Regularized fractional Ornstein-Uhlenbeck processes and their relevance to the modeling of fluid turbulence

Motivated by the modeling of the temporal structure of the velocity field in a highly turbulent flow, we propose and study a linear stochastic differential equation that involves the ingredients of an Ornstein-Uhlenbeck process, supplemented by a fractional Gaussian noise, of parameter $H$, regularized over a (small) time scale $\epsilon >0$. A peculiar correlation between these two plays a key role in the establishment of the statistical properties of its solution. We show that this solution reaches a stationary regime, which marginals, including variance and increment variance, remain bounded when $\epsilon \to 0$. In particular, in this limit, for any $H\in ]0,1[$, we show that the increment variance behaves at small scales as the one of a fractional Brownian motion of same parameter $H$. From the theoretical side, this approach appears especially well suited to deal with the (very) rough case $H<1/2$, including the boundary value $H=0$, and to design simple and efficient numerical simulations.

Figure 1

Variance of the simulated trajectories of the process $X_{\epsilon ,H}$ obtained while integrating the dynamics proposed in Eq. (3) [see Eq. (29) for a discrete version]. We have used the set of parameters $T_0=1, N=2^{28}, \mathrm{\Delta }t=T_0/N, \alpha =50/T_0, \epsilon =10\mathrm{\Delta }t$ and for various values of the parameter $H$. The error bars are estimated as (two times) the standard deviation of the obtained variance over the 150 realizations of the trajectories (see Sec. 5b for details). We have superimposed with a solid line the corresponding theoretical prediction [Eq. (9)] obtained in the stationary regime and in the limit of vanishing $\epsilon$.

Figure 2

Variance $\mathbb{E}{\left({\delta }_{\tau }{X}_{\epsilon ,H}\right)}^2=\mathbb{E}\left[{\left(X_{\epsilon ,H}(t+\tau )-X_{\epsilon ,H}\left(t\right)\right)}^2\right]$ of the increments as a function of the scale $\tau$ in a logarithmic fashion. We have used the same set of parameters as in Fig. 1 [and described in Eq. (9)]. All curves are arbitrarily shifted vertically for the sake of clarity. We represent nine different values of the parameter $H$, from top to bottom $H=0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8$, and 0.9. We superimpose (dashed line) with the same vertical shift our theoretical prediction pointed in Eq. (10).

Figure 3

Representation of the increment variance $\mathbb{E}{\left({\delta }_{\tau }{X}_{\epsilon ,H}\right)}^2$ compensated by the analytical prediction pointed in Eq. (10), as a function of the scales $\tau$. The parameters of the simulation are the same as in Figs. 1 and 2, and we represent, from top to bottom, the results for $H=0.8,0.7,0.6,0.5,0.4,0.3,0.2$. We furthermore superimpose with dashed-lines two characteristics time scales of the problem: the regularizing scale $\epsilon$ (left) and the large time scale ${\alpha }^{-1}$ (right).