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

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 $\lambda_L$ 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 $\phi_L$ for geometrically increasing cut-off scales $L^+=ML$. In fact, proxies $(\tilde\phi_L,\tilde\sigma_L)$ 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 $(\tilde\phi_L,\tilde\sigma_L)$ becomes more transparent -- it is given by a simple It\^{o} 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 It\^{o} calculus. This relies on the characterization of the quadratic variation of the (infinite-dimensional) Gaussian driver.