WIAS Preprint No. 187, (1995)

Cluster formation in a stepping stone model with continuous, hierarchically structured sites


  • Evans, Steven N.
  • Fleischmann, Klaus

2010 Mathematics Subject Classification

  • 60K35 60J60 60B15


  • Interacting diffusion, stepping stone model, cluster formation, clustering, coalescing Lévy process, hierarchical structures, resampling, migration


A stepping stone model with site space a continuous, hierarchical group is constructed via duality with a system of (delayed) coalescing "stable" Lévy processes. This model can be understood as a continuum limit of discrete state-space, two allele, genetics models with hierarchically structured resampling and migration. The existence of a process rescaling limit on suitable large space and time scales is established and interpreted in terms of the dynamics of cluster formation. This paper was inspired by recent work of Klenke.

Appeared in

  • Ann. Probab., vol. 24 (1996), No. 4, pp. 1926--1952

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