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.

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Stochastic Processes and their Applications
School of Mathematics and Statistics

Dawson, D.A, Hochberg, K.J. (Kenneth J.), & Vinogradov, V. (Vladimir). (1996). High-density limits of hierarchically structured branching-diffusing populations. Stochastic Processes and their Applications, 62(2), 191–222. doi:10.1016/0304-4149(96)00052-X