  A linear stochastic continuity equation with non-regular coefficients is
considered. We prove existence and uniqueness of strong solution, in the
probabilistic sense, to the Cauchy problem when the vector field has low
regularity, in which the classical DiPerna-Lions-Ambrosio theory of uniqueness
of distributional solutions does not apply. We solve partially the open problem
that is the case when the vector-field has random dependence. In addition, we
prove a stability result for the solutions.
