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

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1.	Title	Title of document	A Historical Law of Large Numbers for the Marcus-Lushnikov Process
2.	Creator	Author's name, affiliation, country	Stephanie S.M Jacquot; University of Cambridge; United Kingdom
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	historical trees; Marcus-Lushnikov process on trees; limit measure on trees; Smoluchowski coagulation equation; tightness; coupling
3.	Subject	Subject classification	60B10;60J25;60B05;60F15
4.	Description	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.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2010-05-01
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/767
10.	Identifier	Digital Object Identifier	10.1214/EJP.v15-767
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 15
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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