Stochastic mSQG equations with multiplicative transport noises: White noise solutions and scaling limit

We consider the modified Surface Quasi-Geostrophic (mSQG) equation on the 2D torus T², perturbed by multiplicative transport noise. The equation admits the white noise measure on T² 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 Gubinelli and Perkowski (2020).