Stochastic transport by Gaussian noise with regularity greater than 1/2
Franco Flandoli, Francesco Russo
TL;DR
This work analyzes stochastic transport of a passive scalar driven by Gaussian noise with Hölder regularity $H>\tfrac{1}{2}$, comparing the stochastic transport to a mean-field PDE. Using Fourier analysis in the commuting case, it derives a closed mean-field equation $\partial_t \overline{\theta} = \kappa\Delta \overline{\theta} + \dfrac{d\gamma}{dt}(\mathcal{L}\overline{\theta})$, with $\mathcal{L}f = \operatorname{div}(Q\nabla f)$ and $Q=\sum_k\sigma_k\otimes\sigma_k$, and shows that $\dfrac{d\gamma}{dt}$ scales as $t^{2H-1}$, implying reduced dissipation at early times and enhanced diffusion at later times. The variance dynamics is derived, revealing a coupling between the mean-field and fluctuations via $V_t = 2\kappa\Delta V + (\mathcal{L}V) d\gamma/dt + 2\sum_k ( (\sigma_k\cdot\nabla)\overline{\theta})^2 d\gamma$. The paper also discusses a two-scale, non-commutative framework showing a path toward reduction to the commutative case and identifies a commutator remainder that prevents a closed mean-field equation in the fully non-commutative setting, outlining directions for future work in stochastic turbulence modeling and Malliavin-calculus-based analysis.
Abstract
Diffusion with stochastic transport is investigated here when the randomdriving process is a very general Gaussian process, including FractionalBrownian motion. The purpose is the comparison with a deterministic PDE, whichin certain cases represents the equation for the mean value. From thisequation we observe a reduced dissipation property for small times and anenhanced diffusion for large times, with respect to delta correlated noisewhen regularity is higher than the one of Brownian motion, a fact interpretedqualitatively here as a signature of the modified dissipation observed for 2Dturbulent fluids due to the inverse cascade. We give results also for thevariance of the solution and for a scaling limit of a two-component noise input.