KPZ formula for log-infinitely divisible multifractal random measures

Details KPZ formula for log-infinitely divisible multifractal random measures KPZ formula for log-infinitely divisible multifractal random measures

2008-07-07

arxiv.org/abs/0807.1036v2

Mathematics

Probability

Revised version: added the two dimensional case

Scientific paper

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 ${\rm dim}_H$ of a measurable set K and the Hausdorff dimension ${\rm dim}_H^{\rho}$ with respect to \rho of the same set: $\zeta({\rm dim}_H^{\rho}(K))={\r m dim}_H(K)$. 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.

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