  This paper is concerned with an optimal control problem for a mean-field
linear stochastic differential equation with a quadratic functional in the
infinite time horizon. Under suitable conditions, including the
stabilizability, the (strong) exponential, integral, and mean-square turnpike
properties for the optimal pair are established. The keys are to correctly
formulate the corresponding static optimization problem and find the equations
determining the correction processes. These have revealed the main feature of
the stochastic problems which are significantly different from the
deterministic version of the theory.
