KPZ formula for log-infinitely divisible multifractal random measures

Rémi Rhodes; Vincent Vargas

ESAIM: Probability and Statistics (2012)

  • Volume: 15, page 358-371
  • ISSN: 1292-8100

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.

How to cite

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Rhodes, Rémi, and Vargas, Vincent. "KPZ formula for log-infinitely divisible multifractal random measures." ESAIM: Probability and Statistics 15 (2012): 358-371. <http://eudml.org/doc/222457>.

@article{Rhodes2012,
abstract = { 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. },
author = {Rhodes, Rémi, Vargas, Vincent},
journal = {ESAIM: Probability and Statistics},
keywords = {Random measures; Hausdorff dimensions; multifractal processes; infinitely divisible random measure; Hausdorff dimension; Gaussian free field},
language = {eng},
month = {1},
pages = {358-371},
publisher = {EDP Sciences},
title = {KPZ formula for log-infinitely divisible multifractal random measures},
url = {http://eudml.org/doc/222457},
volume = {15},
year = {2012},
}

TY - JOUR
AU - Rhodes, Rémi
AU - Vargas, Vincent
TI - KPZ formula for log-infinitely divisible multifractal random measures
JO - ESAIM: Probability and Statistics
DA - 2012/1//
PB - EDP Sciences
VL - 15
SP - 358
EP - 371
AB - 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.
LA - eng
KW - Random measures; Hausdorff dimensions; multifractal processes; infinitely divisible random measure; Hausdorff dimension; Gaussian free field
UR - http://eudml.org/doc/222457
ER -

References

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