  We propose and study the framework of dissipative statistical solutions for
the incompressible Euler equations. Statistical solutions are
time-parameterized probability measures on the space of square-integrable
functions, whose time-evolution is determined from the underlying Euler
equations. We prove partial well-posedness results for dissipative statistical
solutions and propose a Monte Carlo type algorithm, based on spectral viscosity
spatial discretizations, to approximate them. Under verifiable hypotheses on
the computations, we prove that the approximations converge to a statistical
solution in a suitable topology. In particular, multi-point statistical
quantities of interest converge on increasing resolution. We present several
numerical experiments to illustrate the theory.
