TY - JOUR

T1 - Infinite rate mutually catalytic branching in infinitely many colonies

T2 - Construction, characterization and convergence

AU - Klenke, Achim

AU - Mytnik, Leonid

N1 - Funding Information:
This work is partly funded by the German Israeli Foundation with grant number G-807-227.6/2003.

PY - 2012/12

Y1 - 2012/12

N2 - We construct a mutually catalytic branching process on a countable site space with infinite "branching rate". The finite rate mutually catalytic model, in which the rate of branching of one population at a site is proportional to the mass of the other population at that site, was introduced by Dawson and Perkins (Ann Probab 26(3):1088-1138, 1998). We show that our model is the limit for a class of models and in particular for the Dawson-Perkins model as the rate of branching goes to infinity. Our process is characterized as the unique solution to a martingale problem. We also give a characterization of the process as a weak solution of an infinite system of stochastic integral equations driven by a Poisson noise.

AB - We construct a mutually catalytic branching process on a countable site space with infinite "branching rate". The finite rate mutually catalytic model, in which the rate of branching of one population at a site is proportional to the mass of the other population at that site, was introduced by Dawson and Perkins (Ann Probab 26(3):1088-1138, 1998). We show that our model is the limit for a class of models and in particular for the Dawson-Perkins model as the rate of branching goes to infinity. Our process is characterized as the unique solution to a martingale problem. We also give a characterization of the process as a weak solution of an infinite system of stochastic integral equations driven by a Poisson noise.

KW - Duality

KW - Martingale problem

KW - Mutually catalytic branching

KW - Stochastic differential equations

UR - http://www.scopus.com/inward/record.url?scp=84870482035&partnerID=8YFLogxK

U2 - 10.1007/s00440-011-0376-1

DO - 10.1007/s00440-011-0376-1

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:84870482035

SN - 0178-8051

VL - 154

SP - 533

EP - 584

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 3-4

ER -