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Time-Space Analysis of the Cluster-Formation in Interacting Diffusions
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Klaus Fleischmann, Weierstrass Institute for Applied Analysis and Stochastics
Andreas Greven, Universitat Erlangen-Nurnberg
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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).
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Full text: PDF
Pages: 1-46
Published on: April 8, 1996
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Electronic Journal of Probability. ISSN: 1083-6489 |
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