WIAS Preprint No. 23, (1992)

Diffusive clustering in an infinite system of hierarchically interacting diffusions.


  • Fleischmann, Klaus
  • Greven, Andreas

2010 Mathematics Subject Classification

  • 60K35 60J60 60J15


  • Interacting diffusions, coalescing random walk, hierachial system, cluster formation, universality




We study a countable system of interacting diffusions on the interval [0,1], indexed by a hierarchical group. A particular choice of the interaction guarantees, we are in the diffusive clustering regime. This means clusters of components with values either close to 0 or close to 1 grow on various different scales. However, single components oscillate infinitely often between values close to 0 and close to 1 in such a way that they spend fraction one of their time together and close to the boundary. The processes in the whole class considered and starting with a shift-ergodic initial law have the same qualitative properties (universality).

Appeared in

  • Probab. Theor. Relat. Fields, 98 (1994), pp. 517--566

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