A historical law of large numbers for the Marcus-Lushnikov process

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• Stephanie Jacquot, Statslab
• Wednesday 25 February 2009, 17:00-00:00
• CMS, MR13.

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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 Smoluchowski equation (a system of differential equations describing the evolution of density of coagulating masses) as the number of particles increases to infinity. This process gives at each time the distribution in masses of the particles present in the system, but does not conserve any other information that the particles might contain. In other words, we lose in part the informations contained in the particles, that is their history. We set up a historical analogue of the Marcus-Lushnikov process (built according 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 informations about the times andmasses of the coagulation events that have formed that particle. Hence, our historical process gives us at each time tree particles containing information about the time they formed, their masses and their history of formation that is the particles it contains along with the order in which the particles formed have coagulated with their times of coagulation. 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 Smoluchowski coagulation equation.

This talk is part of the Statistical Laboratory Graduate Seminars series.

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