SUPER BROWNIAN MOTION IS A FRACTAL MEASURE FOR WHICH THE MULTIFRACTAL FORMALISM IS INVALID

Abstract

Whenever Xt is the measure value of super Brownian motion in Rd(d≥3), and [Formula: see text]St is the topological support of Xt it is known7 that there is a positive constant c, depending only on d, such that for every Borel set A, [Formula: see text] There is no such exact measure function for packing measure, but it follows from the precise results in Ref. 16 that the packing dimension as well as the Hausdorff dimension of St is 2. This means that Xt is dimension regular with exact dimension 2. We describe some of the key ideas, written up in Ref. 24, which show that a.s. [Formula: see text] while [Formula: see text] is not empty for 2<β<4. Further Aβ is not dimension regular since dim Aβ=[Formula: see text]. For this reason the multifractal formalism used in Ref. 11 or Ref. 9 is invalid because their function τ(q) cannot exist.