KPZ formula for log-infinitely divisible multifractal random measures

Rémi Rhodes; Vincent Vargas

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arXiv · 0807.1036

KPZ formula for log-infinitely divisible multifractal random measures

Rémi Rhodes·Vincent Vargas

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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 $ρ(x,y)=M([x,y])$ 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 ${\rm dim}_H^ρ$ with respect to ρof the same set: $ζ({\rm dim}_H^ρ(K))={\r m dim}_H(K)$. 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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Rémi Rhodes, Vincent Vargas. 2008-07-27. KPZ formula for log-infinitely divisible multifractal random measures. https://arxiv.org/abs/0807.1036

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