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

## TL;DR

This paper introduces a regularized fractional Ornstein-Uhlenbeck process to model turbulent fluid velocity, capturing roughness and increment behavior similar to fractional Brownian motion, with implications for simulation and analysis.

## Contribution

It proposes a novel linear stochastic differential equation with fractional Gaussian noise, analyzing its properties and relevance to turbulence modeling, especially for rough cases where H<1/2.

## Key findings

- The process reaches a stationary regime with bounded marginals as regularization vanishes.
- Increment variance at small scales matches that of fractional Brownian motion for any H in (0,1).
- The approach is suitable for simulating very rough stochastic processes.

## 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.