On rough 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 twos 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, 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 \(H\), extending thus previous works to the (very) rough case \(H<1/2\).