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.
Mots clés : 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}, 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/} }
TY - JOUR AU - Rhodes, Rémi AU - Vargas, Vincent TI - KPZ formula for log-infinitely divisible multifractal random measures JO - ESAIM: Probability and Statistics PY - 2011 SP - 358 EP - 371 VL - 15 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2010007/ DO - 10.1051/ps/2010007 LA - en ID - PS_2011__15__358_0 ER -
%0 Journal Article %A Rhodes, Rémi %A Vargas, Vincent %T KPZ formula for log-infinitely divisible multifractal random measures %J ESAIM: Probability and Statistics %D 2011 %P 358-371 %V 15 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2010007/ %R 10.1051/ps/2010007 %G en %F PS_2011__15__358_0
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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