A space-time property of a class of measure-valued branching diffusions.

*(English)*Zbl 0641.60060Let \((X_ t)_{t\geq 0}\) 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)].

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 |

##### Keywords:

measure-valued branching diffusion; Hausdorff dimension; exact Hausdorff dimension of random sets
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\textit{E. A. Perkins}, Trans. Am. Math. Soc. 305, No. 2, 743--795 (1988; Zbl 0641.60060)

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