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

**Stephanie S.M Jacquot**

*(University of Cambridge)*

#### 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.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 605-635

Publication Date: May 1, 2010

DOI: 10.1214/EJP.v15-767

#### References

- Norris, J. R. Cluster coagulation.
*Comm. Math. Phys.*209 (2000), no. 2, 407--435. MR1737990 (2002c:82079) - Norris, James. Smoluchowski's coagulation equation: uniqueness, non uniqueness and a hydrodynamic limit for the stochastic coalescent Stochastic Process. Appl. 119 (2009), no. 1, 167--189.
- Fournier, Nicolas; Laurençot, Philippe. Marcus-Lushnikov processes, Smoluchowski's and Flory's models.
*Stochastic Process. Appl.*119 (2009), no. 1, 167--189. MR2485023 (2010f:60275) - Nicolas Fournier and Jean-Sebastien Giet Convergence of the Marcus-Lushnikov process, MSC 2000 : 45K05, 60H30.
- Pollard, David. Convergence of stochastic processes.Springer Series in Statistics.
*Springer-Verlag, New York,*1984. xiv+215 pp. ISBN: 0-387-90990-7 MR0762984 (86i:60074) - Jakubowski, Adam. On the Skorokhod topology.
*Ann. Inst. H. Poincaré Probab. Statist.*22 (1986), no. 3, 263--285. MR0871083 (89a:60008) - Ethier, Stewart N.; Kurtz, Thomas G. Markov processes.Characterization and convergence.Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics.
*John Wiley \& Sons, Inc., New York,*1986. x+534 pp. ISBN: 0-471-08186-8 MR0838085 (88a:60130)

This work is licensed under a Creative Commons Attribution 3.0 License.