On a skewed and multifractal uni-dimensional random field, as a probabilistic representation of Kolmogorov's views on turbulence

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: $u(x)\stackrel{\mathrm{law}}{=}-u(x)\stackrel{\mathrm{law}}{=}u(0)$ (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.

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