  Mean field games are studied in the framework of controlled martingale
problems, and general existence theorems are proven in which the equilibrium
control is Markovian. The framework is flexible enough to include degenerate
volatility, which may depend on both the control and the mean field. The
objectives need not be strictly convex, and the mean field interactions
considered are nonlocal and Wasserstein-continuous. When the volatility is
nondegenerate, continuity assumptions may be weakened considerably. The proofs
first use relaxed controls to establish existence. Then, using a convexity
assumption and measurable selection arguments, strict (non-relaxed) Markovian
equilibria are constructed from relaxed equilibria.
