KPZ formula for log-infinitely divisible multifractal random measures

Author： Rhodes Rémi   Vargas Vincent  

Publisher： Edp Sciences

E-ISSN： 1262-3318|15|issue|358-371

ISSN： 1292-8100

Source： ESAIM: Probability and Statistics, Vol.15, Iss.issue, 2012-01, pp. : 358-371

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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 dimH of a measurable set K and the Hausdorff dimension dimHρ with respect to ρ of the same set: ζ(dimHρ(K)) = dimH(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.

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