High-density limits of hierarchically structured branching-diffusing populations

Donald A. Dawson, Kenneth J. Hochberg, Vladimir Vinogradov

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We develop a general probabilistic approach that enables one to get sharp estimates for the almost-sure short-term behavior of hierarchically structured branching-diffusion processes. This approach involves the thorough investigation of the cluster structure and the derivation of some probability estimates for the sets of rapidly fluctuating realizations. In addition, our approach leads to the derivation of new modulus-of-continuity-type results for measure-valued processes. In turn, the modulus-of-continuity-type results for hierarchical branching-diffusion processes are used to derive upper estimates for the Hausdorff dimension of support.

Original languageEnglish
Pages (from-to)191-222
Number of pages32
JournalStochastic Processes and their Applications
Issue number2
StatePublished - Jul 1996

Bibliographical note

Funding Information:
In this paper, we continue the investigation of path properties of hierarchically structured measure-valued branching-diffusion processes that we began in our two earlier works (Dawson, Hochberg and Vinogradov, 1994, 1995, hereafter referred to as \[DHV1\] and \[DHV2\], respectively). These processes describe populations of individuals undergoing some spatial motion that are affected by one branching mechanism that acts upon individuals and by another, independent, branching mechanism * Corresponding author. 1 Supported by an NSERC grant. 2 Partially supported by an NSERC International Scientific Exchange Award and a grant from the Bar-Ilan University Research Authority. 3 Partially supported by an NSERC grant, NSERC grants of D.A. Dawson and J.N.K. Rao and an internal travel grant from the Department of Mathematics and Computer Science of the University of Northern British Columbia.


  • Hausdorff dimension
  • Hierarchical branching
  • Modulus of continuity
  • Path properties


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