Abstract
This paper is part of our efforts to show how direct application of probabilistic methods, pertaining to central limit general theory, can enlighten us about the convergence to equilibrium of the solutions of the Kac equation. Here, we consider convergence with respect to the following metrics: Kolmogorov’s uniform metric; 1 and 2 Gini’s dissimilarity indices (widely known as 1 and 2 Wasserstein metrics); χ-weighted metrics. Our main results provide new bounds, or improvements on already well-known ones, for the corresponding distances between the solution of the Kac equation and the limiting Gaussian (Maxwellian) distribution. The study is conducted both under the necessary assumption that initial data have finite energy, without assuming existence of moments of order greater than 2, and under the condition that the (2 + δ)-moment of the initial distribution is finite for some δ > 0.
Article PDF
Similar content being viewed by others
References
Bisi M., Carrillo J.A., Toscani G.: Contractive metrics for a Boltzmann equation for granular gases: diffusive equilibria. J. Statist. Phys. 118, 301–331 (2005)
Blair J.M., Edwards C.A., Johnson J.H.: Rational Chebyshev approximations for the inverse of the error function. Math. Comp. 30, 827–830 (1976)
Bobylev A.V.: Exact solutions of the nonlinear Boltzmann equation and the theory of relaxation of a Maxwell gas. Teoret. Mat. Fiz. 60, 280–310 (1984)
Bobylev A.V., Cercignani C., Toscani G.: Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials. J. Statist. Phys. 111, 403–417 (2003)
Carlitz L.: The inverse of the error function. Pacific J. Math. 13, 459–470 (1963)
Carlen E.A., Carvalho M.C., Gabetta E.: Central limit theorem for Maxwellian molecules and truncation of the Wild expansion. Comm. Pure Appl. Math. 53, 370–397 (2000)
Carlen E.A., Carvalho M.C., Gabetta E.: On the relation between rates of relaxation and convergence of Wild sums for solutions of the Kac equation. J. Funct. Anal. 220, 362–387 (2005)
Carlen E., Gabetta E., Regazzini E.: Probabilistic investigations on the explosion of solutions of the Kac equation with infinite energy initial distribution. J. Appl. Probab. 45, 95–106 (2008)
Carlen E.A., Gabetta E., Toscani G.: Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas. Comm. Math. Phys. 199, 521–546 (1999)
Chen L.H.Y., Shao Q.: A non-uniform Berry–Esseen bound via Stein’s method. Probab. Theory Relat. Fields 120, 236–254 (2001)
Cambanis S., Simons G., Stout W.: Inequalities for Ek(X,Y) when the marginals are fixed. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 36, 285–294 (1976)
Chow, Y.S., Teicher, H.: Probability theory. Independence, interchangeability, martingales, 3rd edn. Springer Texts in Statistics-Springer, New York (1997)
Dall’Aglio G.: Sugli estremi dei momenti delle funzioni di ripartizioni doppie. Ann. Scuola Norm. Sup. Pisa 3, 35–74 (1956)
Dolera, E., Gabetta, E., Regazzini, E.: Reaching the best possible rate of convergence to equilibrium for solutions of Kac’s equation via central limit theorem. Ann. Appl. Probab. doi:10.1214/08-AAP538 (2009)
Dolera, E., Regazzini, E.: The role of the central limit theorem in discovering sharp rates of convergence to equilibrium for the solution of the Kac equation. Pubblicazione IMATI-CNR, 33 PV08/28/0
Feller W.: On the Berry–Esseen theorem. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 10, 261–268 (1968)
Gabetta E., Regazzini E.: Some new results for McKean’s graphs with applications to Kac’s equation. J. Statist. Phys. 125, 947–974 (2006)
Gabetta E., Regazzini E.: Central limit theorem for the solution of the Kac equation. Ann. Appl. Probab. 18, 2320–2336 (2008)
Gabetta E., Toscani G., Wennberg B.: Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation. J. Statist. Phys. 81, 901–934 (1995)
Gabetta, E., Toscani, G., Wennberg, B.: The Tanaka functional and exponential convergence for non-cut-off molecules. In: Proceedings of the Second International Workshop on Nonlinear Kinetic Theories and Mathematical Aspects of Hyperbolic Systems (Sanremo, 1994), Transport Theory and Stat. Phys., vol. 25, pp. 543–554 (1996)
Gini, C.: Di una misura della dissomiglianza tra due gruppi di quantità e delle sue applicazioni allo studio delle relazioni statistiche. Atti del R. Istituto Veneto di Scienze Lettere ed Arti (a.a. 1914–1915), vol. 74 (1914)
Kac, M.: Foundations of kinetic theory. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III, pp. 171–197. University of California Press, Berkeley (1956)
Kac M.: Probability and related topics in physical sciences. Interscience Publishers, New York (1959)
Kantorovich L.V.: On one effective method of solving certain classes of extremal problems (Russian). C. R. (Doklady) Acad. Sci. URSS 28, 212–215 (1940)
Kantorovich, L.V.: On mass transportation. (Russian) C. R. (Doklady) Acad. Sci. URSS 37, 227–229 (1942) (English translation in J. Math. Sci. 133(4), 1381–1382, 2006)
Kantorovich, L.V.: On a problem of Monge. (Russian) C. R. (Doklady) Acad. Sci. URSS 3, 225–226 (1948) (English translation in J. Math. Sci. 133(4), 1383, 2006)
McKean H.P. Jr: Speed of approach to equilibrium for Kac’s caricature of a Maxwellian gas. Arch. Rat. Mech. Anal. 21, 343–367 (1966)
McKean H.P. Jr: An exponential formula for solving Boltzmann’s equation for a Maxwellian gas. J. Comb. Theory 2, 358–382 (1967)
Magnus, W., Oberhettinger, F., Soni, R.P.: Formulas and theorems for the special functions of mathematical physics. Third enlarged edition. Die Grundlehren der mathematischen Wissenschaften, Band 52. Springer, New York (1966)
Rachev S.T.: Probability metrics and the stability of stochastic models. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons Ltd., Chichester (1991)
Stroock D.W.: Probability theory, an analytic view. Cambridge University Press, Cambridge (1993)
Tanaka H.: An inequality for a functional of probability distributions and its application to Kac’s one-dimensional model of a Maxwellian gas. Z. Wahrsch. Verw. Gebiete 27, 47–52 (1973)
Tanaka H.: Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrsch. Verw. Gebiete 46, 67–105 (1978)
Tricomi, F.G.: Funzioni ipergeometriche confluenti. Edizioni Cremonese, Roma (1954)
Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Friedlander, S., Serre, D. (eds.) Handbook of mathematical fluid dynamics, vol. I, pp. 71–305. North-Holland, Amsterdam (2002)
Wasserstein L.N.: Markov processes over denumerable products of spaces describing large systems of automata. Problemy Peredači Informacii 5, 64–72 (1969)
Wild E.: On Boltzmann’s equation in the kinetic theory of gases. Proc. Cambridge Philos. Soc. 47, 602–609 (1951)
Zolotarev, V.M.: Modern theory of summation of random variables. Modern Probability and Statistics. VSP, Utrecht (1997)
Author information
Authors and Affiliations
Corresponding author
Additional information
E. Gabetta has been supported by MIUR, grant 2006/015821 and E. Regazzini by MIUR, grant 2006/134525.
Rights and permissions
About this article
Cite this article
Gabetta, E., Regazzini, E. Central limit theorems for solutions of the Kac equation: speed of approach to equilibrium in weak metrics. Probab. Theory Relat. Fields 146, 451–480 (2010). https://doi.org/10.1007/s00440-008-0196-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-008-0196-0
Keywords
- Boltzmann (Kac) equation
- Berry–Esseen inequality
- Central limit theorem
- Gini’s (Tanaka, Wasserstein) metrics
- Kolmogorov’s metric
- χ-Weighted metrics