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A stochastic version of 2D Euler equations with transport type noise in thevorticity is considered, in the framework of Albeverio--Cruzeiro theory [1]where the equation is considered with random initial conditions related to theso called enstrophy measure. The equation is studied by an approximation schemebased on random point vortices. Stochastic processes solving the Eulerequations are constructed and their density with respect to the enstrophymeasure is proved to satisfy a continuity equation in weak form. Relevant incomparison with the case without noise is the fact that here we prove agradient type estimate for the density. Although we cannot prove uniqueness forthe continuity equation, we discuss how the gradient type estimate may berelated to this open problem.

$\rho$-white noise solution to 2D stochastic Euler equations