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KPZ formula for log-infinitely divisible multifractal random measures

Published online by Cambridge University Press:  05 January 2012

Rémi Rhodes  and
Vincent Vargas
Rémi Rhodes
Affiliation:
Université Paris-Dauphine, Ceremade, CNRS, UMR 7534, 75016 Paris, France. rhodes@ceremade.dauphine.fr; vargas@ceremade.dauphine.fr
Vincent Vargas
Affiliation:
Université Paris-Dauphine, Ceremade, CNRS, UMR 7534, 75016 Paris, France. rhodes@ceremade.dauphine.fr; vargas@ceremade.dauphine.fr

Abstract

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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.

Keywords

Random measuresHausdorff dimensionsmultifractal processes

Information

Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 15: Supplement: In honor of Marc Yor , 2011 , pp. 358 - 371
Copyright
© EDP Sciences, SMAI, 2011

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