Markov selections and their regularity for the three-dimensional stochastic Navier-Stokes equations

The martingale problem associated to the three-dimensional Navier–Stokes equations is shown to have a family of solutions satisfying the Markov property. The result 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 showing that every Markov selection has a property of continuous dependence on initial conditions.