  This paper studies some analytical properties of weak solutions of 3D
stochastic primitive equations with periodic boundary conditions. The
martingale problem associated to this model is shown to have a family of
solutions satisfying the Markov property, which is achieved by means of an
abstract selection principle. The Markov property is crucial to extend the
regularity of the transition semigroup from small times to arbitrary times.
Thus, under a regular additive noise, every Markov solution is shown to have a
property of continuous dependence on initial conditions, which follows from
employing the weak-strong uniqueness principle and the Bismut-Elworthy-Li
formula.
