This paper is devoted to the analysis of a stochastic equation describing
the motions of a large class of incompressible linear viscoelastic
fluids in
two-dimensional subject to periodic boundary condition and driven by random
external forces. To do so we distinguish two cases, and for each case a global
existence result of probabilistic weak solution for is expounded in this paper. We
also prove that under suitable hypotheses on the external random forces the solution
turns out to be unique. As concrete examples, we consider the stochastic
equations for the Maxwell and Oldroyd
uids that are of great importance in the
investigation towards the understanding of the elastic turbulence.
