The Annals of Applied Probability

Eternal solutions to Smoluchowski's coagulation equation with additive kernel and their probabilistic interpretations

Jean Bertoin

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The cornerstone of this work, which is partly motivated by the characterization of the so-called eternal additive coalescents by Aldous and Pitman, is an explicit expression for the general eternal solution to Smoluchowski's coagulation equation with additive kernel. This expression points at certain Lévy processes with no negative jumps and more precisely at a stochastic model for aggregation based on such processes, which has been recently considered by Bertoin and Miermont and is known to bear close relations with the additive coalescence. As an application, we show that the eternal solutions can be obtained from some hydrodynamic limit of the stochastic model. We also present a simple condition that ensures the existence of a smooth density for an eternal solution.

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Ann. Appl. Probab., Volume 12, Number 2 (2002), 547-564.

First available in Project Euclid: 17 July 2002

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Primary: 60G51: Processes with independent increments; Lévy processes 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C21: Dynamic continuum models (systems of particles, etc.)

Smoluchowski's coagulation equation additive coalescence Lévy process with no positive jumps


Bertoin, Jean. Eternal solutions to Smoluchowski's coagulation equation with additive kernel and their probabilistic interpretations. Ann. Appl. Probab. 12 (2002), no. 2, 547--564. doi:10.1214/aoap/1026915615.

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