ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF THE FRAGMENTATION EQUATION WITH SHATTERING: AN APPROACH VIA SELF-SIMILAR MARKOV PROCESSES
ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF THE FRAGMENTATION EQUATION WITH SHATTERING: AN APPROACH VIA SELF-SIMILAR MARKOV PROCESSES
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DOI：
10.1214/09-aap622
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发表时间：
2010-04-01
影响因子：
1.8
通讯作者：
Haas, Benedicte
中科院分区：
文献类型：
--
作者：
Haas, Benedicte
这篇文章的主题是一个具有非保守解的碎裂方程，由于强烈的分裂，一些质量被零质量粒子的尘埃所损失。在碎片率有规则变化的假设下，我们描述了解的大时间行为。我们的方法基于概率工具：碎片化方程的解是通过在有限时间内连续达到0的非增自相似马尔可夫过程来构造的。我们的主要概率结果描述了这些过程在非绝灭条件下的渐近行为，并被用于解碎裂方程。我们注意到两个参数显著地影响这些大时间行为：“近-1相对质量”的形成速率(该速率与相应的自相似马尔可夫过程的Levy度量在0附近的行为有关)和大初始粒子的分布。正确地重新密封，然后解收敛到一个非平凡极限，该极限与方程的拟定常解有关。此外，充分描述了这些拟平稳解，或者等价地，自相似马尔可夫过程的拟平稳分布。
The subject of this paper is a fragmentation equation with nonconservative solutions, some mass being lost to a dust of zero-mass particles as a consequence of an intensive splitting. Under some assumptions of regular variation on the fragmentation rate, we describe the large time behavior of solutions. Our approach is based on probabilistic tools: the solutions to the fragmentation equation are constructed via nonincreasing self-similar Markov processes that continuously reach 0 in finite time. Our main probabilistic result describes the asymptotic behavior of these processes conditioned on nonextinction and is then used for the solutions to the fragmentation equation.We note that two parameters significantly influence these large time behaviors: the rate of formation of "nearly-1 relative masses" (this rate is related to the behavior near 0 of the Levy measure associated with the corresponding self-similar Markov process) and the distribution of large initial particles. Correctly resealed, the solutions then converge to a nontrivial limit which is related to the quasi-stationary solutions of the equation. Besides, these quasi-stationary solutions, or, equivalently, the quasi-stationary distributions of the self-similar Markov processes, are fully described.