Global existence of the stochastic Navier-Stokes equations in $L^3$ with small data

We address the global-in-time existence and pathwise uniqueness of solutions for the stochastic incompressible Navier-Stokes equations with a multiplicative noise on the three-dimensional torus. Under natural smallness conditions on the noise, we prove the almost global existence result for small $L^3$ data. Namely, we show that for data sufficiently small, there exists a global-in-time strong $L^3$ solution in a space of probability arbitrarily close to~$1$.