Weak coupling limit of a Brownian particle in the curl of the 2D GFF

In this article, we study the weak coupling limit of the following equation in ${\mathbb{R}}^2$: $$d{X}_t^{\epsilon }=\frac{\widehat{\lambda }}{\sqrt{\log \frac{1}{\epsilon }}}{\omega }^{\epsilon }(X_t^{\epsilon })dt+\nu d{B}_t,X_0^{\epsilon }=0.$$ Here ${\omega }^{\epsilon }={\mathrm{\nabla }}^{\perp }{\rho }_{\epsilon }\ast \xi$ with $\xi$ representing the $2d$ Gaussian Free Field (GFF) and ${\rho }_{\epsilon }$ denoting an appropriate identity. $B_t$ denotes a two-dimensional standard Brownian motion, and $\widehat{\lambda },\nu >0$ are two given constants. We use the approach from \cite{Cannizzaro.2023} to show that the second moment of $X_t^{\epsilon }$ under the annealed law converges to $(c(\nu {)}^2+2{\nu }^2)t$ with a precisely determined constant $c(\nu )>0$, which implies a non-trivial limit of the drift terms as $\epsilon$ vanishes. We also prove that in this weak coupling regime, the sequence of solutions converges in distribution to $\left(\sqrt{\frac{c(\nu {)}^2}{2}+{\nu }^2}\right){\tilde{B}}_t$ as $\epsilon$ vanishes, where ${\tilde{B}}_t$ is a two-dimensional standard Brownian motion.