]

C. C. Heyde, On large deviation probabilities in the case of attraction to a non-normal stable law,, Sankhyā Ser. A, 30 (1968), 253.   Google Scholar

[22]

I. A. Ibragimov and Yu. V. Linnik, "Independent and Stationary Sequences of Random Variables,", Wolters-Noordhoff Publishing, (1971).   Google Scholar

[23]

M. Kac, Foundations of kinetic theory,, in, (1956), 1954.   Google Scholar

[24]

Z. Kielek, An application of convolution iterates to evolution equation in Banach space,, Univ. Iagel. Acta Math., 27 (1988), 247.   Google Scholar

[25]

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

[26]

H. P. McKean, Jr., Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas,, Arch. Rational Mech. Anal., 21 (1966), 343.   Google Scholar

[27]

L. Pareschi and G. Toscani, Self-similarity and power-like tails in nonconservative kinetic models,, J. Statist. Phys., 124 (2006), 747.  doi: 10.1007/s10955-006-9025-y.  Google Scholar

[28]

M. Patriarca, E. Heinsalu and A. Chakraborti, Basic kinetic wealth-exchange models: Common features and open problems,, Eur. Phys. J. B, 73 (2010), 145.   Google Scholar

[29]

A. Pulvirenti and G. Toscani, Asymptotic properties of the inelastic Kac model,, J. Statist. Phys., 114 (2004), 1453.  doi: 10.1023/B:JOSS.0000013964.98706.00.  Google Scholar

[30]

V. Vinogradov, "Refined Large Deviation Limit Theorems,", Pitman Research Notes in Mathematics Series, 315 (1994).   Google Scholar

[31]

V. M. Yakovenko, Statistical mechanics approach to econophysics,, in, ().   Google Scholar

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