Mean-field limit for a class of stochastic ergodic control problems

We study a family of McKean-Vlasov (mean-field) type ergodic optimal control problems with linear control, and quadratic dependence on control of the cost function. For this class of problems we establish existence and uniqueness of an optimal control. We propose an $N$-particles Markovian optimal control problem approximating the McKean-Vlasov one and we prove the convergence in relative entropy, total variation and Wasserstein distance of the law of the former to the law of the latter when $N$ goes to infinity. Some McKean-Vlasov optimal control problems with singular cost function and the relation of these problems with the mathematical theory of Bose-Einstein condensation is also established.