English

KPZ formula for log-infinitely divisible multifractal random measures

Probability 2008-07-28 v2

Abstract

We consider the continuous model of log-infinitely divisible multifractal random measures (MRM) introduced in \cite{bacry} . If M is a non degenerate multifractal measure with associated metric ρ(x,y)=M([x,y])\rho(x,y)=M([x,y]) and structure function \zeta\zet a, we show that we have the following relation between the (Euclidian) Hausdorff dimension dimH{\rm dim}_H of a measurable set K and the Hausdorff dimension dimHρ{\rm dim}_H^{\rho} with respect to \rho of the same set: ζ(dimHρ(K))=m˚dimH(K)\zeta({\rm dim}_H^{\rho}(K))={\r m dim}_H(K). Our results can be extended to higher dimensions in the log normal case: inspired by quantum gravity in dime nsion 2, we consider the 2 dimensional case.

Keywords

Cite

@article{arxiv.0807.1036,
  title  = {KPZ formula for log-infinitely divisible multifractal random measures},
  author = {Rémi Rhodes and Vincent Vargas},
  journal= {arXiv preprint arXiv:0807.1036},
  year   = {2008}
}

Comments

Revised version: added the two dimensional case

R2 v1 2026-06-21T10:58:05.670Z