## The Annals of Probability

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

### The Carrying Dimension of a Stochastic Measure Diffusion

Donald A. Dawson and Kenneth J. Hochberg

#### 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.