Abstract
We introduce a class of kinetic-type equations on the real line, which constitute extensions of the classical Kac caricature. The collisional gain operators are defined by smoothing transformations with rather general properties. By establishing a connection to the central limit problem, we are able to prove long-time convergence of the equation’s solutions toward a limit distribution. For example, we prove that if the initial condition belongs to the domain of normal attraction of a certain stable law ν α, then the limit is a scale mixture of ν α. Under some additional assumptions, explicit exponential rates for the convergence to equilibrium in Wasserstein metrics are calculated, and strong convergence of the probability densities is shown.
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Ambrosio L., Gigli N., Savaré G.: Gradient flows: in metric spaces and in the space of probability measures. Lectures in Mathematics. Birkhäuser, Boston (2008)
Bassetti F., Ladelli L., Regazzini E.: Probabilistic study of the speed of approach to equilibrium for an inelastic Kac model. J. Stat. Phys. 133, 683–710 (2008)
Carlen E.A., Carvalho M.C., Gabetta E.: Central limit theorem for Maxwellian molecules and truncation of the Wild expansion. Commun. Pure Appl. Math. 53, 370–397 (2000)
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., Gabetta E., Toscani G.: Propagation of smoothness and the rate of exponential convergence to equilibrum for a spatially homogeneous Maxwellian gas. Commun. Math. Phys. 305, 521–546 (1999)
Carrillo J.A., Cordier S., Toscani G.: Over-populated tails for conservative-in-the-mean inelastic Maxwell models. Discret. Contin. Dyn. Syst. 24(1), 59–81 (2009)
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. 19, 186–209 (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. Ann. Appl. Probab. (2010). doi:10.1214/09-AAP623
Durrett R., Liggett T.: Fixed points of the smoothing transformation. Z. Wahrsch. Verw. Gebiete 64, 275–301 (1983)
Fristedt B., Gray L.: A Modern Approach to Probability Theory. Birkhäuser, Boston (1997)
Gabetta E., Regazzini E.: Some new results for McKean’s graphs with applications to Kac’s equation. J. Stat. 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., Regazzini E.: Central limit theorem for the solution of the Kac equation: Speed of approach to equilibrium in weak metrics. Probab. Theory Relat. Fields 146, 451–480 (2010)
Ibragimov I.A., Linnik Y.V.: Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff Publishing, Groningen (1971)
Liu Q.: Fixed points of a generalized smoothing transformation and applications to the branching random walk. Adv. Appl. Probab. 30, 85–112 (1998)
Liu Q.: On generalized multiplicative cascades. Stoch. Process. Appl. 86, 263–286 (2000)
Matthes D., Toscani G.: On steady distributions of kinetic models of conservative economies. J. Stat. Phys. 130, 1087–1117 (2008)
McKean H.P. Jr: Speed of approach to equilibrium for Kac’s caricature of a Maxwellian gas. Arch. Ration. Mech. Anal. 21, 343–367 (1966)
McKean H.P. Jr: An exponential formula for solving Boltmann’s equation for a Maxwellian gas. J. Combin. Theory 2, 358–382 (1967)
Prudnikov A.P., Brychkov Yu.A., Marichev O.I.: Integrals and series, vol. 1. Elementary functions. Gordon & Breach Science Publishers, New York (1986)
Pulvirenti A., Toscani G.: Asymptotic properties of the inelastic Kac model. J. Stat. Phys. 114, 1453–1480 (2004)
Rachev S.T.: Probability Metrics and the Stability of Stochastic Models. Wiley, New York (1991)
Sznitman A.S.: Équations de type de Boltzmann, spatialement homogènes. Z. Wahrsch. Verw. Gebiete 66, 559–592 (1986)
von Bahr B., Esseen C.G.: Inequalities for the rth absolute moment of a sum of random variables, 1 ≤ r ≤ 2. Ann. Math. Stat. 36, 299–303 (1965)
Wild E.: On Boltzmann’s equation in the kinetic theory of gases. Proc. Camb. Philos. Soc. 47, 602–609 (1951)
Zolotarev, V.M.: One-dimensional stable distributions. In: Translations of Mathematical Monographs, vol. 65. AMS, Providence (1986)
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F.B.’s research was partially supported by Ministero dell’Istruzione, dell’Università e della Ricerca (MIUR grant 2006/134526). L.L.’s research was partially supported by CNR-IMATI Milano (Italy). D.M. acknowledges support from the Italian MIUR, project “Kinetic and hydrodynamic equations of complex collisional systems”, and from the Deutsche Forschungsgemeinschaft, grant JU 359/7.
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Bassetti, F., Ladelli, L. & Matthes, D. Central limit theorem for a class of one-dimensional kinetic equations. Probab. Theory Relat. Fields 150, 77–109 (2011). https://doi.org/10.1007/s00440-010-0269-8
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DOI: https://doi.org/10.1007/s00440-010-0269-8