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We study the surface quasi-geostrophic equation driven by a generic additivenoise process $W$. By means of convex integration techniques, we establishexistence of weak solutions whenever the stochastic convolution $z$ associatedwith $W$ is well defined and fulfills certain regularity constraints.Quintessentially, we show that the so constructed solutions to the non-linearequation are controlled by $z$ in a linear fashion. This allows us to deducefurther properties of the so constructed solutions, without relying onstructural probabilistic properties such as Gaussianity, Markovianity or amartingale property of the underlying noise $W$.

On convex integration solutions to the surface quasi-geostrophic equation driven by generic additive noise