KPZ formula for log-infinitely divisible multifractal random measures
ESAIM: Probability and Statistics, Tome 15 (2011), pp. 358-371.

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.

DOI : 10.1051/ps/2010007
Classification : 60G57, 28A78, 28A80
Mots-clés : random measures, Hausdorff dimensions, multifractal processes 
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     author = {Rhodes, R\'emi and Vargas, Vincent},
     title = {KPZ formula for log-infinitely divisible multifractal random measures},
     journal = {ESAIM: Probability and Statistics},
     pages = {358--371},
     publisher = {EDP-Sciences},
     volume = {15},
     year = {2011},
     doi = {10.1051/ps/2010007},
     mrnumber = {2870520},
     zbl = {1268.60070},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps/2010007/}
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Rhodes, Rémi; Vargas, Vincent. KPZ formula for log-infinitely divisible multifractal random measures. ESAIM: Probability and Statistics, Tome 15 (2011), pp. 358-371. doi : 10.1051/ps/2010007. http://www.numdam.org/articles/10.1051/ps/2010007/

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