Edwards-Wilkinson limit for a stochastic advection-diffusion PDE

TL;DR

This work proves that, for a diffusion in a Gaussian white-in-time environment, the large-scale fluctuations of the quenched density at diffusive scaling are Gaussian and governed by an additive stochastic heat equation. The authors develop a rigorous solution framework for the SPDE, derive correlation-function PDEs, and obtain uniform moment controls to prove tightness of the rescaled fluctuations. They distinguish compressible and incompressible covariance structures, giving explicit forms for the limiting noise amplitude via $V_{\text{eff}}$ and showing convergence to the Edwards–Wilkinson-type limit in both regimes. The results provide a precise, quantitative description of the first-order correction to the quenched central limit theorem for diffusions in random environments and connect to broader universality classes of Gaussian fluctuations in SPDEs.

Abstract

We consider a diffusion in a Gaussian random environment that is white in time and study the large-scale behavior of the quenched density with respect to the Lebesgue measure. We show that under diffusive rescaling, the fluctuations of the density converge to a Gaussian limit, described by an additive stochastic heat equation. In the case where the environment is divergence-free, our result can be interpreted as computing the scaling limit of the first-order correction to the quenched Central Limit Theorem.

Theorems & Definitions (39)

• Lemma 1.3
• Proposition 1.4
• proof
• Definition 1.5
• Remark 1.6
• Theorem 1.7
• proof
• Proposition 1.8
• proof
• Theorem 1.9