  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.
