Probabilistic View of Explosion in an Inelastic Kac Model

Let {μ(⋅,t):t≥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 −∞ and probability 1/2 “adherent” to +∞ . 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.