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We study the surface quasi-geostrophic equation with an irregular spatialperturbation $$ \partial_{t }\theta+ u\cdot\nabla\theta =-\nu(-\Delta)^{\gamma/2}\theta+ \zeta,\qquadu=\nabla^{\perp}(-\Delta)^{-1}\theta, $$ on $[0,\infty)\times\mathbb{T}^{2}$,with $\nu\geq 0$, $\gamma\in [0,3/2)$ and $\zeta\inB^{-2+\kappa}_{\infty,\infty}(\mathbb{T}^{2})$ for some $\kappa>0$. This coversthe case of $\zeta= (-\Delta)^{\alpha/2}\xi$ for $\alpha<1$ and $\xi$ a spatialwhite noise on $\mathbb{T}^{2}$. Depending on the relation between $\gamma$ and$\alpha$, our setting is subcritical, critical or supercritical in the languageof Hairer's regularity structures \cite{Hai14}. Based on purely analyticaltools from convex integration and without the need of any probabilisticarguments including renormalization, we prove existence of infinitely manyanalytically weak solutions in$L^{p}_{\rm{loc}}(0,\infty;B_{\infty,1}^{-1/2})\capC_{b}([0,\infty);B^{-1/2-\delta}_{\infty,1})\capC^{1}_{b}([0,\infty);B^{-3/2-\delta}_{\infty,1})$ for all $p\in [1,\infty)$ and$\delta>0$. We are able to prescribe an initial as well as a terminal conditionat a finite time $T>0$, and to construct steady state, i.e. time independent,solutions. In all cases, the solutions are non-Gaussian, but we may as wellprescribe Gaussianity at some given times. Moreover, a coming down frominfinity with respect to the perturbation and the initial condition holds.Finally, we show that the our solutions generate statistically stationarysolutions as limits of ergodic averages, and we obtain existence of infinitelymany non-Gaussian time dependent ergodic stationary solutions. We also extendour results to a more general class of singular SPDEs.

A class of supercritical/critical singular stochastic PDEs: existence, non-uniqueness, non-Gaussianity, non-unique ergodicity