# American Institute of Mathematical Sciences

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 and 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, Dover Publications, Inc., Mineola, NY, 2004. [2] F. Bassetti and L. Ladelli, Self similar solutions in one-dimensional kinetic models: A probabilistic view, Ann. Appl. Prob., 22 (2012), 1928-1961. [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), 77-109. doi: 10.1007/s00440-010-0269-8. [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), 683-710. doi: 10.1007/s10955-008-9630-z. [5] F. Bassetti, L. Ladelli and G. Toscani, Kinetic models with randomly perturbed binary collisions, J. Stat. Phys., 142 (2011), 686-709. doi: 10.1007/s10955-011-0136-8. [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), 1-35. [7] B. Basu, B. K. Chackabarti, S. R. Chackavart and K. Gangopadhyay, eds., "Econophysics & Economics of Games, Social Choices and Quantitative Techniques," Springer-Verlag, Milan, 2010. [8] D. Ben-Avraham, E. Ben-Naim, K. Lindenberg and A. Rosas, Self-similarity in random collision processes, Phys. Rev. E, 68 (2003). [9] A. V. Bobylev and C. Cercignani, Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions, J. Statist. Phys., 110 (2003), 333-375. doi: 10.1023/A:1021031031038. [10] A. V. Bobylev, C. Cercignani and I. M. Gamba, Generalized kinetic Maxwell type models of granular gases, in "Mathematical Models of Granular Matter" (eds. G. Capriz, P. Giovine and P. M. Mariano), Lecture Notes in Mathematics, 1937, Springer, Berlin, (2008), 23-57. doi: 10.1007/978-3-540-78277-3_2. [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), 599-644. doi: 10.1007/s00220-009-0876-3. [12] A. Dembo and O. Zeitouni, "Large Deviations Techniques and Applications," Second edition, Applications of Mathematics (New York), Springer-Verlag, New York, 1998. [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), 186-209. doi: 10.1214/08-AAP538. [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), 430-461. doi: 10.1214/09-AAP623. [15] D. Duffie, G. Giroux and G. Manso, Information percolation, American Economic Journal: Microeconomics, 2 (2010), 100-111. [16] B. Fristedt and L. Gray, "A Modern Approach to Probability Theory," Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1997. [17] E. Gabetta and E. Regazzini, Central limit theorem for the solution of the Kac equation, Ann. Appl. Probab., 18 (2008), 2320-2336. doi: 10.1214/08-AAP524. [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), 451-480. doi: 10.1007/s00440-008-0196-0. [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), 1575-1578. [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), 303-308. [21] C. C. Heyde, On large deviation probabilities in the case of attraction to a non-normal stable law, Sankhyā Ser. A, 30 (1968), 253-258. [22] I. A. Ibragimov and Yu. V. Linnik, "Independent and Stationary Sequences of Random Variables," Wolters-Noordhoff Publishing, Groningen, 1971. [23] M. Kac, Foundations of kinetic theory, in "Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955," Vol. III, University of California Press, Berkeley and Los Angeles, (1956), 171-197. [24] Z. Kielek, An application of convolution iterates to evolution equation in Banach space, Univ. Iagel. Acta Math., 27 (1988), 247-257. [25] 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. [26] H. P. McKean, Jr., Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas, Arch. Rational Mech. Anal., 21 (1966), 343-367. [27] L. Pareschi and G. Toscani, Self-similarity and power-like tails in nonconservative kinetic models, J. Statist. Phys., 124 (2006), 747-779. doi: 10.1007/s10955-006-9025-y. [28] M. Patriarca, E. Heinsalu and A. Chakraborti, Basic kinetic wealth-exchange models: Common features and open problems, Eur. Phys. J. B, 73 (2010), 145-153. [29] A. Pulvirenti and G. Toscani, Asymptotic properties of the inelastic Kac model, J. Statist. Phys., 114 (2004), 1453-1480. doi: 10.1023/B:JOSS.0000013964.98706.00. [30] V. Vinogradov, "Refined Large Deviation Limit Theorems," Pitman Research Notes in Mathematics Series, 315, Longman Scientific & Technical, Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1994. [31] V. M. Yakovenko, Statistical mechanics approach to econophysics,, in, ().

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##### References:
 [1] K. B. Athreya and P. E. Ney, "Branching Processes," Reprint of the 1972 original, Dover Publications, Inc., Mineola, NY, 2004. [2] F. Bassetti and L. Ladelli, Self similar solutions in one-dimensional kinetic models: A probabilistic view, Ann. Appl. Prob., 22 (2012), 1928-1961. [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), 77-109. doi: 10.1007/s00440-010-0269-8. [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), 683-710. doi: 10.1007/s10955-008-9630-z. [5] F. Bassetti, L. Ladelli and G. Toscani, Kinetic models with randomly perturbed binary collisions, J. Stat. Phys., 142 (2011), 686-709. doi: 10.1007/s10955-011-0136-8. [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), 1-35. [7] B. Basu, B. K. Chackabarti, S. R. Chackavart and K. Gangopadhyay, eds., "Econophysics & Economics of Games, Social Choices and Quantitative Techniques," Springer-Verlag, Milan, 2010. [8] D. Ben-Avraham, E. Ben-Naim, K. Lindenberg and A. Rosas, Self-similarity in random collision processes, Phys. Rev. E, 68 (2003). [9] A. V. Bobylev and C. Cercignani, Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions, J. Statist. Phys., 110 (2003), 333-375. doi: 10.1023/A:1021031031038. [10] A. V. Bobylev, C. Cercignani and I. M. Gamba, Generalized kinetic Maxwell type models of granular gases, in "Mathematical Models of Granular Matter" (eds. G. Capriz, P. Giovine and P. M. Mariano), Lecture Notes in Mathematics, 1937, Springer, Berlin, (2008), 23-57. doi: 10.1007/978-3-540-78277-3_2. [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), 599-644. doi: 10.1007/s00220-009-0876-3. [12] A. Dembo and O. Zeitouni, "Large Deviations Techniques and Applications," Second edition, Applications of Mathematics (New York), Springer-Verlag, New York, 1998. [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), 186-209. doi: 10.1214/08-AAP538. [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), 430-461. doi: 10.1214/09-AAP623. [15] D. Duffie, G. Giroux and G. Manso, Information percolation, American Economic Journal: Microeconomics, 2 (2010), 100-111. [16] B. Fristedt and L. Gray, "A Modern Approach to Probability Theory," Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1997. [17] E. Gabetta and E. Regazzini, Central limit theorem for the solution of the Kac equation, Ann. Appl. Probab., 18 (2008), 2320-2336. doi: 10.1214/08-AAP524. [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), 451-480. doi: 10.1007/s00440-008-0196-0. [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), 1575-1578. [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), 303-308. [21] C. C. Heyde, On large deviation probabilities in the case of attraction to a non-normal stable law, Sankhyā Ser. A, 30 (1968), 253-258. [22] I. A. Ibragimov and Yu. V. Linnik, "Independent and Stationary Sequences of Random Variables," Wolters-Noordhoff Publishing, Groningen, 1971. [23] M. Kac, Foundations of kinetic theory, in "Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955," Vol. III, University of California Press, Berkeley and Los Angeles, (1956), 171-197. [24] Z. Kielek, An application of convolution iterates to evolution equation in Banach space, Univ. Iagel. Acta Math., 27 (1988), 247-257. [25] 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. [26] H. P. McKean, Jr., Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas, Arch. Rational Mech. Anal., 21 (1966), 343-367. [27] L. Pareschi and G. Toscani, Self-similarity and power-like tails in nonconservative kinetic models, J. Statist. Phys., 124 (2006), 747-779. doi: 10.1007/s10955-006-9025-y. [28] M. Patriarca, E. Heinsalu and A. Chakraborti, Basic kinetic wealth-exchange models: Common features and open problems, Eur. Phys. J. B, 73 (2010), 145-153. [29] A. Pulvirenti and G. Toscani, Asymptotic properties of the inelastic Kac model, J. Statist. Phys., 114 (2004), 1453-1480. doi: 10.1023/B:JOSS.0000013964.98706.00. [30] V. Vinogradov, "Refined Large Deviation Limit Theorems," Pitman Research Notes in Mathematics Series, 315, Longman Scientific & Technical, Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1994. [31] V. M. Yakovenko, Statistical mechanics approach to econophysics,, in, ().
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