  In this paper, we establish the asymptotic stability of the steady-state for
a 1-D stochastic Euler-Poisson equations with Ohmic contact boundary conditions
forced by the Wiener process. We utilize Banach's fixed point theorem and the a
priori energy estimates uniformly in time to ensure the global existence of
solutions around the steady state. In contrast to the deterministic case, the
presence of stochastic forces lead to the lack of temporal derivatives of
momentum, posing challenges for energy estimates. Furthermore, Ohmic contact
boundary conditions pose greater challenges for energy estimates compared to
systems with insulating boundary conditions. To address this issue, we
establish asymptotic stability concerning the spatial derivatives through
weighted energy estimates for the estimates of stochastic integrals, employing
a technique distinct from that of the deterministic case. Furthermore, we
demonstrate the existence of an invariant measure based on the a priori energy
estimates. This invariant measure precisely corresponds to the Dirac measure
generated by the steady state, due to the exponential decay of perturbed
solutions around the steady state.
