The Annals of Probability

Different clustering regimes in systems of hierarchically interacting diffusions

Achim Klenke

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Abstract

We study a system of interacting diffusions

indexed by the hierarchical group $\Xi$, as a genealogical two genotype model [where $x _\xi(t)$ denotes the frequency of, say, type A] with hierarchically determined degrees of relationship between colonies. In the case of short interaction range it is known that the system clusters. That is, locally one genotype dies out. We focus on the description of the different regimes of cluster growth which is shown to depend on the interaction kernel $a(\dot,\dot)$ via its recurrent potential kernel. One of these regimes will be further investigated by mean-field methods. For general interaction range we shall also relate the behavior of large finite systems, indexed by finite subsets $\Xi_n$ of, $\Xi$ to that of the infinite system. On the way we will establish relations between hitting times of random walks and their potentials.

Article information

Source
Ann. Probab., Volume 24, Number 2 (1996), 660-697.

Dates
First available in Project Euclid: 11 December 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1039639358

Digital Object Identifier
doi:10.1214/aop/1039639358

Mathematical Reviews number (MathSciNet)
MR1404524

Zentralblatt MATH identifier
0862.60096

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Interacting diffusions infinite particle systems coalescing random walks

Citation

Klenke, Achim. Different clustering regimes in systems of hierarchically interacting diffusions. Ann. Probab. 24 (1996), no. 2, 660--697. doi:10.1214/aop/1039639358. https://projecteuclid.org/euclid.aop/1039639358


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