A pure jump Markov process associated with Smoluchowski's coagulation equation

• 成果类型: Article
• 署名作者: Deaconu, M; Fournier, N; Tanré, E
• 署名单位: Universite de Lorraine
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2002
• 页码: 1763-1796
• 关键词: fragmentation equations; boltzmann-equation; kac equation; EXISTENCE; cutoff; uniqueness; molecules; models

摘要

The aim of the present paper is to construct a stochastic process, whose law is the solution of the Smoluchowski's coagulation equation. We introduce first a modified equation, dealing with the evolution of the distribution Q(t) (dx) of the mass in the system. The advantage we take on this is that we can perform an unified study for both continuous and discrete models. The integro-partial-differential equation satisfied by {Q(t)}(tgreater than or equal to0) can be interpreted as the evolution equation of the time marginals of a Markov pure jump process. At this end we introduce a nonlinear Poisson driven stochastic differential equation related to the Smoluchowski equation in the following way: if X-t satisfies this stochastic equation, then the law of X-t satisfies the modified Smoluchowski equation. The nonlinear process is richer than the Smoluchowski equation, since it provides historical information on the particles. Existence, uniqueness and pathwise behavior for the solution of this SDE are studied. Finally, we prove that the nonlinear process X can be obtained as the limit of a Marcus-Lushnikov procedure.