  For stochastic systems with nonvanishing noise, i.e., at the desired state
the noise port does not vanish, it is impossible to achieve the global
stability of the desired state in the sense of probability. This bad property
also leads to the loss of stochastic passivity at the desired state if a
radially unbounded Lyapunov function is expected as the storage function. To
characterize a certain (globally) stable behavior for such a class of systems,
the stochastic asymptotic weak stability is proposed in this paper which
suggests the transition measure of the state to be convergent and the
ergodicity. By defining stochastic weak passivity that admits stochastic
passivity only outside a ball centered around the desired state but not in the
whole state space, we develop stochastic weak passivity theorems to ensure that
the stochastic systems with nonvanishing noise can be globally\locally
stabilized in weak sense through negative feedback law. Applications are shown
to stochastic linear systems and a nonlinear process system, and some
simulation are made on the latter further.
