The Annals of Probability

The Carrying Dimension of a Stochastic Measure Diffusion

Donald A. Dawson and Kenneth J. Hochberg

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Abstract

A multiplicative stochastic measure diffusion process in $R^d$ is the continuous analogue of an infinite particle branching Markov process in which the particles move in $R^d$ according to a symmetric stable process of index $\alpha 0 < \alpha \leqslant 2$. The main result of this paper is that there is a random carrying set whose Hausdorff dimension is almost surely less than or equal to $\alpha$. As a corollary it follows that the corresponding random measure is singular for $d > \alpha$. The latter result is also proved by a different approach in the case $d = \alpha$.

Article information

Source
Ann. Probab. Volume 7, Number 4 (1979), 693-703.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aop/1176994991

JSTOR
links.jstor.org

Digital Object Identifier
doi:10.1214/aop/1176994991

Mathematical Reviews number (MathSciNet)
MR537215

Zentralblatt MATH identifier
0411.60084

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J60: Diffusion processes [See also 58J65] 55C10

Keywords
Random measure measure diffusion process Hausdorff dimension

Citation

Dawson, Donald A.; Hochberg, Kenneth J. The Carrying Dimension of a Stochastic Measure Diffusion. Ann. Probab. 7 (1979), no. 4, 693--703. doi:10.1214/aop/1176994991. http://projecteuclid.org/euclid.aop/1176994991.


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