Existence and Uniqueness of Energy Solutions to the Stochastic Diffusive Surface Quasi-Geostrophic Equation with Additive Noise

We continue our study of the dynamics of a nearly inviscid periodic surface quasi-geostrophic equation. Here we consider a slightly diffusive stochastic SQG equation of the form \begin{equation*} \begin{cases} d\theta_t + |D|^{2\delta}\theta_t\,dx + (u_t \cdot \nabla)\theta_t\,dx + |D|^{\delta}dW_t = 0 \\ u_t = \nabla^\perp|D|^{-1}\theta_t. \end{cases} \end{equation*} We construct global energy solutions as introduced by P. Goncalves and M. Jara (2014) for any \(\delta > 0\), so that any small amount of diffusion permits us to construct solutions. We show moreover that pathwise uniqueness of these energy solutions holds in the presence of sufficiently high diffusion \(\delta > \frac32\).