  We consider a stochastic model which describes the motion of a 2D
incompressible fluid in a unbounded domain with viscosity and memory effects.
This model is different from the classical stochastic Navier-Stokes-Voigt
equations due to the absence of the Voigt term $ -\alpha \Delta u_{t}$, and has
a much weaker dissipation than the usual Navier-Stokes-Voigt model since only
the memory viscoelasticity is present. We are interested in the global
well-posedness and long-time behaviors of this model. We first investigate the
well-posedness by using the classical Faedo-Galerkin method. Unlike the general
method of energy estimate, we then split the solution into two parts and get
the low-order and high-order uniform estimates, respectively. Based on the
uniform estimates of far-field values of solutions, we further prove the
existence and uniqueness of random attractors in unbounded domains with a
constructed compact subspace corresponding to memory. Finally, we give the
upper semicontinuity of the attractors when stochastic perturbation approaches
to zero.
