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SUMMARY:A historical law of large numbers for the Marcus-Lushnikov process
  - Stephanie Jacquot\, Statslab
DTSTART:20090225T170000Z
DTEND:20090225T180000Z
UID:TALK17231@talks.cam.ac.uk
CONTACT:Perla Sousi
DESCRIPTION:The Marcus-Lushnikov process is a finite stochastic particle s
 ystem in\nwhich each particle is entirely\ncharacterized by its mass. Each
  pair of particles with masses x and y\nmerges into a single particle at a
 \ngiven rate K(x\; y). Under certain assumptions\, this process converges 
 to\nthe solution to Smoluchowski\nequation (a system of differential equat
 ions describing the evolution of\ndensity of coagulating masses) as\nthe n
 umber of particles increases to infinity. This process gives at\neach time
  the distribution in masses\nof the particles present in the system\, but 
 does not conserve any other\ninformation that the particles\nmight contain
 . In other words\, we lose in part the informations\ncontained in the part
 icles\, that is their\nhistory. We set up a historical analogue of the Mar
 cus-Lushnikov process\n(built according\nthe rules of construction of the 
 usual Markov-Lushnikov process) each\ntime giving what we call the\nhistor
 ical tree of a particle. The historical tree of a particle present\nin the
  Marcus- Lushnikov process\nat a given time t encodes informations about t
 he times andmasses of the\ncoagulation events that have formed that partic
 le.\nHence\, our historical process gives us at each time tree particles\n
 containing information about the time they formed\, their masses and\nthei
 r history\nof formation that is the particles it contains along with the o
 rder in\nwhich the particles formed have\ncoagulated with their times of c
 oagulation. We prove a law of large\nnumbers for the empirical distributio
 n\nof such historical trees. The limit is a natural measure on trees which
 \nis constructed from a solution to\nSmoluchowski coagulation equation.\n
LOCATION:CMS\, MR13
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