Electronic Journal of Probability

A Historical Law of Large Numbers for the Marcus-Lushnikov Process

Stephanie Jacquot

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Abstract

The Marcus-Lushnikov process is a finite stochastic particle system, in which each particle is entirely characterized by its mass. Each pair of particles with masses $x$ and $y$ merges into a single particle at a given rate $K(x,y)$. Under certain assumptions, this process converges to the solution to the Smoluchowski coagulation equation, as the number of particles increases to infinity. The Marcus-Lushnikov process gives at each time the distribution of masses of the particles present in the system, but does not retain the history of formation of the particles. In this paper, we set up a historical analogue of the Marcus-Lushnikov process (built according to the rules of construction of the usual Markov-Lushnikov process) each time giving what we call the historical tree of a particle. The historical tree of a particle present in the Marcus-Lushnikov process at a given time t encodes information about the times and masses of the coagulation events that have formed that particle. We prove a law of large numbers for the empirical distribution of such historical trees. The limit is a natural measure on trees which is constructed from a solution to the Smoluchowski coagulation equation.

Article information

Source
Electron. J. Probab., Volume 15 (2010), paper no. 19, 605-635.

Dates
Accepted: 1 May 2010
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819804

Digital Object Identifier
doi:10.1214/EJP.v15-767

Mathematical Reviews number (MathSciNet)
MR2639735

Zentralblatt MATH identifier
1225.60011

Subjects
Primary: 60B10: Convergence of probability measures
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60B05: Probability measures on topological spaces 60F15: Strong theorems

Keywords
historical trees Marcus-Lushnikov process on trees limit measure on trees Smoluchowski coagulation equation tightness coupling

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Jacquot, Stephanie. A Historical Law of Large Numbers for the Marcus-Lushnikov Process. Electron. J. Probab. 15 (2010), paper no. 19, 605--635. doi:10.1214/EJP.v15-767. https://projecteuclid.org/euclid.ejp/1464819804


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