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 R²: dXₜ^=1^ (Xₜ^) dt+ dBₜ, X₀^=0. Here ^=^_* with representing the 2d Gaussian Free Field (GFF) and _ denoting an appropriate identity. Bₜ denotes a two-dimensional standard Brownian motion, and, >0 are two given constants. We use the approach from Cannizzaro. 2023 to show that the second moment of Xₜ^ under the annealed law converges to (c () ²+2²) t with a precisely determined constant c () >0, which implies a non-trivial limit of the drift terms as vanishes. We also prove that in this weak coupling regime, the sequence of solutions converges in distribution to (c () ²{2+²}) Bₜ as vanishes, where Bₜ is a two-dimensional standard Brownian motion.