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October 2002 A pure jump Markov process associated with Smoluchowski's coagulation equation
Madalina Deaconu, Nicolas Fournier, Etienne Tanré
Ann. Probab. 30(4): 1763-1796 (October 2002). DOI: 10.1214/aop/1039548371


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 $\{Q_t\}_{t\geq 0}$ 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.


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Madalina Deaconu. Nicolas Fournier. Etienne Tanré. "A pure jump Markov process associated with Smoluchowski's coagulation equation." Ann. Probab. 30 (4) 1763 - 1796, October 2002.


Published: October 2002
First available in Project Euclid: 10 December 2002

zbMATH: 1018.60067
MathSciNet: MR1944005
Digital Object Identifier: 10.1214/aop/1039548371

Primary: 60H30 , 60J75 , 60K35

Keywords: Nonlinear stochastic differential equations , Poisson measures , Smoluchowski's coagulation equations

Rights: Copyright © 2002 Institute of Mathematical Statistics


Vol.30 • No. 4 • October 2002
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