Almost global existence for the stochastic Navier-Stokes equations with small $H^{1\mathrm{/}2}$ data

We address the global existence of solutions to the stochastic Navier-Stokes equations with multiplicative noise and with initial data in $H^{1\mathrm{/}2}({\mathbb{T}}^3)$. We prove that the solution exists globally in time with probability arbitrarily close to~$1$ if the initial data and noise are sufficiently small. If the noise is not assumed to be small, then the solution is global on a sufficiently small deterministic time interval with probability arbitrarily close to~$1$.