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SUMMARY: A historical law of large numbers for the Marcus-Lushnikov proces
 s - Stephanie Jacquot   (Statslab)
DTSTART:20090225T170000Z
DTEND:20090225T180000Z
UID:TALK23559@talks.cam.ac.uk
DESCRIPTION:The Marcus-Lushnikov process is a finite stochastic article sy
 stem in which each particle is entirely\ncharacterized by its mass. Each p
 air of particles with masses x and y merges into a single particle at a\ng
 iven rate K(x\; y). Under certain assumptions\, this process converges to 
 the solution to Smoluchowski\nequation (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 di
 stribution in masses of the particles present in the system\, but does not
  conserve any other information that the particles\nmight contain.\nIn oth
 er words\, we lose in part the informations\ncontained in the particles\, 
 that is their history. We set up a historical analogue of the Marcus-Lushn
 ikov process (built according the rules of construction of the usual Marko
 v-Lushnikov process) each time giving what we call the historical tree of 
 a particle.\nThe historical tree of a particle present in the Marcus- Lush
 nikov process at a given time t encodes informations about the times andma
 sses of the coagulation events that have formed that particle.\nHence\, ou
 r historical process gives us at each time tree particles containing infor
 mation about the time they formed\, their masses and their history of form
 ation 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.\n
LOCATION:CMS\, MR13
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