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Probabilistic Investigations on the Explosion of Solutions of the KAC Equation with Infinite Energy Initial Distribution

Part of: Time-dependent statistical mechanics (dynamic and nonequilibrium) Limit theorems

Published online by Cambridge University Press:  14 July 2016

Eric Carlen ,
Ester Gabetta  and
Eugenio Regazzini
Eric Carlen*
Affiliation:
Rutgers University
Ester Gabetta*
Affiliation:
Università degli Studi di Pavia
Eugenio Regazzini*
Affiliation:
Università degli Studi di Pavia and IMATI-CNR
*
∗Postal address: Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA.
∗∗Postal address: Dipartimento di Matematica ‘F. Casorati’, Università degli Studi di Pavia, via Ferrata 1, I-27100 Pavia, Italy.
∗∗Postal address: Dipartimento di Matematica ‘F. Casorati’, Università degli Studi di Pavia, via Ferrata 1, I-27100 Pavia, Italy.
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Abstract

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Gabetta and Regazzini (2006b) have shown that finiteness of the initial energy (second moment) is necessary and sufficient for the solution of the Kac's model Boltzmann equation to converge weakly (Cb-convergence) to a probability measure on R. Here, we complement this result by providing a detailed analysis of what does actually happen when the initial energy is infinite. In particular, we prove that such a solution converges vaguely (C0-convergence) to the zero measure (which is identically 0 on the Borel sets of R). More precisely, we prove that the total mass of the limiting distribution splits into two equal masses (of value ½ each), and we provide quantitative estimates on the rate at which such a phenomenon takes place. The methods employed in the proofs also apply in the context of sums of weighted independent and identically distributed random variables x̃1, x̃2, …, where these random variables have an infinite second moment and zero mean. Then, with Tn := ∑j=1ηnλj,nx̃j, with max1 ≤ j ≤ ηnλj,n → 0 (as n → +∞), and ∑j=1ηnλj,n2 = 1, n = 1, 2, …, the classical central limit theorem suggests that T should in some sense converge to a ‘normal random variable of infinite variance’. Again, in this setting we prove quantitative estimates on the rate at which the mass splits into adherent masses to -∞ and +∞, or to ∞, that are analogous to those we have obtained for the Kac equation. Although the setting in this case is quite classical, we have not uncovered any previous results of a similar type.

Keywords

Boltzmann equationcentral limit theorem

MSC classification

Secondary: 60F05: Central limit and other weak theorems 82C40: Kinetic theory of gases
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 1 , March 2008 , pp. 95 - 106
Copyright
Copyright © Applied Probability Trust 2008 

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