{
  "id": "1712.00332",
  "version": "v1",
  "published": "2017-12-01T14:23:16.000Z",
  "updated": "2017-12-01T14:23:16.000Z",
  "title": "On a skewed and multifractal uni-dimensional random field, as a probabilistic representation of Kolmogorov's views on turbulence",
  "authors": [
    "Laurent Chevillard",
    "Christophe Garban",
    "Rémi Rhodes",
    "Vincent Vargas"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We construct, for the first time to our knowledge, a one-dimensional stochastic field $\\{u(x)\\}_{x\\in \\mathbb{R}}$ which satisfies the following axioms which are at the core of the phenomenology of turbulence mainly due to Kolmogorov: (i) Homogeneity and isotropy: $u(x) \\overset{\\mathrm{law}}= -u(x) \\overset{\\mathrm{law}}=u(0)$ (ii) Negative skewness (i.e. the $4/5^{\\mbox{\\tiny th}}$-law): \\\\ $\\mathbb{E}{(u(x+\\ell)-u(x))^3} \\sim_{\\ell \\to 0+} - C \\, \\ell\\,,$ \\, for some constant $C>0$ (iii) Intermittency: $\\mathbb{E}{|u(x+\\ell)-u(x) |^q} \\asymp_{\\ell \\to 0} |\\ell|^{\\xi_q}\\,,$ for some non-linear spectrum $q\\mapsto \\xi_q$ Since then, it has been a challenging problem to combine axiom (ii) with axiom (iii) (especially for Hurst indexes of interest in turbulence, namely $H<1/2$). In order to achieve simultaneously both axioms, we disturb with two ingredients a underlying fractional Gaussian field of parameter $H\\approx \\frac 1 3 $. The first ingredient is an independent Gaussian multiplicative chaos (GMC) of parameter $\\gamma$ that mimics the intermittent, i.e. multifractal, nature of the fluctuations. The second one is a field that correlates in an intricate way the fractional component and the GMC without additional parameters, a necessary inter-dependence in order to reproduce the asymmetrical, i.e. skewed, nature of the probability laws at small scales.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-01T14:23:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K40",
      "76F55",
      "76F30"
    ],
    "keywords": [
      "multifractal uni-dimensional random field",
      "kolmogorovs views",
      "probabilistic representation",
      "turbulence",
      "one-dimensional stochastic field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}