KPZ formula for log-infinitely divisible multifractal random measures

Abstract

We consider the continuous model of log-infinitely divisible multifractal random measures (MRM) introduced in [E. Bacry et al. Comm. Math. Phys. 236 (2003) 449-475]. If M is a non degenerate multifractal measure with associated metric ρ(x,y) = M([x,y]) and structure function ζ, we show that we have the following relation between the (Euclidian) Hausdorff dimension dim H of a measurable set K and the Hausdorff dimension dim Hρ with respect to ρ of the same set: ζ(dim Hρ(K)) = dim H(K). Our results can be extended to all dimensions: inspired by quantum gravity in dimension 2, we focus on the log normal case in dimension 2. © 2011 EDP Sciences, SMAI.