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1996 Time-Space Analysis of the Cluster-Formation in Interacting Diffusions
Klaus Fleischmann, Andreas Greven
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Electron. J. Probab. 1: 1-46 (1996). DOI: 10.1214/EJP.v1-6


A countable system of linearly interacting diffusions on the interval [0,1], indexed by a hierarchical group is investigated. A particular choice of the interactions guarantees that we are in the diffusive clustering regime, that is spatial clusters of components with values all close to 0 or all close to 1 grow in various different scales. We studied this phenomenon in [FG94]. In the present paper we analyze the evolution of single components and of clusters over time. First we focus on the time picture of a single component and find that components close to 0 or close to 1 at a late time have had this property for a large time of random order of magnitude, which nevertheless is small compared with the age of the system. The asymptotic distribution of the suitably scaled duration a component was close to a boundary point is calculated. Second we study the history of spatial 0- or 1-clusters by means of time scaled block averages and time-space-thinning procedures. The scaled age of a cluster is again of a random order of magnitude. Third, we construct a transformed Fisher-Wright tree, which (in the long-time limit) describes the structure of the space-time process associated with our system. All described phenomena are independent of the diffusion coefficient and occur for a large class of initial configurations (universality).


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Klaus Fleischmann. Andreas Greven. "Time-Space Analysis of the Cluster-Formation in Interacting Diffusions." Electron. J. Probab. 1 1 - 46, 1996.


Accepted: 8 April 1996; Published: 1996
First available in Project Euclid: 25 January 2016

zbMATH: 0891.60094
MathSciNet: MR1386298
Digital Object Identifier: 10.1214/EJP.v1-6

Primary: 60K35
Secondary: 60J15 , 60J60

Keywords: clustering , delayed coalescing random walk with immigration , ensemble of log-coalescents , Infinite particle system , Interacting diffusion , low dimensional systems , transformed Fisher-Wright tree

Vol.1 • 1996
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