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Central limit theorem for a class of one-dimensional kinetic equations
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  • Published: 24 February 2010

Central limit theorem for a class of one-dimensional kinetic equations

  • Federico Bassetti1,
  • Lucia Ladelli2 &
  • Daniel Matthes3 

Probability Theory and Related Fields volume 150, pages 77–109 (2011)Cite this article

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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Author information

Authors and Affiliations

  1. Università degli Studi di Pavia, via Ferrata 1, 27100, Pavia, Italy

    Federico Bassetti

  2. Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, Italy

    Lucia Ladelli

  3. Technische Universität Wien, Wiedner Hauptstraße 8-10/E101, 1040, Wien, Austria

    Daniel Matthes

Authors
  1. Federico Bassetti
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  2. Lucia Ladelli
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  3. Daniel Matthes
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Corresponding author

Correspondence to Federico Bassetti.

Additional information

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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  • Received: 23 October 2008

  • Revised: 29 January 2010

  • Published: 24 February 2010

  • Issue Date: June 2011

  • DOI: https://doi.org/10.1007/s00440-010-0269-8

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Keywords

  • Central limit theorem
  • Domain of normal attraction
  • Stable law
  • Kac model
  • Smoothing transformations

Mathematics Subject Classification (2000)

  • 60F05
  • 82C40
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