ESAIM: Probability and Statistics

Research Article

KPZ formula for log-infinitely divisible multifractal random measures

Rémi Rhodesa1 and Vincent Vargasa1

a1 Université Paris-Dauphine, Ceremade, CNRS, UMR 7534, 75016 Paris, France.;


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.

(Received August 17 2009)

(Online publication January 5 2012)

Key Words:

  • Random measures;
  • Hausdorff dimensions;
  • multifractal processes

Mathematics Subject Classification:

  • 60G57;
  • 28A78;
  • 28A80
  • [1] E. Bacry and J.F. Muzy, Log-infinitely divisible multifractal processes. Comm. Math. Phys. 236 (2003) 449–475. [OpenURL Query Data]  [Google Scholar]
  • [2] E. Bacry, A. Kozhemyak and J.-F. Muzy, Continuous cascade models for asset returns. J. Econ. Dyn. Control 32 (2008) 156–199. [OpenURL Query Data]  [Google Scholar]
  • [3] J. Barral and B.B. Mandelbrot, Multifractal products of cylindrical pulses. Probab. Theory Relat. Fields 124 (2002) 409–430. [OpenURL Query Data]  [Google Scholar]
  • [4] I. Benjamini and O. Schramm, KPZ in one dimensional random geometry of multiplicative cascades. Com. Math. Phys. 289 (2009) 653–662. [OpenURL Query Data]  [Google Scholar]
  • [5] P. Billingsley, Ergodic Theory and Information. Wiley New York (1965).
  • [6] B. Castaing, Y. Gagne and E. Hopfinger, Velocity probability density functions of high Reynolds number turbulence. Physica D 46 (1990) 177–200. [OpenURL Query Data]  [Google Scholar]
  • [7] F. David, Conformal Field Theories Coupled to 2-D Gravity in the Conformal Gauge. Mod. Phys. Lett. A 3 (1988).
  • [8] J. Duchon, R. Robert and V. Vargas, Forecasting volatility with the multifractal random walk model, to appear in Mathematical Finance, available at
  • [9] B. Duplantier and S. Sheffield, in preparation (2008).
  • [10] K.J. Falconer, The geometry of fractal sets. Cambridge University Press (1985). [Google Scholar]
  • [11] U. Frisch, Turbulence. Cambridge University Press (1995). [Google Scholar]
  • [12] J.-P. Kahane, Sur le chaos multiplicatif. Ann. Sci. Math. Québec 9 (1985) 105–150. [OpenURL Query Data]  [Google Scholar]
  • [13] V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Fractal structure of 2D-quantum gravity. Modern Phys. Lett A 3 (1988) 819–826. [OpenURL Query Data]  [Google Scholar]
  • [14] G. Lawler, Conformally Invariant Processes in the Plane. A.M.S (2005).
  • [15] Q. Liu, On generalized multiplicative cascades. Stochastic Processes their Appl.. 86 (2000) 263–286.
  • [16] B.B. Mandelbrot, A possible refinement of the lognormal hypothesis concerning the distribution of energy in intermittent turbulence, Statistical Models and Turbulence, La Jolla, CA, Lecture Notes in Phys. No. 12. Springer (1972) 333–335.
  • [17] B.B. Mandelbrot, Multiplications aléatoires et distributions invariantes par moyenne pondérée aléatoire. CRAS, Paris 278 (1974) 289–292, 355–358.
  • [18] B. Rajput and J. Rosinski, Spectral representations of infinitely divisible processes. Probab. Theory Relat. Fields 82 (1989) 451–487. [OpenURL Query Data]  [Google Scholar]
  • [19] R. Robert and V. Vargas, Gaussian Multiplicative Chaos revisited, available on arxiv at the URL, to appear in the Annals of Probability.
  • [20] S. Sheffield, Gaussian free fields for mathematicians. Probab. Theory Relat. Fields 139 (1989) 521–541. [OpenURL Query Data]  [Google Scholar]