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

Abstract:

If $d > \alpha$, it is shown that the $d$-dimensional branching diffusion of index $\alpha$, studied by Dawson and others, distributes its mass over a random support in a uniform manner with respect to the Hausdorff ${\phi _\alpha }$-measure, where ${\phi _\alpha }(x) = {x^\alpha }\log \log 1/x$. More surprisingly, it does so for all positive times simultaneously. Slightly less precise results are obtained in the critical case $d = \alpha$. In particular, the process is singular at all positive times a.s. for $d \geqslant \alpha$.