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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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 17 July 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1026915615

Digital Object Identifier
doi:10.1214/aoap/1026915615

Mathematical Reviews number (MathSciNet)
MR1910639

Zentralblatt MATH identifier
1030.60036

Subjects
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.)

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

Citation

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. https://projecteuclid.org/euclid.aoap/1026915615


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