Averaging Dynamics and Wong-Zakai approximations for a Fast-Slow Navier-Stokes System Driven by fractional Brownian Motion

TL;DR

This paper investigates the limiting behavior of a coupled Navier-Stokes system with fractional Brownian noise, revealing different limit equations depending on the Hurst parameter, using rough path theory.

Contribution

It introduces a novel analysis of a coupled Navier-Stokes system driven by fractional Brownian motion, showing how the limit depends on the Hurst parameter and applying rough path techniques.

Findings

Convergence to Navier-Stokes with Itô-Stokes drift for H<1/2

Limit with transport noise for H>1/2

Different limiting behaviors based on Hurst parameter

Abstract

We study a slow-fast system of coupled two- and three-dimensional Navier-Stokes equations in which the fast component is perturbed by an additive fractional Brownian noise with Hurst parameter $H>\frac{1}{3}$. The system is analyzed using rough path theory, and the limiting behaviour strongly depends on the value of $H$. We prove convergence in law of the slow component to a Navier-Stokes system with an additional It\^o-Stokes drift when $H<\frac{1}{2}$. In contrast, for $H\in (\frac{1}{2},1)$, the limit equation features only a transport noise driven by a rough path.