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 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 L for geometrically increasing cut-off scales L^+=ML. In fact, proxies (L, L) for the corrector and flux corrector were introduced incrementally and the residuum fL estimated. In this short supplementary note, we reproduce the arguments of the above paper in the continuum setting of M 1. This has the advantage that the definition of the proxies (L, L) becomes more transparent -- it is given by a simple It\^o SDE with L acting as a time variable. It also has the advantage that the residuum fL, 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.