Well-posedness and Hurst parameter estimation for fluid equations driven by fractional transport noise

TL;DR

This paper investigates the well-posedness of a 2D fluid equation driven by fractional Brownian noise and introduces a new method for estimating the Hurst parameter from the flow's statistical properties.

Contribution

It develops a novel sewing lemma adapted for transport structures, enabling analysis of stochastic PDEs driven by fractional noise, and proposes a Hurst parameter estimator.

Findings

Established existence and uniqueness of solutions for the stochastic fluid equation.

Developed a new estimator for the Hurst parameter based on quadratic functionals.

Provided a flexible analytical framework for a broader class of stochastic PDEs.

Abstract

We study a two-dimensional incompressible vorticity equation on the torus driven by transport-type fractional Brownian noise with Hurst parameter $H \in (1/2,1)$. The model captures persistent, long-range correlated forcing consistent with inertial-range scaling laws and fractional Brownian approximations of turbulent fluctuations. A central ingredient of our approach is a version of the sewing lemma adapted to a class of integrands that includes, but is not limited to, transport-type structures. This result provides a flexible tool for constructing the Young integral and serves as a basis for analysing a wider class of stochastic partial differential equations. Using this approach, we establish existence and uniqueness of solutions via a fixed point argument and investigate statistical properties of the flow. In particular, we study quadratic functionals of the solution and derive an estimator for the Hurst parameter $H$.