  For a prescribed deterministic kinetic energy we use convex integration to
construct analytically weak and probabilistically strong solutions to the 3D
incompressible Navier-Stokes equations driven by a linear multiplicative
stochastic forcing. These solutions are defined up to an arbitrarily large
stopping time and have deterministic initial values, which are part of the
construction. Moreover, by a suitable choice of different kinetic energies
which coincide on an interval close to time 0, we obtain non-uniqueness.

Analysis of PDEs;Probability