Probabilistic investigations on the explosion of solutions of the Kac equation with infinite energy initial distribution
Probabilistic investigations on the explosion of solutions of the Kac equation with infinite energy initial distribution
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具有无限能量初始分布的Kac方程解爆炸的概率研究
DOI：
10.1239/jap/1208358954
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发表时间：
2008
影响因子：
1
通讯作者：
E. Regazzini
中科院分区：
文献类型：
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作者：
E. Carlen;E. Gabetta;E. Regazzini
Gabetta和Regazzini（2006 B）已经证明，初始能量（二阶矩）的有限性对于Kac模型Boltzmann方程的解弱收敛（C B -收敛）到R上的概率测度是必要且充分的。在这里，我们通过详细分析初始能量无限大时实际发生的情况来补充这一结果。特别是，我们证明了这样的解决方案模糊收敛（C 0 -收敛）的零措施（这是相同的0上的Borel集R）。更确切地说，我们证明了极限分布的总质量分裂成两个相等的质量（每个质量的值为1/2），并且我们提供了对这种现象发生的速率的定量估计。证明中采用的方法也适用于加权独立同分布随机变量x 1，x 2，.的和，其中这些随机变量具有无穷大的二阶矩和零均值。然后，设Tn：= ∑ j =1 η n λ j，n x j，其中max 1 ≤ j ≤ η n λ j，n → 0（当n → +∞时），且∑ j =1 η n λ j，n2 = 1，n = 1，2，.，经典中心极限定理表明T在某种意义下收敛于“无穷方差的正态随机变量”。同样，在这种情况下，我们证明了质量分裂成粘附质量到-∞和+∞或到∞的速率的定量估计，这与我们对Kac方程获得的结果类似。虽然在这种情况下的设置是相当经典的，我们还没有发现任何类似类型的以前的结果。
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 ( C b -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 ( C 0 -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 x 1 , x 2 , …, where these random variables have an infinite second moment and zero mean. Then, with T n := ∑ j =1 η n λ j , n x j , with max 1 ≤ j ≤ η n λ j , n → 0 (as n → +∞), and ∑ j =1 η n λ j , n 2 = 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.