The author considers a discrete linear dynamical system evolving in a stochastic environment and modelled by the following autonomous difference equation \(x_{n+1}=Ax_ n+Bu_{n+1},\quad x_ 0=Bu_ 0,\) where \(A\in {\mathcal B}[H,H]\) and \(B\in {\mathcal B}[U,H]\). Here U and H are separable nontrivial Hilbert spaces; \({\mathcal B}[X,Y]\) denotes the Banach space of all bounded linear transformations of X into Y, \(\{x_ i; i\geq 0\}\) denotes an H-valued state sequence such that \(x_ 0\) is an \({\mathcal R}(B)\) H-valued second order random variable. The input disturbance sequence \(\{u_ i; i\geq 0\}\) is assumed to be an U-valued second order wide sense stationary white noise, with correlation operator \(R=R^*=E\{u_ i\circ u_ i\}\geq 0\in {\mathcal B}_ 1[U]\) for all \(i\geq 0\). Mean square stability conditions are investigated, including a comparison with the deterministic stability problem. The particular case of compact operators is considered in some detail.