PUB - Publikationen an der Universität Bielefeld

We are concerned with the three-dimensional incompressible Navier- Stokes equations driven by an additive stochastic forcing of trace class. First, for every divergence free initial condition in L2 we establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions, solving one of the open problems in the field. This result, in partic-ular, implies nonuniqueness in law. Second, we prove nonuniqueness of the associated Markov processes in a suitably chosen class of analytically weak solutions satisfying a relaxed form of an energy inequality. Translated to the deterministic setting, we obtain nonuniqueness of the associated semiflows.