The Annals of Applied Probability

Stochastic interacting particle systems and nonlinear kinetic equations

Andreas Eibeck and Wolfgang Wagner

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We present the stochastic approach to nonlinear kinetic equations (without gradient terms) in a unifying general framework, which covers many interactions important in applications, such as coagulation, fragmentation, inelastic collisions, as well as source and efflux terms. We provide conditions for the existence of corresponding stochastic particle systems in the sense of regularity (nonexplosion) of a jump process with unbounded intensity. Using an appropriate space of measure-valued functions, we prove relative compactness of the sequence of processes and characterize the weak limits in terms of solutions to the nonlinear equation. As a particular application, we derive existence theorems for Smoluchowski's coagulation equation with fragmentation, efflux and source terms, and for the Boltzmann equation with inelastic collisions.

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Ann. Appl. Probab., Volume 13, Number 3 (2003), 845-889.

First available in Project Euclid: 6 August 2003

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Primary: 60K40: Other physical applications of random processes 65C35: Stochastic particle methods [See also 82C80]

Stochastic particle systems regularity of jump processes kinetic equations existence of solutions coagulation fragmentation source and efflux dissipative collisions


Eibeck, Andreas; Wagner, Wolfgang. Stochastic interacting particle systems and nonlinear kinetic equations. Ann. Appl. Probab. 13 (2003), no. 3, 845--889. doi:10.1214/aoap/1060202829.

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