Weak vorticity formulation of 2D Euler equations with white noise initial condition

The 2D Euler equations with random initial condition distributed as a certain Gaussian measure are considered. The theory developed by S. Albeverio and A.-B. Cruzeiro is revisited, following the approach of weak vorticity formulation. A solution is constructed as a limit of random point vortices. This allows to prove that it is also limit of L^\infty-vorticity solutions. The result is generalized to initial measures that have a continuous bounded density with respect to the original Gaussian measure.