On a Skewed and Multifractal Unidimensional 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: 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. This necessary inter-dependence is added in order to reproduce the asymmetrical, i.e., skewed, nature of the probability laws at small scales.