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A space-time property of a class of measure-valued branching diffusions. (English) Zbl 0641.60060
Let $$(X_ t)_{t\geq 0}$$ denote the so-called measure-valued branching diffusion on $$R^ d$$ whose spatial diffusion is governed by a symmetric stable process of index $$\alpha\in [0,2]$$, starting with a fixed finite measure. $$[X_ t$$ can be approximated by large populations of small branching and moving particles, cf. e.g. D. A. Dawson, Z. Wahrscheinlichkeitstheor. Verw. Geb. 40, 125-145 (1977; Zbl 0343.60001)]. It is known that, for any fixed $$t>0$$, $$X_ t$$ is carried by (random $$F_{\sigma}$$-) sets of Hausdorff dimension $$\min\{\alpha,d\}$$, but not by smaller sets.
The author extends these results in two important directions:
Theorem A. Let $$d>\alpha$$. There are constants $$0<c\leq C<\infty$$, depending on $$\alpha$$ and d, such that the following holds almost surely: For every $$t>0$$, there exists a random Borel set $$\Lambda_ t$$ with $$X_ t(R^ d\setminus \Lambda_ t)=0$$ and $c \Phi_{\alpha}-m(A\cap \Lambda_ t)\leq X_ t(A)\leq C \Phi_{\alpha}-m(A\cap \Lambda_ t),\quad A\in {\mathfrak B}\quad d.$ Here $$\Phi_{\alpha}-m$$ denotes the Hausdorff $$\Phi_{\alpha}$$-measure with $$\Phi_{\alpha}(s)=s^{\alpha}\log \log 1/s.$$
Thus $$X_ t$$ spreads its mass over its “support” in a very uniform manner (with respect to geometric $$\Phi_{\alpha}$$-measure) and does so for all $$t>0$$ simultaneously.
Because of the immense heuristical and technical expenditure there are up to now only a few papers which are concerned with the exact Hausdorff dimension of random sets. (Thus, e.g., the author uses nonstandard analysis to be more compact.) In case $$\alpha =2$$ the form of $$\Phi_{\alpha}(s)$$ coincides with that for Brownian motion in $$R^ d$$, $$d\geq 3$$ [P. Lévy, Giorn. Ist. Ital. Attuari 16, 1-37 (1953; Zbl 0053.101), E. Ciesielsky and S. J. Taylor, Trans. Am. Math. Soc. 103, 434-450 (1962; Zbl 0121.130)] - which is not mentioned in the paper.
In the critical case $$d=\alpha$$ (= 1 or 2) the results are less precise. The conjecture is $\Phi (s)=s^{\alpha}\log (1/s) \log\log\log(1/s),$ which would coincide in case $$\alpha =2$$ with that for Brownian motion in $$R^ 2$$ [S. J. Taylor, Proc. Camb. Philos. Soc. 60, 253-258 (1964; Zbl 0149.131)].
Although there is a close relationship with the random recursive constructions of Falconer, Graf, Mauldin and Williams, the exact Hausdorff dimension of their random fractal sets is given by $$\Phi (s)=s^{\alpha}(\log \log 1/s)^{1-\alpha /d}$$ [S. Graf, R. D. Mauldin and S. C. Williams, Mem. Am. Math. Soc. 71, No.381, 121 p. (1988; Zbl 0641.60003)].
Reviewer: U.Zähle

##### MSC:
 60G57 Random measures 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60J60 Diffusion processes 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 28E05 Nonstandard measure theory
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