Title: Global optimality of softmax policy gradient with single hidden layer neural networks in the mean-field regime. Abstract: We study the problem of policy optimization for infinite-horizon discounted Markov Decision Processes with softmax policy and nonlinear function approximation trained with policy gradient algorithms. We concentrate on the training dynamics in the mean-field regime, modeling e.g. the behavior of wide single hidden layer neural networks, when exploration is encouraged through entropy regularization. The dynamics of these models is established as a Wasserstein gradient flow of distributions in parameter space.  We further prove global optimality of the fixed points of this dynamics  under mild conditions on their initialization.