Electronic Journal of Probability

Multidimensional Multifractal Random Measures

Rémi Rhodes and Vincent Vargas

Full-text: Open access

Abstract

We construct and study space homogeneous and isotropic random measures (MMRM) which generalize the so-called MRM measures constructed by previous authors. Our measures satisfy an exact scale invariance equation and are therefore natural models in dimension 3 for the dissipation measure in a turbulent flow.

Article information

Source
Electron. J. Probab., Volume 15 (2010), paper no. 9, 241-258.

Dates
Accepted: 1 January 2010
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819794

Digital Object Identifier
doi:10.1214/EJP.v15-746

Mathematical Reviews number (MathSciNet)
MR2609587

Zentralblatt MATH identifier
1201.60046

Subjects
Primary: 60G57: Random measures
Secondary: 28A78: Hausdorff and packing measures 28A80: Fractals [See also 37Fxx]

Keywords
Random measures Multifractal processes

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Rhodes, Rémi; Vargas, Vincent. Multidimensional Multifractal Random Measures. Electron. J. Probab. 15 (2010), paper no. 9, 241--258. doi:10.1214/EJP.v15-746. https://projecteuclid.org/euclid.ejp/1464819794


Export citation

References

  • E. Bacry, J.F. Muzy. Log-infinitely divisible multifractal processes. Comm. Math. Phys. 236 (2003), 449–475.
  • J. Barral, B.B. Mandelbrot. Multifractal products of cylindrical pulses. Probab. Theory Relat. Fields 124 (2002), 409–430.
  • B. Castaing, Y. Gagne, E.J. Hopfinger. Velocity probability density-functions of high Reynolds-number turbulence. Physica D 46 (1990), 177–200.
  • B. Castaing, Y. Gagne, M. Marchand. Conditional velocity pdf in 3-D turbulence. J. Phys. II France 4 (1994), 1–8.
  • P. Chainais. Multidimensional infinitely divisible cascades. Application to the modelling of intermittency in turbulence. European Physical Journal B 51 (2006), 229–243.
  • B. Duplantier, S. Sheffield. Liouville Quantum Gravity and KPZ. Available on arxiv at the URL http://arxiv.org/abs/0808.1560
  • U. Frisch. Turbulence, Cambridge University Press (1995).
  • T. Gneiting. Criteria of Polya type for radial positive definite functions. Proceedings of the American Mathematical Society 129 (2001), 2309–2318.
  • F. Hiai, D. Petz. The semicircle law, free random variables and Entropy, A.M.S. (2000).
  • J.P. Kahane. Positive martingales and random measures. Chi. Ann. Math 8B (1987), 1–12.
  • J.P. Kahane. Sur le chaos multiplicatif. Ann. Sci. Math. Quebec 9 (1985), 105–150.
  • V.G. Knizhnik, A.M. Polyakov, A.B. Zamolodchikov. Fractal structure of 2D-quantum gravity. Modern Phys. Lett A 3 (1988), 819–826.
  • W. Rudin. An extension theorem for positive-definite functions. Duke Math Journal 37 (1970), 49–53.
  • B. Rajput, J. Rosinski. Spectral representations of infinitely divisible processes. Probab. Theory Relat. Fields 82 (1989), 451–487.
  • R. Robert, V. Vargas. Gaussian Multiplicative Chaos revisited. To appear in the Annals of Probability, available on arxiv at the URL http://arxiv.org/abs/0807.1030
  • R. Rhodes, V. Vargas. KPZ formula for log-infinitely divisible multifractal random measures. To appear in ESAIM:PS, available on arxiv at the URL http://arxiv.org/abs/0807.1036
  • F. Schmitt, D. Lavallee, D. Schertzer, S. Lovejoy. Empirical determination of universal multifractal exponents in turbulent velocityfields. Phys. Rev. Lett. 68 (1992), 305–308.
  • Z.S. She, E. Leveque. Universal scaling laws in fully developed turbulence. Phys. Rev. Lett. 72 (1994), 336–339.
  • G. Stolovitzky, P. Kailasnath, K.R. Sreenivasan. Kolmogorov's Refined Similarity Hypotheses. Phys. Rev. Lett. 69 (1992), 1178–1181.
  • D.W. Stroock, S.R.S. Varadhan. Multidimensionnal Diffusion Processes Grundlehren der Mathematischen Wissenschaft 233, Springer, Berlin et al., (1979).