The Gaussian free-field as a stream function: continuum version of the scale-by-scale homogenization result

TL;DR

The paper studies a two-dimensional drift-diffusion with a time-independent divergence-free Gaussian drift that decorrelates on large scales, a regime where standard stochastic homogenization is borderline and leads to near-superdiffusive behavior. It reformulates the scale-by-scale homogenization in the continuum limit $M\downarrow 1$ by letting $s=\ln L$ act as a time variable and introducing Itô SDEs for proxy fields $(\tilde{\phi},\tilde{\sigma})$ along with a residuum $f$ that is a martingale, enabling precise control via quadratic variations. The authors establish a homogeneous evolution for the residuum and derive explicit moment bounds for the proxies and $f$, thereby re-deriving the annealed second-moment asymptotics of CMOW in this continuum setting. This continuum formulation clarifies the scale-wise homogenization and provides an efficient Itô-calculus-based framework that links variance decomposition in Gaussian fields with Polchinski-type evolution, enhancing understanding of anomalous diffusion in 2D Gaussian environments.

Abstract

This note is about a drift-diffusion process $X$ with a time-independent, divergence-free drift $b$, where $b$ is a smooth Gaussian field that decorrelates over large scales. In two space dimensions, this just fails to fall into the standard theory of stochastic homogenization, and leads to a borderline super-diffusive behavior. In a previous paper by Chatzigeorgiou, Morfe, Otto, and Wang (2022), precise asymptotics of the annealed second moments of $X$ were derived by characterizing the asymptotics of the effective diffusivity $λ_L$ in terms of an artificially introduced large-scale cut-off $L$. The latter was carried out by a scale-by-scale homogenization, and implemented by monitoring the corrector $φ_L$ for geometrically increasing cut-off scales $L^+=ML$. In fact, proxies $(\tildeφ_L,\tildeσ_L)$ for the corrector and flux corrector were introduced incrementally and the residuum $f_L$ estimated. In this short supplementary note, we reproduce the arguments of the above paper in the continuum setting of $M\downarrow 1$. This has the advantage that the definition of the proxies $(\tildeφ_L,\tildeσ_L)$ becomes more transparent -- it is given by a simple Itô SDE with $\ln L$ acting as a time variable. It also has the advantage that the residuum $f_L$, which is a martingale, can be efficiently and precisely estimated by Itô calculus. This relies on the characterization of the quadratic variation of the (infinite-dimensional) Gaussian driver.

Theorems & Definitions (4)

• Proposition 1.1: see CMOW
• proof : Proof of CMOW revisited.
• proof : Proof of CMOW revisited.
• proof : Proof of Proposition \ref{['proposition']}.