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:

• (i) Homogeneity and isotropy: $u(x) \overset{{\mathrm {law}}}{=} -u(x) \overset{{\mathrm {law}}}{=}u(0)$

• (ii) Negative skewness (i.e., the 4 / 5th-law): ${\mathbb {E}\left[ (u(x+\ell )-u(x))^3\right] } \sim _{\ell \rightarrow 0+} C_3' \, \ell \,,$   for some constant $C_3'<0$

• (iii) Intermittency: ${\mathbb {E}\left[ |u(x+\ell )-u(x) |^q\right] } \sim _{\ell \rightarrow 0} C_q|\ell |^{\xi _q}\,,$   for some nonlinear spectrum $q\mapsto \xi _q$ and constants $C_q>0$

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.

Appendices

Proofs of Several Lemmas

1.1 Concerning the Analytical Properties of the Asymptotic form of Variance and Increments Variance

Let us here show the lemma entering in Sects. 3.3 and 3.4 necessary to give a meaning to the asymptotic expressions of variance and increment variance (Eqs. 3.8 and 3.13).

Lemma A.1

For $H\in ]0,1[/\{1/2\}$ and $|h|>0$, the function $(\phi \star \phi )(h)$ is differentiable and its derivative is given by

$$\begin{aligned} (\phi \star \phi )'(h) =&\int \varphi (x)\varphi '(x+h)\frac{1}{|x|^{\frac{1}{2}-H}}\frac{1}{|x+h|^{\frac{1}{2}-H}} \mathrm{d}x\nonumber \\&+ (H-1/2)\,{\mathrm { P.V. }}\int \varphi (x)\varphi (x+h)\frac{1}{|x|^{\frac{1}{2}-H}}\frac{x+h}{|x+h|^{\frac{5}{2}-H}} \mathrm{d}x, \end{aligned}$$

(A.1)

where we have defined the principal value integral (P.V.) that can be written using a convergent integral:

$$\begin{aligned}&{\mathrm { P.V. }}\int \varphi (x)\varphi (x+h)\frac{1}{|x|^{\frac{1}{2}-H}}\frac{x+h}{|x+h|^{\frac{5}{2}-H}}\mathrm{d}x\\&\quad =\int _0^{\infty }\frac{\varphi (x)}{x^{\frac{3}{2}-H}}\left[ \frac{\varphi (x-h)}{|x-h|^{\frac{1}{2}-H}}-\frac{\varphi (x+h)}{|x+h|^{\frac{1}{2}-H}}\right] \mathrm{d}x. \end{aligned}$$

Furthermore, for $H>1/2$, $(\phi \star \phi )'(h)$ is continuous, bounded over $\mathbb {R}$ and $(\phi \star \phi )'(0)=0$, and we have for $H\in ]0,1[/\{1/2\}$ the following equivalent at the origin

$$\begin{aligned} (\phi \star \phi )'(h)\mathrel {\mathop {\sim }\limits _{h\rightarrow 0^+}^{}}(H-1/2)\varphi ^2(0){{\,\mathrm{sign}\,}}(h)|h|^{2H-1}\,{\mathrm { P.V. }}\int \frac{1}{|x|^{\frac{1}{2}-H}}\frac{x+1}{|x+1|^{\frac{5}{2}-H}} \mathrm{d}x. \end{aligned}$$

Proof

To prove the expression for the derivative (A.1), regularize the singularity and pass to the limit. To get the equivalent, rescale the dummy integration variable by |h| in the second term of the RHS of (A.1) and take the limit. Remark that this equivalent is also correct for $H>1/2$, since the first term of the RHS of (A.1) behaves as h at the origin (using the fact that $\varphi$ is even), i.e.,

$$\begin{aligned}&\int \varphi (x)\varphi '(x+h)\frac{1}{|x|^{\frac{1}{2}-H}}\frac{1}{|x+h|^{\frac{1}{2}-H}} \mathrm{d}x\\&\quad \mathrel {\mathop {\sim }\limits _{h\rightarrow 0}^{}}h \left[ \int \varphi (x)\varphi ''(x)\frac{1}{|x|^{1-2H}} \mathrm{d}x+ (H-1/2)\,{\mathrm { P.V. }}\int \varphi (x)\varphi '(x)\frac{x}{|x|^{3-2H}} \mathrm{d}x\right] , \end{aligned}$$

and so tends to 0 when $h\rightarrow 0$ faster than the second term. $\square$

1.2 Concerning the Analytical Properties of the Asymptotic form of the Third Moment of Increments

Similarly, let us here show the lemma entering in Sect. 3.5 necessary to give a meaning to the asymptotic expression of the third moment of increments as $\epsilon \rightarrow 0$ (Eq. 3.17) and to its equivalent at small scales $\ell \rightarrow 0$ (Eq. 3.18).

Lemma A.2

For $H\in ]1/6,1[$ and $\forall h$, $(\Phi _\ell \star \Phi ^2_\ell )(h)$ is a continuous and bounded function of its argument. For $H\in ]0,1/6]$, $(\Phi _\ell \star \Phi ^2_\ell )$ has an additional singularity at $h=\ell$ given by

$$\begin{aligned} (\Phi _\ell \star \Phi ^2_\ell )(h) \underset{h \rightarrow \ell }{\sim } {\left\{ \begin{array}{ll} d_H\varphi ^3(0) |h-\ell |^{3H-\frac{1}{2}},&{} \; \text {if} \; H<1/6\\ 2\varphi ^3(0) \ln \frac{1}{|h-\ell |},&{} \; \text {if} \; H=1/6, \\ \end{array}\right. } \end{aligned}$$

where $d_H$ is a constant independent of $\varphi (0)$ that we can compute. Furthermore, for any $H\in ]0,1[/\{1/2\}$ we have the following equivalent at small arguments

$$\begin{aligned}&(\Phi _\ell \star \Phi ^2_\ell )(h)\\&\quad \mathrel {\mathop {\sim }\limits _{h\rightarrow 0}^{}} h\int \left[ (1/2-H)\varphi (x)-x\varphi '(x)\right] \frac{x}{|x|^{5/2-H}}\left[ \frac{\varphi (x-\ell )}{|x-\ell |^{1/2-H}}-\frac{\varphi (x+\ell )}{|x+\ell |^{1/2-H}}\right] \\&\qquad \quad \times \left[ \frac{\varphi (x-\ell )}{|x-\ell |^{1/2-H}}+\frac{\varphi (x+\ell )}{|x+\ell |^{1/2-H}}-\frac{2\varphi (x)}{|x|^{1/2-H}}\right] \mathrm{d}x. \end{aligned}$$

Proof

We have

$$\begin{aligned} (\Phi _\ell \star \Phi ^2_\ell )(h) =&\int \left[ \frac{\varphi (x+\ell /2)}{|x+\ell /2|^{1/2-H}}-\frac{\varphi (x-\ell /2)}{|x-\ell /2|^{1/2-H}}\right] \\&\times \left[ \frac{\varphi (x+h+\ell /2)}{|x+h+\ell /2|^{1/2-H}}-\frac{\varphi (x+h-\ell /2)}{|x+h-\ell /2|^{1/2-H}}\right] ^2\mathrm{d}x. \end{aligned}$$

Notice that $(\Phi _\ell \star \Phi ^2_\ell )(h)$ can be written with the following convenient form

$$\begin{aligned} (\Phi _\ell \star \Phi ^2_\ell )(h) =&\int \frac{\varphi (x-h)}{|x-h|^{1/2-H}}\left[ \frac{\varphi (x-\ell )}{|x-\ell |^{1/2-H}}-\frac{\varphi (x+\ell )}{|x+\ell |^{1/2-H}}\right] \\&\times \left[ \frac{\varphi (x-\ell )}{|x-\ell |^{1/2-H}}+\frac{\varphi (x+\ell )}{|x+\ell |^{1/2-H}}-\frac{2\varphi (x)}{|x|^{1/2-H}}\right] \mathrm{d}x, \end{aligned}$$

which shows that $(\Phi _\ell \star \Phi ^2_\ell )$ is continuous and bounded for $H\in ]1/6,1[$ and $\forall h$. For $H\in ]0,1/6]$, $(\Phi _\ell \star \Phi ^2_\ell )$ has an additional singularity at $h=\ell$. The proposed equivalent for $h\rightarrow 0$ follows from the Taylor Series of the first ratio entering in the integral (the first contribution to this development vanishes by symmetry).

Let us now take a look at the additional singularity when $H\le 1/6$. From this former expression, we see that $(\Phi _\ell \star \Phi ^2_\ell )(h)$ as the same singularity when h goes to $\ell$ as

$$\begin{aligned} \int _{|x| \;\leqslant \;1} \frac{\varphi (x+\ell -h)}{|x+\ell -h|^{1/2-H}}\frac{\varphi ^2(x)}{|x|^{1-2H}}\mathrm{d}x. \end{aligned}$$

If $H< \frac{1}{6}$, then it is equal to take, for instance, $h<\ell$,

$$\begin{aligned}&|h-\ell |^{3H-1/2}\int _{|y| \;\leqslant \;\frac{1}{|h-\ell |}} \frac{\varphi [|h-\ell |(y+1)]}{|y+1|^{1/2-H}}\frac{\varphi ^2[|h-\ell |y]}{|y|^{1-2H}}\mathrm{d}y\\&\quad \underset{|h-\ell | \rightarrow 0}{\sim } d_H\varphi ^3(0)|h-\ell |^{3H-\frac{1}{2}}, \end{aligned}$$

where

$$\begin{aligned} d_H=\int _{{\mathbb {R}}} \frac{1}{|y+1|^{\frac{1}{2}-H }|y|^{1-2H}} \mathrm{d}y. \end{aligned}$$

If $H= \frac{1}{6}$, then it is equal to

$$\begin{aligned} \int _{|y| \;\leqslant \;\frac{1}{|h-\ell |}} \frac{\varphi [|h-\ell |(y+1)]}{|y+1|^{1/3}}\frac{\varphi ^2[|h-\ell |y]}{|y|^{2/3}}\mathrm{d}y\underset{|h-\ell | \rightarrow 0}{\sim } 2 \varphi ^3(0)\ln \frac{1}{|h-\ell |}. \end{aligned}$$

$\square$

Lemma A.3

For $H\in ]1/6,1[/\{1/2\}$ and $\forall h$, $f_H$ is a continuous and bounded function of its argument. For $H\in ]0,1/6]$, $f_H$ has an additional singularity at $h=1$ given by

$$\begin{aligned} f_H(h) \underset{h \rightarrow 1}{\sim } {\left\{ \begin{array}{ll} d_H |h-1|^{3H-\frac{1}{2}},&{} \; \text {if} \; H<1/6\\ 2 \ln \frac{1}{|h-1|},&{} \; \text {if} \; H=1/6, \\ \end{array}\right. } \end{aligned}$$

where $d_H$ is the same constant entering in Lemma A.2. Furthermore, for $H\in ]0,1[/\{1/2\}$ we have the following equivalent at small arguments

$$\begin{aligned} f_H(h)&\mathrel {\mathop {\sim }\limits _{h\rightarrow 0}^{}} -(H-1/2)h\int \frac{x}{|x|^{5/2-H}}\left[ \frac{1}{|x-1|^{1/2-H}}-\frac{1}{|x+1|^{1/2-H}}\right] \\&\quad \times \left[ \frac{1}{|x-1|^{1/2-H}}+\frac{1}{|x+1|^{1/2-H}}-\frac{2}{|x|^{1/2-H}}\right] \mathrm{d}x \end{aligned}$$

and the following equivalent at large arguments

$$\begin{aligned} f_H(h)&\mathrel {\mathop {\sim }\limits _{h\rightarrow \infty }^{}} -(H-1/2)h^{H-3/2}\int x\left[ \frac{1}{|x-1|^{1/2-H}}-\frac{1}{|x+1|^{1/2-H}}\right] \\&\quad \times \left[ \frac{1}{|x-1|^{1/2-H}}+\frac{1}{|x+1|^{1/2-H}}-\frac{2}{|x|^{1/2-H}}\right] \mathrm{d}x. \end{aligned}$$

Proof

Noticing once again that $f_H$ can be written with the following convenient form

$$\begin{aligned} f_H(h) =&\int \frac{1}{|x-h|^{1/2-H}}\left[ \frac{1}{|x-1|^{1/2-H}}-\frac{1}{|x+1|^{1/2-H}}\right] \\&\times \left[ \frac{1}{|x-1|^{1/2-H}}+\frac{1}{|x+1|^{1/2-H}}-\frac{2}{|x|^{1/2-H}}\right] \mathrm{d}x, \end{aligned}$$

proofs are then similar to those of Lemma A.2. The proposed equivalent $h\rightarrow \infty$ follows from the factorization of h in the first ratio and then doing a Taylor Series. $\square$

Hypercontractivity on Wiener Chaos

This section summarizes results exposed in [29], especially Theorem 2.7.2 and Corollary 2.8.14.

Consider a real separable Hilbert space ${\mathbb {H}}$ with inner product $(\cdot ,\cdot )_{{\mathbb {H}}}$ and norm $\Vert \cdot \Vert _{{\mathbb {H}}}$. An isonormal Gaussian process X over ${\mathbb {H}}$ is a centered Gaussian process $\{X(h);h\in {\mathbb {H}}\}$ defined on some probability space $(\Omega ,{\mathcal {F}},\mathbb {P})$ such that $\mathbb {E}[X(h)X(g)]=(h,g)_{{\mathbb {H}}}$. The nth Wiener chaos ${\mathcal {H}}_n$ of X is the closed linear subspace of $L^2(\Omega ,{\mathcal {F}},\mathbb {P})$ generated by the random variables $H_n(X(h))$ with $h\in {\mathbb {H}}$ and $H_n$ the nth Hermite polynomial.

Theorem B.1

For all $q>0$ and $p\;\geqslant \;1$, there exists a constant $0<k(p,q)<\infty$ (depending only on q and p) such that

$$\begin{aligned} \frac{1}{k(p,q)}\mathbb {E}[|F|^q]^{1/q}\;\leqslant \;\mathbb {E}[F^2]^{1/2}\;\leqslant \;k(p,q)\mathbb {E}[|F|^q]^{1/q} \end{aligned}$$

for all F elements of the pth Wiener chaos of X.

In the case when the isonormal Gaussian process X is a white noise over $L^2(\mathbb {R})$, it is not hard to check that pth Wiener chaos coincides with multiple integrals of the type (see details in [29, section 2.7])

$$\begin{aligned} \int _{\mathbb {R}^p}f(x_1,\dots ,x_p)W(\mathrm{d}x_1)\dots W(\mathrm{d}x_p) \end{aligned}$$

for symmetric functions $f\in L^2(\mathbb {R}^p)$. In particular, one can see that if $\varphi , \psi$ are piecewise smooth functions with compact support, then the random variable defined by

$$\begin{aligned} u(x):=\int \varphi (x-y)Y(y)W(\mathrm{d}y),\quad \text { with }\quad Y(y):=\int \psi (y-z)W(\mathrm{d}z) \end{aligned}$$

belongs to the second Wiener chaos.

Fig. 3

Numerical estimation of the function $f_{\epsilon ,H}(h)$, at a given approximation $\epsilon =10^{-5}$ (see text). a From top to bottom, $H=0.3, 0.35, 0.38, 0.4, 0.5$. b From top to bottom, $H=0.5, 0.6, 0.7, 0.8, 0.9$

Numerics

1.1 Estimation of the Function $f_H(h)$ and Its Sign

We represent in Fig. 3 the results of the numerical integration of a approximation $f_{\epsilon ,H}$ of the function $f_{H}$ (Eq. 3.19) entering in the third moment of the increments, namely

$$\begin{aligned} f_{\epsilon ,H}(h)=&\int \left[ \frac{1}{|x+1/2|_\epsilon ^{\frac{1}{2}-H}}-\frac{1}{|x-1/2|_\epsilon ^{\frac{1}{2}-H}}\right] \nonumber \\&\times \left[ \frac{1}{|x+h+1/2|_\epsilon ^{\frac{1}{2}-H}}-\frac{1}{|x+h-1/2|_\epsilon ^{\frac{1}{2}-H}}\right] ^2\mathrm{d}x, \end{aligned}$$

(C.1)

where enters the regularized norm $|x|_\epsilon ^2 = |x|^2+\epsilon ^2$. The numerical integration is made using adaptively a Newton–Cotes 5/9 point rule, as described in Ref. [30]. For this estimation, we use $\epsilon =10^{-5}$, and we checked (data not shown) that it is representative of the limit value $\epsilon \rightarrow 0$. We study a large set of values for H, and check that indeed $f_{\epsilon ,1/2}(h)=0$ for $h>0$. This numerical study confirms the assumption on the sign of $f_H$ made in Eq. 1.13 that is positive for $H<1/2$ and negative for $H>1/2$.

Fig. 4

Numerical estimation of the statistical properties of $u_\epsilon$ (1.3) (using continuous lines); numerical details are provided in the text. a Estimation and representation of $S_2(\ell ) = \mathbb {E}\left( \delta _\ell u_\epsilon \right) ^2$ and the absolute value of $S_3(\ell ) = \mathbb {E}\left( \delta _\ell u_\epsilon \right) ^3$ as a function of the scale $\ell$, in a logarithmic fashion. b Estimation of the skewness ${\mathcal {S}}(\ell )$ (Eq. C.2). c Estimation of the flatness ${\mathcal {F}}(\ell )$ (Eq. C.3). In a–c we superimpose the estimations of these statistical quantities using $u_\epsilon ^{\tiny {\text{ g }}}$ defined in (2.5) (dotted line) instead of $u_\epsilon$ (represented with a continuous line). Moreover, in a, we add using a dotted–dashed line the logarithmic behavior of the absolute value of the third-order moment of the increments of $u_\epsilon$, i.e., $S_3(\ell )$. In a and c, we, furthermore, represent the expected power-law behaviors using dashed lines. d Logarithmic representations of the probability density functions (PDFs) of the increments $\delta _\ell u_\epsilon$ (renormalized by their respective standard deviation). Curves are arbitrary shifted vertically for clarity. From top to bottom, we have used $\log _{10}(\ell /L)=-7.2, -6, -4.8, -3.5, -2.3, -1, 0.2$

1.2 Simulation of the Random Process and Estimation of Its Statistical Properties

We here present a method to simulate the proposed random field $u_\epsilon$ defined in (1.3) in a periodic fashion, such that we can work with the discrete Fourier transform. To do so, discretize the interval [0, 1] over N collocation points. For full benefit of the fast Fourier transform (FFT) algorithm, choose N to be a power of 2. This defines the numerical resolution of the simulation, i.e., $\mathrm{d}x=1/N$. Choose, for example, as a cutoff function $\varphi _L$ a Gaussian shape, i.e., $\varphi _L(x) = e^{-\frac{x^2}{2L^2}}$. The precise shape of this function only matters at large scales; statistics at small scales are independent on it, besides its value at the origin. Choose as a regularized norm $|x|_\epsilon ^2 = |x|^2+\epsilon ^2$. Once again, the precise definition of the regularized norm does not matter since Theorem 1.2 ensures that the statistical properties of $u_\epsilon$ are independent of the regularization procedure when $\epsilon \rightarrow 0$.

Consider then two independent white fields W and ${\widehat{W}}$, each of them made of N independent realizations of a zero-average Gaussian variable of variance $\mathrm{d}x$. Define the deterministic kernels $\phi _\epsilon$ (1.5) (replace the norm |.| entering in $\phi$ by its regularized form $|.|_\epsilon$) and $k_\epsilon$ (1.6) in a periodic fashion. Take ${\widehat{X}} = k_\epsilon *{\widehat{W}}$, and use W as the remaining white field entering in the construction of $u_\epsilon$ (1.3). Convolutions are then efficiently performed in the Fourier space.

We represent in Fig. 1 an instance of the process $u_\epsilon$ (1.3), as obtained by the aforementioned numerical method. We have used for the simulation the following set of parameters: $N= 2^{20}$, $L=1/3$, $\epsilon =2\mathrm{d}x$, and the values $\gamma =\sqrt{0.025}/2$ and $H=1/3+4\gamma ^2$. These chosen values for the parameters $\gamma$ and H correspond to what is observed in turbulence (see Remark 1.7).

To go further in the characterization of the statistical properties of the field $u_\epsilon$, we perform an additional simulation at a higher resolution $N=2^{31}$ in order to estimate in a reliable way the behaviors at small scales and represent in Fig. 4 the results of our estimations. We have chosen as a cutoff length scale $L=2^{-6}$ and as a regularizing scale $\epsilon =2\mathrm{d}x$. Once again, values of the parameters are those which are realistic of turbulence, i.e., $\gamma =\sqrt{0.025}/2$ and $H=1/3+4\gamma ^2$ (see Remark 1.7). For the sake of comparison, we have furthermore made our estimations on the underlying fractional Gaussian field $u_\epsilon ^{\tiny {\text{ g }}}$ defined in (2.5).

We begin in Fig. 4a with the estimation of the second-order $S_2(\ell ) = \mathbb {E}\left( \delta _\ell u_\epsilon \right) ^2$ and third-order $S_3(\ell ) = \mathbb {E}\left( \delta _\ell u_\epsilon \right) ^2$ structure functions as a function of the scale. As far is concerned the second-order structure function, we observe for $u_\epsilon$ a power-law behavior at small scales, i.e., $S_2(\ell )\sim \ell ^{\xi (2)}$, where the function $\xi (q)$ is defined in (1.11), consistently with Theorem 1.2. As for $u_\epsilon ^{\tiny {\text{ g }}}$, we also observe a power-law behavior at small scales, that we know to be $S_2(\ell )\sim \ell ^{2H}$. Let us remark that since $\xi (2)$ is very close to 2H, it is difficult to see a difference in between these power-laws. Concerning the third-order structure function $S_3$, we recall that it vanishes for $u_\epsilon ^{\tiny {\text{ g }}}$, since densities are symmetric (and Gaussian). We represent the respective $S_3$ for $u_\epsilon$ also in Fig. 4a. Since it is expected to be negative, we more precisely represent the behavior of its absolute value in a logarithmic fashion. We indeed see that it behaves at small scales as the power-law $S_3(\ell )\sim \ell ^{\xi (3)}$, where by construction given the chosen values for H and $\gamma ^2$ we have $\xi (3)=1$. It is again consistent with Proposition 1.6.

To see more clearly the sign of $S_3$ and how it compares with the scaling of $S_2$, we represent in Fig. 4b the result of our estimation for the skewness factor ${\mathcal {S}}(\ell )$ of the increments given by

$$\begin{aligned} {\mathcal {S}}(\ell ) = \frac{\mathbb {E}\left( \delta _\ell u_\epsilon \right) ^3}{\left[ \mathbb {E}\left( \delta _\ell u_\epsilon \right) ^2\right] ^{3/2}}. \end{aligned}$$

(C.2)

We see that the present process is indeed skewed at small scales, being close to zero close to the large-scale L, and growing toward values close to -2 at small scales. Remark that the quantity ${\mathcal {S}}(\ell )$is expected to behave as a power-law of exponent $\xi (3)-\frac{3}{2}\xi (2)$ at small scales. Remark also that it is indeed negative, as required by the phenomenology of turbulence (Sect. 2.1). In comparison, we see that the Skewness factor for the Gaussian process $u_\epsilon ^{\tiny {\text{ g }}}$ is close to zero at any scales, as expected from symmetric statistical laws.

We represent in Fig. 4c the result of our estimation for the Flatness factor ${\mathcal {F}}(\ell )$ of the increments given by

$$\begin{aligned} {\mathcal {F}}(\ell ) = \frac{\mathbb {E}\left( \delta _\ell u_\epsilon \right) ^4}{\left[ \mathbb {E}\left( \delta _\ell u_\epsilon \right) ^2\right] ^{2}}. \end{aligned}$$

(C.3)

Whereas ${\mathcal {F}}(\ell )$ is independent on the scale for the Gaussian process $u_\epsilon ^{\tiny {\text{ g }}}$ (and equal to 3), we see that it behaves as a power-law of exponent $\xi (4)-2\xi (2)$ at small scales.

Finally, we represent in Fig. 4d the histograms of the values of the increments $\delta _\ell u_\epsilon$ for several scales given in the caption. We see that whereas the histogram of the increments of the process $u_\epsilon$ is close to a Gaussian function at large scales $\ell \sim L$, they develop heavier and heavier tails at smaller scales, with a noticeable asymmetry.