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

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 $H$.

• The paper demonstrates a novel Young integration scheme tailored for fractional transport noise in 2D fluid equations.
• It proves the well-posedness of the vorticity equation via a fixed point method that leverages semigroup regularization.
• A rigorous estimator for the Hurst parameter is developed, linking theoretical turbulence modeling to practical data calibration.

Well-Posedness and Hurst Parameter Estimation for Fluid Equations Driven by Fractional Transport Noise

Introduction and Context

This paper rigorously investigates the stochastic two-dimensional (2D) incompressible vorticity equation on the torus, subject to persistent, long-range-correlated fractional Brownian (fBm) transport noise with Hurst parameter $H \in (1/2, 1)$. The core motivation arises from the statistical structure of 2D turbulence, particularly the phenomenology of inertial-range scaling and the representation of non-Markovian, temporally correlated forcing in turbulence modeling. The work also capitalizes on the natural analogy between temporal increments in turbulent flows and fBm, with $H$ controlling the memory and regularity of the noise, and thus having direct physical relevance.

A central technical contribution is the development of a sewing lemma and Young integration scheme explicitly adapted to transport-type (gradient-driven) stochastic integrals, encompassing (but not limited to) the forms arising from the considered SPDE. This abstraction allows a unifying treatment of both the stochastic vorticity equation and a larger class of nonlinear SPDEs with fractional noise, improving previous frameworks that often depended on rough path theory or were limited to Markovian (white-in-time) noise.

Mathematical Setting and Main Results

Stochastic Vorticity Equation

The primary equation is: $$d\omega_t + u_t \cdot \nabla \omega_t\,dt + \mathcal{L}_\xi \omega_t\,dW^H_t = \Delta \omega_t\,dt,$$ with $\omega_t$ the vorticity, $\mathcal{L}_\xi \omega := \xi \cdot \nabla \omega$ ($\xi$ a time-independent, solenoidal vector field), and $W^H$ an fBm process with $H > 1/2$. The interplay of the nonlinear drift, Laplacian dissipation, and non-Markovian stochastic transport (the latter representing temporally persistent unresolved forcings from turbulence phenomenology) fundamentally distinguishes the analysis from the classical settings.

Analytic Framework and Young Integration

A significant innovation is a version of the sewing lemma and the corresponding Young integral for semigroups with transport-type integrands. This approach exploits analytic semigroup smoothing and time-regularity to establish existence, continuity, and optimal regularity of the stochastic convolutions, crucial for formulating and solving the stochastic vorticity equation in a mild sense.

Specifically, given $Y$ with sufficient time and spatial regularity, the stochastic integral

$I_t = \int_0^t S_{t-r} Y_r\,dW^H_r$

is constructed as the unique limit (in an appropriate Sobolev regularity) of dyadic Riemann sums, with explicit Lipschitz dependence on the data. The regularization parameter relationship, dictated by the interplay between the Hölder exponent $H$0 and the space gain from heat semigroup smoothing, is made precise via delicate estimates.

Well-Posedness via Fixed Point

Well-posedness in the functional space $H$1 for $H$2 is obtained by recasting the mild solution map into a fixed-point framework. The mapping's contractivity for sufficiently small $H$3 is quantitatively established using the nonlinear drift's structure, semigroup regularization, and the previously mentioned Young-integral estimates.

Existence and uniqueness thus follow from Banach's contraction principle. The construction is robust for a large class of fluid evolution equations

$H$4

under stated analyticity, boundedness, and differentiability hypotheses on $H$5 and $H$6.

Statistical Estimation of the Hurst Parameter

A novel contribution is the rigorous, strongly consistent estimator for $H$7 from solution trajectories. The methodology leverages the self-similarity of fBm and the quadratic scaling of its increments. For an observed process $H$8, with $H$9 a test function (typically a Fourier mode), it is shown that asymptotically, the rescaled quadratic variation of the increments over dyadic partitions converges a.s. to a deterministic limit depending only on the energy of the integrand and $$d\omega_t + u_t \cdot \nabla \omega_t\,dt + \mathcal{L}_\xi \omega_t\,dW^H_t = \Delta \omega_t\,dt,$$0. Consequently, the estimator

$$d\omega_t + u_t \cdot \nabla \omega_t\,dt + \mathcal{L}_\xi \omega_t\,dW^H_t = \Delta \omega_t\,dt,$$1

converges almost surely to the true $$d\omega_t + u_t \cdot \nabla \omega_t\,dt + \mathcal{L}_\xi \omega_t\,dW^H_t = \Delta \omega_t\,dt,$$2, given sufficient solution regularity. The argument holds for the full SPDE solution due to the drift and multiplicative structure's negligible influence on small-scale increments, further substantiated via a careful analysis of the rescaled quadratic variation.

Implications and Impact

Theoretical Advancements

• Non-Markovian Turbulence Models: This work rigorously links classical turbulence scaling heuristics and stochastic fluid equations with temporal correlations parameterized by $$d\omega_t + u_t \cdot \nabla \omega_t\,dt + \mathcal{L}_\xi \omega_t\,dW^H_t = \Delta \omega_t\,dt,$$3. The role of $$d\omega_t + u_t \cdot \nabla \omega_t\,dt + \mathcal{L}_\xi \omega_t\,dW^H_t = \Delta \omega_t\,dt,$$4 as a quantifier of memory and long-range dependence is secured both in the sense of well-posedness and statistical inference.
• Unified Treatment of SPDEs with Fractional Noise: The adapted sewing lemma and analytic approach bypass the need for rough path machinery (which is overkill for $$d\omega_t + u_t \cdot \nabla \omega_t\,dt + \mathcal{L}_\xi \omega_t\,dW^H_t = \Delta \omega_t\,dt,$$5 and not always appropriate for transport-type nonlinearities), yielding a flexible solution theory.
• Parameter Estimation Theory: The generic estimator for $$d\omega_t + u_t \cdot \nabla \omega_t\,dt + \mathcal{L}_\xi \omega_t\,dW^H_t = \Delta \omega_t\,dt,$$6 is robust to the presence of nonlinear drift, as shown by the pathwise asymptotics, establishing a concrete pathway between theory and statistical inference for SPDEs.

Practical and Future Directions

• Data-Driven Model Calibration: The consistency of the $$d\omega_t + u_t \cdot \nabla \omega_t\,dt + \mathcal{L}_\xi \omega_t\,dW^H_t = \Delta \omega_t\,dt,$$7 estimator enables the use of time series from systems described by such fluid equations to tune the memory properties of the stochastic parameterization—directly relevant for subgrid modeling in geophysical flows.
• General SPDE Analysis: The technical innovations here are poised for extension to three-dimensional flows, active scalar models, multi-layer QG equations, or other SPDEs where correlated (not white) stochastic effects are physically relevant (e.g., climate modeling, oceanography).
• Beyond $$d\omega_t + u_t \cdot \nabla \omega_t\,dt + \mathcal{L}_\xi \omega_t\,dW^H_t = \Delta \omega_t\,dt,$$8: While this work focuses on the tractable Young regime, the analytic machinery may inform future advances for $$d\omega_t + u_t \cdot \nabla \omega_t\,dt + \mathcal{L}_\xi \omega_t\,dW^H_t = \Delta \omega_t\,dt,$$9 upon merging with rough driver techniques and pathwise methods for rough SPDEs.

Conclusion

This paper establishes a robust, analytic solution theory for 2D incompressible fluid equations driven by fractional transport-type noise, supported by a self-contained and flexible approach to stochastic integration in the Young regime. The connection to turbulence phenomenology is rigorously formalized, and a practical, strongly consistent estimator for the Hurst parameter is provided. These developments connect stochastic fluid dynamics, turbulence modeling, and statistical inference in a unified framework, opening the way to data-driven analysis and the deployment of such models in realistic multiscale environments.