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 \cite{bacry} . If M is a non degenerate multifractal measure with associated metric $\rho (x,y)=M([x,y])$ and structure function $\zet a$, we show that we have the following relation between the (Euclidian) Hausdorff dimension ${\dim }_H$ of a measurable set K and the Hausdorff dimension ${\dim }_H^{\rho }$ with respect to ρof the same set: $\zeta ({\dim }_H^{\rho }(K))={\mathring{m}dim}_H(K)$. Our results can be extended to higher dimensions in the log normal case: inspired by quantum gravity in dime nsion 2, we consider the 2 dimensional case.

Revised version: added the two dimensional case