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This paper addresses semistable stochastic Linear-Quadratic Consensus (LQC) problems motivated by the recently developed Optimal Semistable Control (OSC) and semistable H2 control problems. OSC deals with linear-quadratic optimal semistabilization. In the framework of OSC, the closed-loop system is not asymptotically stable, but semistable. Semistability is the property that every trajectory of the closed-loop system converges to a Lyapunov stable equilibrium point determined by the system initial conditions. Hence, the limiting state of the closed-loop system is not a fixed point a priori, but a continuum of equilibria. In such a sense, OSC can be viewed as an optimal regulation problem with nondeterministic, nonzero set-points. In this paper, we consider stochastic OSC for optimal consensus seeking under white noise and random distribution of initial conditions. We show that the distinct feature of the proposed semistable stochastic LQC problem is the possibility of nonuniqueness of the solutions and hence, cannot be treated by using the methods developed for the classical LQR control theory. We develop a new framework for semistable stochastic LQC and suggest an alternative constrained optimization method to solve it. To this end, necessary and sufficient conditions for semistability and optimal consensus seeking under white noise and random distribution of initial conditions are derived in the paper.