SCALING LIMIT OF AN ADAPTIVE CONTACT PROCESS

• 成果类型: Article
• 署名作者: Casanova, Adrian Gonzalez; Tobias, Andras; Valesin, Daniel
• 署名单位: Universidad Nacional Autonoma de Mexico; Budapest University of Technology & Economics; University of Warwick
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/23-AOP1662
• 发表日期: 2024
• 页码: 296-349

摘要

We introduce and study an interacting particle system evolving on the ddimensional torus (Z/NZ)d. Each vertex of the torus can be either empty or occupied by an individual of type lambda is an element of (0, infinity). An individual of type lambda dies with rate one and gives birth at each neighboring empty position with rate lambda; moreover, when the birth takes place, the newborn individual is likely to have the same type as the parent but has a small probability of being a mutant. A mutant child of an individual of type lambda has type chosen according to a probability kernel. We consider the asymptotic behavior of this process when N -> infinity and, simultaneously, the mutation probability tends to zero fast enough that mutations are sufficiently separated in time so that the amount of time spent on configurations with more than one type becomes negligible. We show that, after a suitable time scaling and deletion of the periods of time spent on configurations with more than one type, the process converges to a Markov jump process on (0, infinity), whose rates we characterize.