The multifractal structure of super-Brownian motion

We calculate the multifractal spectrum and mass exponents for super-Brownian motion in three or more dimensions. The former is trivial for points of unusually high density but not for points in the support of unusually low density. This difference is due to the presence of sets of points in the support (of positive dimension) about which there are asymptotically large empty annuli. This behaviour is quite different from that of ordinary Brownian motion and invalidates the multifractal formalism in the physics literature. The mass exponents for packing and Hausdorff measure are distinct, and both are piecewise linear