Scaling Limits of Stochastic Transport Equations on Manifolds

TL;DR

The paper analyzes scaling limits for stochastic transport equations on compact Riemannian manifolds, extending prior torus results to curved spaces with transport noise that is white in time and colored in space. By tuning the noise so that the diagonal covariance converges to the identity while the covariance operator vanishes in $L^2$, the authors derive universal heat-type limits: white-initial data lead to a stochastic heat equation $\partial_t u = \frac{1}{2}\Delta u + (-\Delta)^{1/2}\xi$, whereas $L^2$ initial data yield the deterministic heat equation $\partial_t u = \frac{1}{2}\Delta u$, with quantitative convergence rates for the stochastic part. The approach combines heat-kernel techniques on manifolds, Hodge/Leray projections, and a generator-convergence/tightness framework to establish limits via properly scaled $Q^{(N)}$-Wiener noise, including explicit constructions using mollified white noise. The results provide a rigorous link between stochastic transport on curved spaces and eddy-diffusion-type limits, with potential applications to geophysical fluid models and other SPDEs with transport noise on manifolds.

Abstract

In this work, we generalize some results on scaling limits of stochastic transport equations on the torus, developed recently by Flandoli, Galeati and Luo in Galeati (2020); Flandoli and Luo (2020); Flandoli et al. (2024), to manifolds. We consider the stochastic transport equations driven by colored space-time noise (smooth in space, white in time) on a compact Riemannian manifold without boundary. Then we study the scaling limits of stochastic transport equations, tuning the noise in such a way that the space covariance of the noise on the diagonal goes to the identity matrix but the covariance operator itself goes to zero. This includes the large scale analysis regime with diffusive scaling. We obtain different scaling limits depending on the initial data. With space white noise as initial data, the solutions to the stochastic transport equations converge in distribution to the solution to a stochastic heat equation with additive noise. With square integrable initial data, the solutions to the stochastic transport equations converge to the solution to the deterministic heat equation, and we provide quantitative estimates on the convergence rate.

Theorems & Definitions (76)

• Proposition 2.1
• Remark 2.2
• Theorem 2.3: Hodge decomposition
• Lemma 2.4
• proof
• Theorem 2.5
• proof
• Proposition 2.6
• proof
• Lemma 2.7