KPZ formula for log-infinitely divisible multifractal random measures

Rhodes, Rémi; Vargas, Vincent

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

Type

Article accepté pour publication ou publié

External document link

http://hal.archives-ouvertes.fr/hal-00293878/en/

Date

2011

Journal name

ESAIM. Probability and Statistics

Volume

Publisher

EDP Sciences

Pages

358-371

Abstract (EN)

We consider the continuous model of log-infinitely divisible multifractal random measures (MRM) introduced in [1]. 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 mea- surable set K and the Hausdorff dimension dimρ H with respect to ρ of the same set: ζ (dimρ (K)) = dimH (K). Our results can be extended to higher dimensions in the log normal case: inspired by quantum gravity in dimension 2, we consider the 2 dimensional case.

Subjects / Keywords

Multifractal processes.; Random measures; Hausdorff dimensions