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

We construct, for the first time to our knowledge, a one-dimensional stochastic field ${u(x)}_{xin mathbb{R}}$ 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 $qmapsto xi_q$ 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 $Happrox 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.