Scaling Limits of Stochastic Transport Equations on Manifolds

TL;DR

This paper extends the analysis of stochastic transport equations' scaling limits from the torus to manifolds, revealing different limiting behaviors depending on initial data and noise scaling, including convergence to heat equations.

Contribution

It generalizes existing results on stochastic transport equations' scaling limits to the setting of compact Riemannian manifolds, including new convergence results and quantitative estimates.

Findings

Solutions with white noise initial data converge to stochastic heat equations.

Solutions with square integrable initial data converge to deterministic heat equations.

Quantitative estimates on convergence rates are provided.

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.