The Carrying Dimension of a Stochastic Measure Diffusion

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 $d = \alpha$.