We consider the modified Surface Quasi-Geostrophic (mSQG) equation on the 2D torus $\mathbb{ T}^2$, perturbed by multiplicative transport noise. The equation admits the white noise measure on $\mathbb{ T}^2$ as the invariant measure. We first prove the existence of white noise solutions to the stochastic equation via the method of point vortex approximation, then, under a suitable scaling limit of the noise, we show that the solutions converge weakly to the unique stationary solution of the dissipative mSQG equation driven by space-time white noise. The weak uniqueness of the latter equation is also proved by following Gubinelli and Perkowski's approach in \cite{ GP-18}.
Categories: math.PR