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

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