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 two iconic examples of a stochastic perturbation: either an additive or a linear multiplicative noise driven by a Wiener process. In both cases, we develop a stochastic counterpart of the convex integration method introduced recently by Buckmaster and Vicol. This permits to construct probabilistically strong and analytically weak solutions defined up to a suitable stopping time. In addition, these solutions fail 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 $[0,\infty)$. Finally, we show that their law is distinct from the law of solutions obtained by Galerkin approximation. In particular, non-uniqueness in law holds on an arbitrary time interval $[0,T]$, $T>0$.