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} dtheta_t + |D|^{2delta}theta_t,dx + (u_t cdot abla)theta_t,dx + |D|^{delta}dW_t = 0 u_t = abla^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 $delta > frac32$.