×

zbMATH — the first resource for mathematics

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
PDF BibTeX Cite
Full Text: DOI
References:
[1] Robert M. Anderson and Salim Rashid, A nonstandard characterization of weak convergence, Proc. Amer. Math. Soc. 69 (1978), no. 2, 327 – 332. · Zbl 0393.03047
[2] Robert B. Ash, Real analysis and probability, Academic Press, New York-London, 1972. Probability and Mathematical Statistics, No. 11. · Zbl 0249.28001
[3] J. Theodore Cox and David Griffeath, Occupation times for critical branching Brownian motions, Ann. Probab. 13 (1985), no. 4, 1108 – 1132. · Zbl 0582.60091
[4] Nigel J. Cutland, Nonstandard measure theory and its applications, Bull. London Math. Soc. 15 (1983), no. 6, 529 – 589. · Zbl 0529.28009
[5] D. A. Dawson, Stochastic evolution equations and related measure processes, J. Multivariate Anal. 5 (1975), 1 – 52. · Zbl 0299.60050
[6] D. A. Dawson, The critical measure diffusion process, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 40 (1977), no. 2, 125 – 145. · Zbl 0343.60001
[7] Donald A. Dawson and Kenneth J. Hochberg, The carrying dimension of a stochastic measure diffusion, Ann. Probab. 7 (1979), no. 4, 693 – 703. · Zbl 0411.60084
[8] Donald A. Dawson and Thomas G. Kurtz, Applications of duality to measure-valued diffusion processes, Advances in filtering and optimal stochastic control (Cocoyoc, 1982) Lect. Notes Control Inf. Sci., vol. 42, Springer, Berlin, 1982, pp. 91 – 105. · Zbl 0496.60057
[9] William Feller, Diffusion processes in genetics, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley and Los Angeles, 1951, pp. 227 – 246.
[10] William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966. · Zbl 0077.12201
[11] Theodore E. Harris, The theory of branching processes, Die Grundlehren der Mathematischen Wissenschaften, Bd. 119, Springer-Verlag, Berlin; Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. · Zbl 1037.60001
[12] John Hawkes, A lower Lipschitz condition for the stable subordinator, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 17 (1971), 23 – 32. · Zbl 0193.45002
[13] Richard A. Holley and Daniel W. Stroock, Generalized Ornstein-Uhlenbeck processes and infinite particle branching Brownian motions, Publ. Res. Inst. Math. Sci. 14 (1978), no. 3, 741 – 788. · Zbl 0412.60065
[14] I. Iscoe, A weighted occupation time for a class of measure-valued branching processes, Probab. Theory Relat. Fields 71 (1986), no. 1, 85 – 116. · Zbl 0555.60034
[15] Frank B. Knight, Essentials of Brownian motion and diffusion, Mathematical Surveys, vol. 18, American Mathematical Society, Providence, R.I., 1981. · Zbl 0458.60002
[16] Peter A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113 – 122. · Zbl 0312.28004
[17] Albert T. Bharucha-Reid , Probabilistic analysis and related topics. Vol. 2, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. · Zbl 0435.00012
[18] Reimers (1986), Univ. of British Columbia, Ph.D. dissertation.
[19] Sylvie Roelly-Coppoletta, A criterion of convergence of measure-valued processes: application to measure branching processes, Stochastics 17 (1986), no. 1-2, 43 – 65. · Zbl 0598.60088
[20] C. A. Rogers, Hausdorff measures, Cambridge University Press, London-New York, 1970. · Zbl 0204.37601
[21] C. A. Rogers and S. J. Taylor, Functions continuous and singular with respect to a Hausdorff measure., Mathematika 8 (1961), 1 – 31. · Zbl 0145.28701
[22] John B. Walsh, An introduction to stochastic partial differential equations, École d’été de probabilités de Saint-Flour, XIV — 1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986, pp. 265 – 439.
[23] Shinzo Watanabe, A limit theorem of branching processes and continuous state branching processes, J. Math. Kyoto Univ. 8 (1968), 141 – 167. · Zbl 0159.46201
[24] Zähle (1984), The fractal carrying dimension of a critical multiplicative measure diffusion process, Technical Report N/84/79, Friedrich-Schiller-Universifät Jena. · Zbl 0551.60082
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.