Non-uniqueness in law of Leray solutions to 3D forced stochastic Navier-Stokes equations

This paper concerns the forced stochastic Navier-Stokes equation driven by additive noise in the three dimensional Euclidean space. By constructing an appropriate forcing term, we prove that there exist distinct Leray solutions in the probabilistically weak sense. In particular, the joint uniqueness in law fails in the Leray class. The non-uniqueness also displays in the probabilistically strong sense in the local time regime, up to stopping times. Furthermore, we discuss the optimality from two different perspectives: sharpness of the hyper-viscous exponent and size of the external force. These results in particular yield that the Lions exponent is the sharp viscosity threshold for the uniqueness/non-uniqueness in law of Leray solutions. Our proof utilizes the self-similarity and instability programme developed by Jia \v{S}ver\'{a}k [42,43] and Albritton-Bru\'{e}-Colombo [1], together with the theory of martingale solutions including stability for non-metric spaces and gluing procedure.