Nonuniqueness in law of stochastic 3D Navier-Stokes equations

We consider the stochastic Navier-Stokes equations in three dimensions and prove that the law of analytically weak solutions is not unique. In particular, we focus on three examples of a stochastic perturbation: an additive, a linear multiplicative and a nonlinear noise of cylindrical type, all driven by a Wiener process. In these settings, we develop a stochastic counterpart of the convex integration method introduced recently by Buckmaster and Vicol. This permits us to construct probabilistically strong and analytically weak solutions defined up to a suitable stopping time. In addition, these solutions fail to satisfy the corresponding energy inequality at a prescribed time with a prescribed probability. Then we introduce a general probabilistic construction used to extend the convex integration solutions beyond the stopping time and in particular to the whole time interval [0, infinity). Finally, we show that their law is distinct from the law of solutions obtained by Galerkin approximation. In particular, nonuniqueness in law holds on an arbitrary time interval [0, T], T > 0.