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

Abstract

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 a 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.