Probabilistic View of Explosion in an Inelastic Kac Model

TL;DR

This paper investigates the explosion phenomenon in an inelastic Kac model, showing that for certain heavy-tailed initial data, the solution's distribution diverges to infinity with probability 1/2 on each side, and provides explicit explosion rates.

Contribution

It introduces a probabilistic interpretation of explosion in the inelastic Kac model and derives explicit rates for this divergence phenomenon.

Findings

Heavy-tailed initial data lead to improper limiting distributions with divergence to infinity.

The explosion occurs with probability 1/2 on each side, indicating a symmetric divergence.

An explicit expression for the rate of explosion is provided.

Abstract

Let $\{\mu(\cdot,t):t\geq0\}$ be the family of probability measures corresponding to the solution of the inelastic Kac model introduced in Pulvirenti and Toscani [\textit{J. Stat. Phys.} \textbf{114} (2004) 1453-1480]. It has been proved by Gabetta and Regazzini [\textit{J. Statist. Phys.} \textbf{147} (2012) 1007-1019] 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.