非弹性Kac模型中爆炸的概率观
Probabilistic View of Explosion in an Inelastic Kac Model
摘要：
令$\{μ(\cdot,t):t\geq0\}$为对应于Pulvirenti和Toscani [ \textit{J. Stat. Phys.} \textbf{114}（2004）1453-1480]中引入的非弹性Kac模型的解的概率测度族。Gabetta和Regazzini [ \textit{J. Statist. Phys.} \textbf{147}（2012）1007-1019]已经证明，当且仅当初始数据的适当对称化形式属于特定稳定定律的标准吸引区域时，解弱收敛到平衡。本文证明了，对于比上述数据更有尾的初始数据，极限分布是不适当的，因为它有概率1/2“粘附”于$-\infty$，概率1/2“粘附”于$+\infty$ .它是解释在这个意义上这种现象是服从一种爆炸，主要结果包括在一个明确的表达率这样的爆炸。在介绍这些陈述之前，先讨论了证明其有效性的假设的必要性。这使有机会作出调整的一部分，证明载于上述文件的Gabetta和Regazzini。
英文摘要：
Let $\{μ(\cdot,t):t\geq0\}$ 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.
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