KPZ formula for log-infinitely divisible multifractal random measures
ESAIM: Probability and Statistics, Volume 15  (2011), p. 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 : https://doi.org/10.1051/ps/2010007
Classification:  60G57,  28A78,  28A80
Keywords: random measures, Hausdorff dimensions, multifractal processes
@article{PS_2011__15__358_0,
author = {Rhodes, R\'emi and Vargas, Vincent},
title = {KPZ formula for log-infinitely divisible multifractal random measures},
journal = {ESAIM: Probability and Statistics},
publisher = {EDP-Sciences},
volume = {15},
year = {2011},
pages = {358-371},
doi = {10.1051/ps/2010007},
zbl = {1268.60070},
mrnumber = {2870520},
language = {en},
url = {http://www.numdam.org/item/PS_2011__15__358_0}
}

Rhodes, Rémi; Vargas, Vincent. KPZ formula for log-infinitely divisible multifractal random measures. ESAIM: Probability and Statistics, Volume 15 (2011) , pp. 358-371. doi : 10.1051/ps/2010007. http://www.numdam.org/item/PS_2011__15__358_0/

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