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 an Ornstein-Uhlenbeck process, supplemented by a fractional Gaussian noise, of parameter H, regularized over a (small) time scale ε>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 ε→0. In particular, in this limit, for any H∈]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.