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

Abstract

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.