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

Abstract: In this article, we study the weak coupling limit of the following equation in $\mathbb{R}^2$: $$dX_t^\varepsilon=\frac{\hat{\lambda}}{\sqrt{\log\frac1\varepsilon}}\omega^\varepsilon(X_t^\varepsilon)dt+\nu dB_t,\quad X_0^\varepsilon=0. $$ Here $\omega^\varepsilon=\nabla^{\perp}\rho_\varepsilon*\xi$ with $\xi$ representing the $2d$ Gaussian Free Field (GFF) and $\rho_\varepsilon$ denoting an appropriate identity. $B_t$ denotes a two-dimensional standard Brownian motion, and $\hat{\lambda},\nu>0$ are two given constants. We use the approach from \cite{Cannizzaro.2023} to show that the second moment of $X_t^\varepsilon$ 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 $\varepsilon$ vanishes. We also prove that in this weak coupling regime, the sequence of solutions converges in distribution to $\left(\sqrt{\frac{c(\nu)^2}{2}+\nu^2}\right)\widetilde{B}_t$ as $\varepsilon$ vanishes, where $\widetilde{B}_t$ is a two-dimensional standard Brownian motion.