Dynamical Fractional and Multifractal Fields

Abstract

Motivated by the modeling of three-dimensional fluid turbulence, we define and study a class of stochastic partial differential equations (SPDEs) that are randomly stirred by a spatially smooth and uncorrelated in time forcing term. To reproduce the fractional, and more specifically multifractal, regularity nature of fully developed turbulence, these dynamical evolutions incorporate an homogenous pseudo-differential linear operator of degree 0 that takes care of transferring energy that is injected at large scales in the system, towards smaller scales according to a cascading mechanism. In the simplest situation which concerns the development of fractional regularity in a linear and Gaussian framework, we derive explicit predictions for the statistical behaviors of the solution at finite and infinite time. Doing so, we realize a cascading transfer of energy using linear, although non local, interactions. These evolutions can be seen as a stochastic version of recently proposed systems of forced waves intended to model the regime of weak wave turbulence in stratified and rotational flows. To include multifractal, i.e. intermittent, corrections to this picture, we get some inspiration from the Gaussian multiplicative chaos, which is known to be multifractal, to motivate the introduction of an additional quadratic interaction in these dynamical evolutions. Because the theoretical analysis of the obtained class of nonlinear SPDEs is much more demanding, we perform numerical simulations and observe the non-Gaussian and in particular skewed nature of their solution.

Proof of Proposition 5

Let us start with proposing Lemma 1, similarly to the Lemma 2.1 of Ref. [52], but here for complex Gaussian variables:

Lemma 1

Consider a complex zero average Gaussian random variable Z, a function $F:{\mathbb {C}}\rightarrow {\mathbb {C}}$ and its derivative $F'$ that grows at most exponentially. We have

$$\begin{aligned} {\mathbb {E}}\left[ ZF(Z)\right] ={\mathbb {E}}(Z^2){\mathbb {E}}\left[ F'(Z)\right] . \end{aligned}$$

(74)

More generally, considering the collection of $(n+1)$ complex Gaussian variables $(Z,Z_1,\ldots ,Z_n)$ and the function $F:{\mathbb {C}}^n\rightarrow {\mathbb {C}}$. We have the following Gaussian integration by parts formula

$$\begin{aligned} {\mathbb {E}}\left[ ZF(Z_1,\ldots ,Z_n)\right] =\sum _{k=1}^n{\mathbb {E}}(ZZ_k){\mathbb {E}}\left[ \frac{\partial F}{\partial x_k}(Z_1,\ldots ,Z_n)\right] . \end{aligned}$$

(75)

Proof of Proposition 5

Concerning the average of $v_{H,\gamma }$ (Eq. 59), we make use of Eq. (75) and obtain

$$\begin{aligned} {\mathbb {E}}\left[ e^{\gamma {\widetilde{P}}_0u_0(t,y)} u_0(t,y)\right] =\gamma {\mathbb {E}}\left[ u_0(t,y){\widetilde{P}}_0u_0(t,y)\right] e^{\frac{\gamma ^2}{2}{\mathbb {E}}\left[ {\widetilde{v}}_0^2(t,y)\right] }=0, \end{aligned}$$

(76)

because for any positions, ${\mathbb {E}}\left[ u_0(t,y)u_0(t,z)\right] =0$ (Eq. 15), which shows that ${\mathbb {E}}v_{H,\gamma }=0$.

The calculation of the variance is done in a similar way, and requires the following step: Making use of Eq. (75), we obtain

$$\begin{aligned}&{\mathbb {E}}\left[ u_0(t,y_1) u_0^*(t,y_2)e^{\gamma \left( {\widetilde{P}}_0u_0(t,y_1)+{\widetilde{P}}_0u_0^*(t,y_2)\right) }\right] ={\mathcal {C}}_{u_0} (t,y_1-y_2)e^{\gamma ^2{\mathcal {C}}_{{\widetilde{v}}_0}(t,y_1-y_2)}\nonumber \\&\qquad +\gamma {\mathbb {E}}\left[ u_0(t,y_1) {\widetilde{P}}_0u_0^*(t,y_2)\right] {\mathbb {E}}\left[ u_0^*(t,y_2)e^{\gamma \left( {\widetilde{P}}_0u_0(t,y_1)+{\widetilde{P}}_0u_0^*(t,y_2)\right) }\right] \nonumber \\&\quad = \left( {\mathcal {C}}_{u_0}(t,y_1-y_2) + \gamma ^2 {\mathbb {E}}\left[ u_0(t,y_1) {\widetilde{P}}_0u_0^*(t,y_2)\right] {\mathbb {E}}\left[ u_0^*(t,y_2) {\widetilde{P}}_0u_0(t,y_1)\right] \right) e^{\gamma ^2\mathcal C_{{\widetilde{v}}_0}(t,y_1-y_2)}. \end{aligned}$$

(77)

Using the odd symmetry of the function ${\widetilde{P}}_0(x)=- {\widetilde{P}}_0(-x)$, notice that

$$\begin{aligned} {\mathcal {K}}(t,y_1-y_2)\equiv {\mathbb {E}}\left[ u_0(t,y_1) {\widetilde{P}}_0u_0^*(t,y_2)\right]&=\int {\widetilde{P}}_0(y_2-z){\mathcal {C}}_{u_0}(t,y_1-z)dz\nonumber \\&=-{\mathbb {E}}\left[ u_0^*(t,y_2) {\widetilde{P}}_0u_0(t,y_1)\right] \nonumber \\&=-{\mathcal {K}}^*(t,y_2-y_1) \end{aligned}$$

(78)

and we obtain

$$\begin{aligned} {\mathbb {E}}\left[ \left| v_{H,\gamma }(t,x)\right| ^2\right] = \int e^{2i\pi ky}\frac{1}{|k|_{1/L}^{2H+1}}\left[ {\mathcal {C}}_{u_0}(t,y)-\gamma ^2 {\mathcal {K}}^2(t,y)\right] e^{\gamma ^2\mathcal C_{{\widetilde{v}}_0}(t,y)}dkdy. \end{aligned}$$

(79)

Remark now that

$$\begin{aligned} \int _0^{+\infty }(P_H\star P_H)'(y)e^{\gamma ^2{\mathcal {C}}_{{\widetilde{v}}_0}(t,y)}dy =&-(P_H\star P_H)(0)e^{\gamma ^2{\mathcal {C}}_{{\widetilde{v}}_0}(t,0)} \nonumber \\&- \int _0^{+\infty }(P_H\star P_H)(y)\gamma ^2\mathcal C'_{{\widetilde{v}}_0}(t,y)e^{\gamma ^2\mathcal C_{{\widetilde{v}}_0}(t,y)}dy, \end{aligned}$$

(80)

where we have introduced the correlation product $\star$, defined by, for any appropriate real functions f and g,

$$\begin{aligned} (f\star g)(y)=\int f(x)g(x+y)dx = \int e^{2i\pi ky} {\widehat{f}}^*(k){\widehat{g}}(k)dk, \end{aligned}$$

(81)

such that

$$\begin{aligned}&{\mathbb {E}}\left[ \left| v_{H,\gamma }(t,x)\right| ^2\right] + \frac{{\mathcal {C}}_f(0)}{2|c|}\int _0^{+\infty }(P_H\star P_H)'(y)e^{\gamma ^2{\mathcal {C}}_{{\widetilde{v}}_0}(t,y)}dy \end{aligned}$$

(82)

$$\begin{aligned}&\quad =\int (P_H\star P_H)(y) {\mathcal {C}}_{u_0}(t,y) e^{\gamma ^2{\mathcal {C}}_{{\widetilde{v}}_0}(t,y)}dy -\frac{{\mathcal {C}}_f(0)}{2|c|}(P_H\star P_H)(0)e^{\gamma ^2{\mathcal {C}}_{{\widetilde{v}}_0}(t,0)} \end{aligned}$$

(83)

$$\begin{aligned}&\qquad -\gamma ^2 \int (P_H\star P_H)(y) \left[ \mathcal K^2(t,y)+\frac{{\mathcal {C}}_f(0)}{4|c|}\mathcal C'_{{\widetilde{v}}_0}(t,|y|)\right] e^{\gamma ^2\mathcal C_{{\widetilde{v}}_0}(t,y)}dy. \end{aligned}$$

(84)

Recall that ${\mathcal {C}}_{u_0}(t,y)$ behaves as a Dirac function as $t\rightarrow \infty$, weighted by an appropriate factor, as is stated in Eq. (16). Hence, it is clear that the contribution given in Eq. (83) will vanish as $t\rightarrow \infty$. Similarly, in the same limit, the function ${\mathcal {K}}(t,y)$ (Eq. 78) converges towards ${\widetilde{P}}_0(-y)$, again weighted by an appropriate factor, such that, using the asymptotic expression of ${\mathcal {C}}_{{\widetilde{v}}_0}$ (Eq. 28), we obtain pointwise

$$\begin{aligned} \lim _{t\rightarrow \infty } \left[ {\mathcal {K}}^2(t,y)+\frac{\mathcal C_f(0)}{4|c|}{\mathcal {C}}'_{{\widetilde{v}}_0}(t,|y|)\right] =\frac{\mathcal {C}^2_f(0)}{4|c|^2}\left[ {\widetilde{P}}_0^2(y)-|\ln _+'(|y|)| \right] +\frac{{\mathcal {C}}_f(0)}{4|c|}{\widetilde{h}}'(|y|), \end{aligned}$$

(85)

where we have denoted by $|\ln _+'(|y|)|$ the derivative of the smoothly-truncaded logarithm $\ln _+$ evaluated at |y|, that is expected to behave as 1/|y| in the vicinity of the origin. $\square$

Let us focus on the second contribution displayed in Eq. (84). As we have already observed, the function $(P_H\star P_H)(y)$ is a bounded function of its argument for any $H>0$, and its rapid decrease away from the origin ensures integrability when the dummy variable y goes towards infinite values. Notice that

$$\begin{aligned} {\widetilde{P}}_0(y) =-i \int e^{2i\pi ky}\frac{k}{|k|^{3/2}_{1/L}}dk\mathrel {\mathop {\sim }\limits _{y\rightarrow 0}^{}}\frac{y}{|y|^{3/2}}, \end{aligned}$$

(86)

which says that the first term of the RHS of Eq. (85) grows at most logarithmically near the origin, which is integrable. Thus, the integral entering in Eq. (84) exists as $t\rightarrow \infty$ if the remaining singular term, i.e. $|y|^{-\gamma ^2{\mathcal {C}}_f(0)/|c|}$, is integrable, i.e. $\gamma ^2<|c|/{\mathcal {C}}_f(0)$.

To conclude, concerning the limit at large time of the variance ${\mathbb {E}}\left[ \left| v_{H,\gamma }(t,x)\right| ^2\right]$, let us examine the second term of the LHS of Eq. (82). It is easy to see that near the origin, whereas $(P_H\star P_H)(y)$ remains bounded for any $H>0$, its derivative will behave as $(P_H\star P_H)'(y)\sim y^{2H-1}$. Thus, as $t\rightarrow \infty$, this contribution is finite for $2H-1-\gamma ^2{\mathcal {C}}_f(0)/|c|>-1$, i.e. $\gamma ^2<2H|c|/{\mathcal {C}}_f(0)$. Hence, for $\gamma ^2<\min (1,2H)|c|/{\mathcal {C}}_f(0)$, the variance is finite and its expression is given by

$$\begin{aligned}&\lim _{t\rightarrow \infty }{\mathbb {E}}\left[ \left| v_{H,\gamma }(t,x)\right| ^2\right] =- \frac{{\mathcal {C}}_f(0)}{2|c|}\int _0^{+\infty }(P_H\star P_H)'(y)\left| \frac{L}{y}\right| _+^{\gamma ^2\frac{{\mathcal {C}}_f(0)}{|c|}}e^{\gamma ^2{\widetilde{h}}(y)}dy\nonumber \\&\quad -\gamma ^2\frac{{\mathcal {C}}_f(0)}{4|c|} \int (P_H\star P_H)(y) \left[ \frac{{\mathcal {C}}_f(0)}{|c|}\left[ {\widetilde{P}}_0^2(y)-|\ln _+'(|y|)| \right] +{\widetilde{h}}'(|y|)\right] \left| \frac{L}{y}\right| _+^{\gamma ^2\frac{\mathcal C_f(0)}{|c|}}e^{\gamma ^2{\widetilde{h}}(y)}dy, \end{aligned}$$

(87)

where the notation $|y|_+=\exp \left( \ln _+|y|\right)$ is introduced.

To see the behavior at small scales of the second-order structure function, consider the function

$$\begin{aligned} {\mathcal {P}}_{H,\ell }(x)=P_{H}(x+\ell )-P_{H}(x)=\int e^{2i\pi kx}\frac{e^{2i\pi k\ell }-1}{|k|_{1/L}^{H+1/2}}dk, \end{aligned}$$

(88)

such that we can conveniently write the velocity increment as

$$\begin{aligned} \delta _\ell v_{H,\gamma }(t,x)=\int {\mathcal {P}}_{H,\ell }(x-y)e^{\gamma {\widetilde{P}}_0u_0(t,y)} u_0(t,y)dy. \end{aligned}$$

(89)

Previous calculations concerning the variance apply and we get

$$\begin{aligned}&\lim _{t\rightarrow \infty }{\mathbb {E}}\left[ \left| \delta _\ell v_{H,\gamma }(t,x)\right| ^2\right] =- \frac{{\mathcal {C}}_f(0)}{2|c|}\int _0^{+\infty }( {\mathcal {P}}_{H,\ell }\star {\mathcal {P}}_{H,\ell })'(y)\left| \frac{L}{y}\right| _+^{\gamma ^2\frac{{\mathcal {C}}_f(0)}{|c|}}e^{\gamma ^2{\widetilde{h}}(y)}dy\nonumber \\&\quad -\gamma ^2\frac{{\mathcal {C}}_f(0)}{4|c|} \int ( \mathcal P_{H,\ell }\star {\mathcal {P}}_{H,\ell })(y) \left[ \frac{\mathcal C_f(0)}{|c|}\left[ {\widetilde{P}}_0^2(y)-|\ln _+'(|y|)| \right] +{\widetilde{h}}'(|y|)\right] \left| \frac{L}{y}\right| _+^{\gamma ^2\frac{\mathcal C_f(0)}{|c|}}e^{\gamma ^2{\widetilde{h}}(y)}dy. \end{aligned}$$

(90)

Notice that

$$\begin{aligned} ( {\mathcal {P}}_{H,\ell }\star {\mathcal {P}}_{H,\ell })(y)=\int e^{2i\pi ky}\frac{\left| e^{2i\pi k\ell }-1\right| ^2}{|k|_{1/L}^{2H+1}}dk, \end{aligned}$$

(91)

such that

$$\begin{aligned} ( {\mathcal {P}}_{H,\ell }\star {\mathcal {P}}_{H,\ell })(\ell y)\mathrel {\mathop {\sim }\limits _{\ell \rightarrow 0^+}^{}}\ell ^{2H}\int e^{2i\pi ky}\frac{\left| e^{2i\pi k}-1\right| ^2}{|k|^{2H+1}}dk, \end{aligned}$$

(92)

this equivalence at small scales making sense only for $H\in ]0, 1[$. Similarly, we have

$$\begin{aligned} ( {\mathcal {P}}_{H,\ell }\star {\mathcal {P}}_{H,\ell })'(\ell y)\mathrel {\mathop {\sim }\limits _{\ell \rightarrow 0^+}^{}}\ell ^{2H-1}g_H(y), \end{aligned}$$

(93)

where

$$\begin{aligned} g_H(y)=\int 2i\pi k e^{2i\pi ky}\frac{\left| e^{2i\pi k}-1\right| ^2}{|k|^{2H+1}}dk. \end{aligned}$$

(94)

Once having rescaled the dummy variable y entering in the integrals at the RHS of Eq. (90), we can see that the first term will be order $\ell ^{2H-\gamma ^2\mathcal C_f(0)/|c|}$, and thus will dominate the second term that goes to zero as $\ell ^{2H+1-\gamma ^2{\mathcal {C}}_f(0)/|c|}$. Doing so, we get the equivalent behavior of the second-order structure function at small scales, which reads

$$\begin{aligned} \lim _{t\rightarrow \infty }{\mathbb {E}}\left[ \left| \delta _\ell v_{H,\gamma }(t,x)\right| ^2\right] \mathrel {\mathop {\sim }\limits _{\ell \rightarrow 0^+}^{}}- \frac{\mathcal C_f(0)}{2|c|}\ell ^{2H}\left( \frac{\ell }{L}\right) ^{-\gamma ^2\frac{\mathcal C_f(0)}{|c|}}e^{\gamma ^2{\widetilde{h}}(0)}\int _0^{+\infty }\frac{g_H(y)}{|y|^{\gamma ^2\frac{\mathcal C_f(0)}{|c|}}}dy. \end{aligned}$$

(95)

It remains to determine the range of parameters such that the equivalence given in Eq. (95) makes sense, and hence check the integrability of the remaining integral that enters in it. Although the behavior of the function $g_H$ defined in Eq. (94) at small and large arguments can be tricky to establish, its integrability is pretty much straightforward. Indeed, with notation $a=\gamma ^2\frac{{\mathcal {C}}_f(0)}{|c|}$, using the equality

$$\begin{aligned} \int _0^{+\infty }\frac{g_H(y)}{|y|^{a}}dy=- (2\pi )^{a}\Gamma \left( 1 - a\right) \cos \left( \frac{a\pi }{2}\right) \int \frac{\left| e^{2i\pi k}-1\right| ^2}{|k|^{2H+1-a}}dk, \end{aligned}$$

it is clear that the equivalent (Eq. 95) makes sense for $H\in ]0,1[$ and $\gamma ^2\frac{{\mathcal {C}}_f(0)}{|c|}<\min (1,2H)$, and is indeed positive. As a further check, we can note that the expression (Eq. 95) indeed coincides with the equivalence obtained for fractional Gaussian fields (Eq. 22) when $\gamma =0$.

Let us now calculate the third order structure function. We have, making use of the definition and symmetries of the function ${\mathcal {K}}$ (Eq. 78),

$$\begin{aligned}&{\mathbb {E}}\left[ u_0(t,y_1) u_0^*(t,y_2)u_0(t,y_3)e^{\gamma \left( {\widetilde{P}}_0u_0(t,y_1)+{\widetilde{P}}_0u_0^*(t,y_2)+{\widetilde{P}}_0u_0(t,y_3)\right) }\right] \nonumber \\&\quad ={\mathcal {C}}_{u_0}(t,y_1-y_2){\mathbb {E}}\left[ u_0(t,y_3)e^{\gamma \left( {\widetilde{P}}_0u_0(t,y_1)+{\widetilde{P}}_0u_0^*(t,y_2)+{\widetilde{P}}_0u_0(t,y_3)\right) }\right] \nonumber \\&\qquad +\gamma {\mathcal {K}}(t,y_1-y_2){\mathbb {E}}\left[ u_0^*(t,y_2)u_0(t,y_3)e^{\gamma \left( {\widetilde{P}}_0u_0(t,y_1)+{\widetilde{P}}_0u_0^*(t,y_2)+{\widetilde{P}}_0u_0(t,y_3)\right) }\right] \nonumber \\&\quad ={\mathcal {C}}_{u_0}(t,y_1-y_2)\left[ \gamma {\mathcal {K}}(t,y_3-y_2)e^{\gamma ^2\left[ {\mathcal {C}}_{{\widetilde{v}}_0}(t,y_1-y_2)+{\mathcal {C}}_{{\widetilde{v}}_0}(t,y_3-y_2)\right] }\right] \nonumber \\&\qquad +\gamma {\mathcal {K}}(t,y_1-y_2)\left[ {\mathcal {C}}_{u_0}(t,y_3-y_2)+\gamma ^2\left[ {\mathcal {K}}^*(t,y_2-y_1)+{\mathcal {K}}^*(t,y_2-y_3)\right] {\mathcal {K}}(t,y_3-y_2)\right] \nonumber \\&\qquad \times e^{\gamma ^2\left[ {\mathcal {C}}_{{\widetilde{v}}_0}(t,y_1-y_2)+{\mathcal {C}}_{{\widetilde{v}}_0}(t,y_3-y_2)\right] } \nonumber \\&\quad =\gamma \left[ {\mathcal {C}}_{u_0}(t,y_1-y_2){\mathcal {K}}(t,y_3-y_2)+{\mathcal {C}}_{u_0}(t,y_3-y_2){\mathcal {K}}(t,y_1-y_2)\right] e^{\gamma ^2\left[ {\mathcal {C}}_{{\widetilde{v}}_0}(t,y_1-y_2)+{\mathcal {C}}_{{\widetilde{v}}_0}(t,y_3-y_2)\right] }\nonumber \\&\qquad -\gamma ^3{\mathcal {K}}(t,y_1-y_2)\left[ {\mathcal {K}}(t,y_1-y_2)+\mathcal K(t,y_3-y_2)\right] {\mathcal {K}}(t,y_3-y_2) e^{\gamma ^2\left[ \mathcal C_{{\widetilde{v}}_0}(t,y_1-y_2)+\mathcal C_{{\widetilde{v}}_0}(t,y_3-y_2)\right] }, \end{aligned}$$

(96)

such that

$$\begin{aligned} {\mathbb {E}}\left[ \delta _\ell v_{H,\gamma }|\delta _\ell v_{H,\gamma }|^2\right]&=2\gamma \int {\mathcal {P}}_{H,\ell }(y_2-z_1) {\mathcal {P}}_{H,\ell }(y_2) {\mathcal {P}}_{H,\ell }(y_2-z_3) \\&\times \left[ {\mathcal {C}}_{u_0}(t,z_1)-\gamma ^2\mathcal K^2(t,z_1)\right] {\mathcal {K}}(t,z_3)e^{\gamma ^2\left[ \mathcal C_{{\widetilde{v}}_0}(t,z_1)+{\mathcal {C}}_{{\widetilde{v}}_0}(t,z_3)\right] }dz_1 dy_2 dz_3. \end{aligned}$$

Let us introduce the following function

$$\begin{aligned} h_{H,\ell }(z_1,z_3)&=\int {\mathcal {P}}_{H,\ell }(y_2-z_1) {\mathcal {P}}_{H,\ell }(y_2) {\mathcal {P}}_{H,\ell }(y_2-z_3) dy_2\\&=\int e^{-2i\pi (k_1z_1+k_3z_3)} \frac{\left( e^{2i\pi k_1\ell }-1\right) \left( e^{-2i\pi (k_1+k_3)\ell }-1\right) \left( e^{2i\pi k_3\ell }-1\right) }{|k_1|_{1/L}^{H+1/2}|k_1+k_3|_{1/L}^{H+1/2}|k_3|_{1/L}^{H+1/2}}dk_1dk_3\\&=-h_{H,\ell }(-z_1,-z_3), \end{aligned}$$

such that

$$\begin{aligned}&{\mathbb {E}}\left[ \delta _\ell v_{H,\gamma }|\delta _\ell v_{H,\gamma }|^2\right] =\nonumber \\&\quad 2\gamma \int h_{H,\ell }(z_1,z_3) \left[ \mathcal C_{u_0}(t,z_1)-\gamma ^2{\mathcal {K}}^2(t,z_1)\right] \mathcal K(t,z_3)e^{\gamma ^2\left[ {\mathcal {C}}_{{\widetilde{v}}_0}(t,z_1)+\mathcal C_{{\widetilde{v}}_0}(t,z_3)\right] }dz_1dz_3. \end{aligned}$$

(97)

Using the same ideas to determine the limiting value as $t\rightarrow \infty$ of the variance (Eq. 79), remark that

$$\begin{aligned} \int _0^{+\infty } \partial _{z_1}h_{H,\ell }(z_1,z_3)e^{\gamma ^2{\mathcal {C}}_{{\widetilde{v}}_0}(t,z_1)}dz_1 =&-h_{H,\ell }(0,z_3)e^{\gamma ^2{\mathcal {C}}_{{\widetilde{v}}_0}(t,0)} \nonumber \\&- \int _0^{+\infty }h_{H,\ell }(z_1,z_3)\gamma ^2\mathcal C'_{{\widetilde{v}}_0}(t,z_1)e^{\gamma ^2\mathcal C_{{\widetilde{v}}_0}(t,z_1)}dz_1, \end{aligned}$$

(98)

as we obtained in Eq. (80). Doing so, we determine the proper quantity that eventually dominates at small scales, and we obtain

$$\begin{aligned} \lim _{t\rightarrow \infty }{\mathbb {E}}\left[ \delta _\ell v_{H,\gamma }|\delta _\ell v_{H,\gamma }|^2\right]&\mathrel {\mathop {\sim }\limits _{\ell \rightarrow 0^+}^{}}\gamma \frac{{\mathcal {C}}^2_f(0)}{2|c|^2}\ell ^{3H}\left( \frac{\ell }{L}\right) ^{-2\gamma ^2\frac{{\mathcal {C}}_f(0)}{|c|}}e^{2\gamma ^2{\widetilde{h}}(0)} \\&\times \int _{(z_1,z_3)\in {\mathbb {R}}^+\times {\mathbb {R}}}g_H(z_1,z_3)\frac{1}{|z_1|^{\frac{\gamma ^2\mathcal C_f(0)}{|c|}}}\frac{z_3}{|z_3|^{\frac{3}{2}+\frac{\gamma ^2\mathcal C_f(0)}{|c|}}}dz_1dz_3, \end{aligned}$$

where we have introduced the function

$$\begin{aligned} g_H(z_1,z_3)=-2i\pi \int e^{-2i\pi (k_1z_1+k_3z_3)} \frac{k_1\left( e^{2i\pi k_1}-1\right) \left( e^{-2i\pi (k_1+k_3)}-1\right) \left( e^{2i\pi k_3}-1\right) }{|k_1|^{H+1/2}|k_1+k_3|^{H+1/2}|k_3|^{H+1/2}}dk_1dk_3.\nonumber \\ \end{aligned}$$

(99)

Additionally, we will need the following exact Fourier transforms,

$$\begin{aligned} \int e^{-2i\pi k_1z_1} \frac{1_{z_1\ge 0}}{|z_1|^a}dz_1 =(2\pi )^{a-1} \Gamma (1 - a) \left[ \sin (a \pi /2) -i\cos (a \pi /2)\frac{k_1}{|k_1|}\right] \frac{1}{|k_1|^{1-a}}, \end{aligned}$$

(100)

for $0<a<1$, and

$$\begin{aligned} \int e^{-2i\pi k_3z_3} \frac{z_3}{|z_3|^{\frac{3}{2}+a}}dz_3 =-i (2 \pi )^{a+1/2} \frac{1/4 + a/2 }{\Gamma (a+3/2)\sin (\pi (a/2+1/4))} \frac{k_3}{|k_3|^{3/2 - a}},\nonumber \\ \end{aligned}$$

(101)

for $0<a<3/2$, and the identity

$$\begin{aligned} \left( e^{2i\pi k_1}-1\right) \left( e^{-2i\pi (k_1+k_3)}-1\right)&\left( e^{2i\pi k_3}-1\right) \\&=-2i\left[ \sin \left( 2\pi (k_1+k_3)\right) -\sin (2\pi k_1)-\sin (2\pi k_3)\right] . \end{aligned}$$

Using symmetries, it can be shown that the real part of Eq. (100) does not contribute, only remaining

$$\begin{aligned}&\int _{(z_1,z_3)\in {\mathbb {R}}^+\times {\mathbb {R}}}g_H(z_1,z_3)\frac{1}{|z_1|^{\frac{\gamma ^2{\mathcal {C}}_f(0)}{|c|}}} \frac{z_3}{|z_3|^{\frac{3}{2}+\frac{\gamma ^2{\mathcal {C}}_f(0)}{|c|}}}dz_1dz_3\nonumber \\&\quad =4\pi A_\gamma \int \frac{k_3\left[ \sin \left( 2\pi (k_1+k_3)\right) -\sin (2\pi k_1)-\sin (2\pi k_3)\right] }{|k_1|^{H+1/2-\frac{\gamma ^2\mathcal C_f(0)}{|c|}}|k_1+k_3|^{H+1/2}|k_3|^{H+2-\frac{\gamma ^2\mathcal C_f(0)}{|c|}}}dk_1dk_3, \end{aligned}$$

(102)

with $A_\gamma \in {\mathbb {R}}$ a real multiplicative constant that can be obtained from the multiplicative contributions displayed in Eqs. (100) and  (101). The sign of the remaining contribution of the RHS of Eq. (102) expressed as a double integral over the dummy variables $k_1$ and $k_3$ is not obvious, neither whether it vanishes or not. Nonetheless, it gives a condition on $\gamma$, to ensure its finiteness. Inspecting the integrability properties of this term, we find that the integral exists along the diagonal $k_1=k_3$ if

$$\begin{aligned} \frac{\gamma ^2{\mathcal {C}}_f(0)}{|c|}<\min (1,3H/2). \end{aligned}$$

(103)

Doing so, we have thus shown that, under the condition provided in Eq. (103), the third-order moment of the increments of the process $v_{H,\gamma }$, as it is defined in Eq. (97), is finite, and does not vanish in an obvious manner. It is furthermore real, and it behaves at small scale as

$$\begin{aligned} \lim _{t\rightarrow \infty }{\mathbb {E}}\left[ \delta _\ell v_{H,\gamma }|\delta _\ell v_{H,\gamma }|^2\right]&\mathrel {\mathop {\sim }\limits _{\ell \rightarrow 0^+}^{}}d_{H,\gamma }\ell ^{3H}\left( \frac{\ell }{L}\right) ^{-2\gamma ^2\frac{\mathcal C_f(0)}{|c|}} \end{aligned}$$

with

$$\begin{aligned} d_{H,\gamma }=\gamma \frac{\mathcal C^2_f(0)}{2|c|^2}e^{2\gamma ^2{\widetilde{h}}(0)} \int _{(z_1,z_3)\in {\mathbb {R}}^+\times {\mathbb {R}}}g_H(z_1,z_3)\frac{1}{|z_1|^{\frac{\gamma ^2\mathcal C_f(0)}{|c|}}}\frac{z_3}{|z_3|^{\frac{3}{2}+\frac{\gamma ^2\mathcal C_f(0)}{|c|}}}dz_1dz_3, \end{aligned}$$

(104)

where the function $g_H(z_1,z_3)$ is defined in Eq. (99).

Let us finally determine the behavior at small scales of the statistics at high-order considering $q\in {\mathbb {N}}^*$,

$$\begin{aligned} {\mathbb {E}}\left[ |\delta _\ell v_{H,\gamma }|^{2q}\right]&=\int \prod _{i=1}^q {\mathcal {P}}_{H,\ell }(x-y_i){\mathcal {P}}_{H,\ell }(x-z_i)\prod _{i=1}^q dy_idz_i\nonumber \\&\times {\mathbb {E}}\left[ \prod _{i=1}^q u_0(t,y_i) u_0^*(t,z_i)e^{\gamma \sum _{i=1}^q{\widetilde{P}}_0u_0(t,y_i)+{\widetilde{P}}_0u_0^*(t,z_i)}\right] , \end{aligned}$$

(105)

where the operator ${\mathcal {P}}_{H,\ell }$ is defined in Eq. (88). The determination of the exact expression of the correlator entering in Eq. (105) can be done using some combinatorial analysis, although it can become cumbersome. Instead, in a first approach, let us evaluate the spectrum of exponents that governs the decrease towards 0 as $\ell \rightarrow 0$. In particular, intermittent corrections are eventually governed by a term of the form

$$\begin{aligned} {\mathbb {E}}\left[ e^{\gamma \sum _{i=1}^q{\widetilde{P}}_0u_0(t,y_i)+{\widetilde{P}}_0u_0^*(t,z_i)}\right] = e^{\gamma ^2\left[ \sum _{i=1}^q{\mathcal {C}}_{{\widetilde{v}}_0}(t,y_i-z_i)+ \sum _{i<j=1}^q{\mathcal {C}}_{{\widetilde{v}}_0}(t,y_i-z_j) +{\mathcal {C}}^*_{{\widetilde{v}}_0}(t,y_j-z_i)\right] }, \end{aligned}$$

contributing at small scales as

$$\begin{aligned} \lim _{t\rightarrow \infty } {\mathbb {E}}\left[ e^{\gamma \sum _{i=1}^q{\widetilde{P}}_0u_0(t,\ell y_i)+{\widetilde{P}}_0u_0^*(t,\ell z_i)}\right]&\mathrel {\mathop {\sim }\limits _{\ell \rightarrow 0^+}^{}}\left( \frac{\ell }{L}\right) ^{-q^2\gamma ^2\frac{{\mathcal {C}}_f(0)}{|c|}}e^{q^2\gamma ^2{\widetilde{h}}(0)} \\&\times \prod _{i=1}^q \frac{1}{|y_i-z_i|^{\gamma ^2\frac{\mathcal C_f(0)}{|c|}}} \prod _{i<j=1}^q \frac{1}{|(y_i-z_j)(y_j-z_i)|^{\gamma ^2\frac{\mathcal C_f(0)}{|c|}}}, \end{aligned}$$

whereas contributions from the fractional part will be of the order of $\ell ^{2qH}$. Once again, the determination of the appropriate range of values for $\gamma$ is tricky to get at this stage because we have to compute in an exact fashion the expectation entering in the RHS of Eq. (105). To do so, we have to generalize the calculations made in Eqs. (77) and (96), using combinatorial developments such as those proposed in Ref. [52] (see their Lemma 2.2). Such a calculation is beyond the scope of the present article. We nonetheless expect the additional condition $\gamma ^2\frac{\mathcal C_f(0)}{|c|}<2H/q$.