Infinitesimal Invariance of Completely Random Measures for 2D Euler Equations

We consider suitable weak solutions of 2-dimensional Euler equations on bounded domains, and show that the class of completely random measures is infinitesimally invariant for the dynamics. Space regularity of samples of these random fields falls outside of the well-posedness regime of the PDE under consideration, so it is necessary to resort to stochastic integrals with respect to the candidate invariant measure in order to give a definition of the dynamics. Our findings generalize and unify previous results on Gaussian stationary solutions of Euler equations and point vortices dynamics. We also discuss difficulties arising when attempting to produce a solution flow for Euler’s equations preserving independently scattered random measures.