  In this paper, we establish the asymptotic stability of steady-state for
$3$-D stochastic Euler-Poisson equations with insulating boundary conditions
forced by the Wiener process. We use Banach's fixed point theorem and the a
priori energy estimates uniformly in time to get the global existence of
solutions around the steady state. Compared with the deterministic case, due to
the stochastic forces, the momentum does not have the temporal derivative,
which causes troubles to the energy estimates. We establish the asymptotic
stability about the spatial derivatives by the weighted energy estimates for
the estimates of stochastic integrals, employing a technique distinct from the
deterministic case. The existence of invariant measure generated by the
solutions to the evolving semiconductor equations is shown from the a priori
energy estimates. Furthermore, the invariant measure is exactly the Dirac
measure generated by the steady state, in which the exponential decay of
perturbed solutions around the steady state plays an essential role.
