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We consider the stochastic Navier--Stokes equations in three dimensions andprove that the law of analytically weak solutions is not unique. In particular,we focus on three examples of a stochastic perturbation: an additive, a linearmultiplicative and a nonlinear noise of cylindrical type, all driven by aWiener process. In these settings, we develop a stochastic counterpart of theconvex integration method introduced recently by Buckmaster and Vicol. Thispermits to construct probabilistically strong and analytically weak solutionsdefined up to a suitable stopping time. In addition, these solutions fail thecorresponding energy inequality at a prescribed time with a prescribedprobability. Then we introduce a general probabilistic construction used toextend the convex integration solutions beyond the stopping time and inparticular to the whole time interval $[0,\infty)$. Finally, we show that theirlaw is distinct from the law of solutions obtained by Galerkin approximation.In particular, non-uniqueness in law holds on an arbitrary time interval$[0,T]$, $T>0$.

Non-uniqueness in law of stochastic 3D Navier--Stokes equations