The Langevin algorithm is a standard method to minimize, in the space of probability measures, the sum of a linear functional and the entropy. In this talk, motivated by the analysis of noisy gradient descent to compute grid-free regularized Wasserstein barycenters, we consider the « Mean-Field Langevin Dynamics », a nonlinear generalization of the Langevin dynamics that minimizes the sum of a *convex* functional and the entropy. We show that, under a certain uniform log-Sobolev assumption, the dynamics converges exponentially fast to global minimizers (this result was also proven independently in [Nitanda et al. 2022]). We also present the "simulated annealing » variant of this dynamics, and show that for a suitable noise decay, it converges in value to the global minimizer of the convex functional. As a consequence of these abstract results, we obtain a convergence rate for noisy gradient descent to compute regularized Wasserstein barycenters.