Quantitative convergence rates for scaling limit of SPDEs with transport noise

TL;DR

This paper establishes quantitative convergence rates for the scaling limits of stochastic 2D fluid equations with transport noise, demonstrating how stochastic effects influence deterministic viscous equations and related models.

Contribution

It provides explicit convergence rate estimates using analytic and probabilistic methods, and applies these ideas to control blow-up probabilities and analyze mixing and dissipation properties.

Findings

Quantitative convergence rates for stochastic to deterministic fluid equations.

Explicit noise choices reduce blow-up probability in Keller-Segel model.

Identification of mixing and dissipation enhancement effects.

Abstract

We consider on the torus the scaling limit of stochastic 2D (inviscid) fluid dynamical equations with transport noise to deterministic viscous equations. Quantitative estimates on the convergence rates are provided by combining analytic and probabilistic arguments, especially heat kernel properties and maximal estimates for stochastic convolutions. Similar ideas are applied to the stochastic 2D Keller-Segel model, yielding explicit choice of noise to ensure that the blow-up probability is less than any given threshold. Our approach also gives rise to some mixing property for stochastic linear transport equations and dissipation enhancement in the viscous case.