Infinite rate mutually catalytic branching in infinitely many colonies: construction, characterization and convergence

• 成果类型: Article
• 署名作者: Klenke, Achim; Mytnik, Leonid
• 署名单位: Johannes Gutenberg University of Mainz; Technion Israel Institute of Technology
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-011-0376-1
• 发表日期: 2012
• 页码: 533-584
• 关键词: times

摘要

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.