KPZ formula for log-infinitely divisible multifractal random measures
ESAIM: Probability and Statistics, Volume 15  (2011), p. 358-371

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.

DOI : https://doi.org/10.1051/ps/2010007
Classification:  60G57,  28A78,  28A80
Keywords: 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},
     publisher = {EDP-Sciences},
     volume = {15},
     year = {2011},
     pages = {358-371},
     doi = {10.1051/ps/2010007},
     zbl = {1268.60070},
     mrnumber = {2870520},
     language = {en},
     url = {http://www.numdam.org/item/PS_2011__15__358_0}
}
Rhodes, Rémi; Vargas, Vincent. KPZ formula for log-infinitely divisible multifractal random measures. ESAIM: Probability and Statistics, Volume 15 (2011) , pp. 358-371. doi : 10.1051/ps/2010007. http://www.numdam.org/item/PS_2011__15__358_0/

[1] E. Bacry and J.F. Muzy, Log-infinitely divisible multifractal processes. Comm. Math. Phys. 236 (2003) 449-475. | MR 2021198 | Zbl 1032.60046

[2] E. Bacry, A. Kozhemyak and J.-F. Muzy, Continuous cascade models for asset returns. J. Econ. Dyn. Control 32 (2008) 156-199. | MR 2381693 | Zbl 1181.91338

[3] J. Barral and B.B. Mandelbrot, Multifractal products of cylindrical pulses. Probab. Theory Relat. Fields 124 (2002) 409-430. | MR 1939653 | Zbl 1014.60042

[4] I. Benjamini and O. Schramm, KPZ in one dimensional random geometry of multiplicative cascades. Com. Math. Phys. 289 (2009) 653-662. | MR 2506765 | Zbl 1170.83006

[5] P. Billingsley, Ergodic Theory and Information. Wiley New York (1965). | MR 192027 | Zbl 0141.16702

[6] B. Castaing, Y. Gagne and E. Hopfinger, Velocity probability density functions of high Reynolds number turbulence. Physica D 46 (1990) 177-200. | Zbl 0718.60097

[7] F. David, Conformal Field Theories Coupled to 2-D Gravity in the Conformal Gauge. Mod. Phys. Lett. A 3 (1988). | MR 981529

[8] J. Duchon, R. Robert and V. Vargas, Forecasting volatility with the multifractal random walk model, to appear in Mathematical Finance, available at http://arxiv.org/abs/0801.4220. | MR 2881881 | Zbl 1279.60051

[9] B. Duplantier and S. Sheffield, in preparation (2008).

[10] K.J. Falconer, The geometry of fractal sets. Cambridge University Press (1985). | MR 867284 | Zbl 0587.28004

[11] U. Frisch, Turbulence. Cambridge University Press (1995). | MR 1428905 | Zbl 0832.76001

[12] J.-P. Kahane, Sur le chaos multiplicatif. Ann. Sci. Math. Québec 9 (1985) 105-150. | MR 829798 | Zbl 0596.60041

[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. | MR 947880

[14] G. Lawler, Conformally Invariant Processes in the Plane. A.M.S (2005). | MR 2129588 | Zbl 1074.60002

[15] Q. Liu, On generalized multiplicative cascades. Stochastic Processes their Appl.. 86 (2000) 263-286. | MR 1741808 | Zbl 1028.60087

[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. | Zbl 0227.76081

[17] B.B. Mandelbrot, Multiplications aléatoires et distributions invariantes par moyenne pondérée aléatoire. CRAS, Paris 278 (1974) 289-292, 355-358. | Zbl 0276.60096

[18] B. Rajput and J. Rosinski, Spectral representations of infinitely divisible processes. Probab. Theory Relat. Fields 82 (1989) 451-487. | MR 1001524 | Zbl 0659.60078

[19] R. Robert and V. Vargas, Gaussian Multiplicative Chaos revisited, available on arxiv at the URL http://arxiv.org/abs/0807.1036v1, to appear in the Annals of Probability. | MR 2642887 | Zbl 1191.60066

[20] S. Sheffield, Gaussian free fields for mathematicians. Probab. Theory Relat. Fields 139 (1989) 521-541. | MR 2322706 | Zbl 1132.60072