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 Gubinelli and Perkowski (2020).

我们考虑 2D 环面上的修正表面准地转 (mSQG) 方程 ${\mathbb{吨}}^2$，受到乘法传输噪声的干扰。该方程承认白噪声测度${\mathbb{吨}}^2$作为不变测度。我们首先通过点涡旋近似的方法证明了随机方程的白噪声解的存在，然后，在噪声的适当缩放限制下，我们证明了解弱收敛到耗散 mSQG 方程驱动的唯一平稳解通过时空白噪声。后一个方程的弱唯一性也通过遵循 Gubinelli 和 Perkowski 在 Gubinelli and Perkowski (2020) 中的方法得到了证明。