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Journal of Statistical Physics

, Volume 154, Issue 5, pp 1292–1324 | Cite as

Probabilistic View of Explosion in an Inelastic Kac Model

  • Andrea Bonomi
  • Eleonora Perversi
  • Eugenio Regazzini
Article

Abstract

Let \(\{\mu (\cdot ,t):t\ge 0\}\) be the family of probability measures corresponding to the solution of the inelastic Kac model introduced in Pulvirenti and Toscani (J Stat Phys 114:1453–1480, 2004). It has been proved by Gabetta and Regazzini (J Stat Phys 147:1007–1019, 2012) that the solution converges weakly to equilibrium if and only if a suitable symmetrized form of the initial data belongs to the standard domain of attraction of a specific stable law. In the present paper it is shown that, for initial data which are heavier-tailed than the aforementioned ones, the limiting distribution is improper in the sense that it has probability \(1/2\) “adherent” to \(-\infty \) and probability \(1/2\) “adherent” to \(+\infty \). It is explained in which sense this phenomenon is amenable to a sort of explosion, and the main result consists in an explicit expression of the rate of such an explosion. The presentation of these statements is preceded by a discussion about the necessity of the assumption under which their validity is proved. This gives the chance to make an adjustment to a portion of a proof contained in the above-mentioned paper by Gabetta and Regazzini.

Keywords

Central limit theorem Explosion of solution Inelastic Kac model Skorokhod representation theorem 

Mathematics Subject Classification

60F05 82C40 60B10 

Notes

Acknowledgments

We would like to thank two anonymous referees for giving several valuable comments regarding the presentation of the main result. The research of Eleonora Perversi and Eugenio Regazzini has been partially supported by MIUR-2012AZS52J-003.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Andrea Bonomi
    • 1
  • Eleonora Perversi
    • 1
  • Eugenio Regazzini
    • 1
    • 2
  1. 1.Dipartimento di MatematicaUniversità degli Studi di PaviaPaviaItaly
  2. 2.CNR-IMATIMilanItaly

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