  We consider the stochastic electrokinetic flow in a smooth bounded domain
$\mathcal{D}$, modelled by a Nernst-Planck-Navier-Stokes system with a blocking
boundary conditions for ionic species concentrations, perturbed by
multiplicative noise. Several results are established in this paper. In both
$2d$ and $3d$ cases, we establish the global existence of weak martingale
solution which is weak in both PDEs and probability sense, and also the
existence and uniqueness of the maximal strong pathwise solution which is
strong in PDEs and probability sense. Particularly, we show that the maximal
pathwise solution is global one in $2d$ case without the restriction of
smallness of initial data.
