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Time-Space Analysis of the Cluster-Formation in Interacting Diffusions

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1.	Title	Title of document	Time-Space Analysis of the Cluster-Formation in Interacting Diffusions
2.	Creator	Author's name, affiliation, country	Klaus Fleischmann; Weierstrass Institute for Applied Analysis and Stochastics
2.	Creator	Author's name, affiliation, country	Andreas Greven; Universitat Erlangen-Nurnberg
3.	Subject	Discipline(s)	Mathematics
3.	Subject	Keyword(s)	interacting diffusion, clustering, infinite particle system, delayed coalescing random walk with immigration, transformed Fisher-Wright tree, low dimensional systems, ensemble of log-coalescents
3.	Subject	Subject classification	Primary 60K35; Secondary 60J60, 60J15
4.	Description	Abstract	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).
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	1996-04-08
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/6
10.	Identifier	Digital Object Identifier	10.1214/EJP.v1-6
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 1
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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