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

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{l}\mathrm{a}\mathrm{w}$}= -u(x) \overset{$\mathrm{l}\mathrm{a}\mathrm{w}$}=u(0)$(ii)Negativeskewness(i\mathrm{.}e\mathrm{.}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: $\mathbb{E}\mathrm{\mid }u(x+\mathrm{\ell })-u(x){\mathrm{\mid }}^q{\asymp }_{\mathrm{\ell }\to 0}\mathrm{\mid }\mathrm{\ell }{\mathrm{\mid }}^{{\xi }_q},$ 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 $H<1\mathrm{/}2$). 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.