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