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hausdorff dimensions

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KPZ formula for log-infinitely divisible multifractal random measures
Rémi Rhodes, Vincent Vargas
Journal:

ESAIM: Probability and Statistics / Volume 15 / 2011

Published online by Cambridge University Press:

05 January 2012, pp. 358-371

Print publication:

2011
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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.