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^ε=λ̂√log1εω^ε(Xₜ^ε)dt+ν dBₜ, X₀^ε=0. Here ω^ε=∇⊥ρ_ε*ξ with ξ representing the $2d$ Gaussian Free Field (GFF) and ρ_ε denoting an appropriate identity. Bₜ denotes a two-dimensional standard Brownian motion, and λ̂,ν>0 are two given constants. We use the approach from {Cannizzaro.2023} to show that the second moment of Xₜ^ε under the annealed law converges to (c(ν)²+2ν²)t with a precisely determined constant c(ν)>0, which implies a non-trivial limit of the drift terms as ε vanishes. We also prove that in this weak coupling regime, the sequence of solutions converges in distribution to (√c(ν)²/2+ν²)Bₜ as ε vanishes, where Bₜ is a two-dimensional standard Brownian motion.