Summary
We study a countable system of interacting diffusions on the interval [0,1], indexed by a hierarchical group. A particular choice of the interaction guaranties, 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).
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Fleischmann, K., Greven, A. Diffusive clustering in an infinite system of hierarchically interacting diffusions. Probab. Th. Rel. Fields 98, 517–566 (1994). https://doi.org/10.1007/BF01192837
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DOI: https://doi.org/10.1007/BF01192837