Oscillating solutions to the mean-field Langevin descent-ascent flow

We present a counterexample to the statement of convergence of the mean-field Langevin descent-ascent flow on $\mathbb{R}^2$. We consider payoff functions that are shaped as a double well in each coordinate, and for which the deterministic dynamics admits a limit cycle. When the coupling between the two coordinates is sufficiently strong and the entropic regularization sufficiently small, we show that the mean-field dynamics remains close to this cyclic behavior, and in particular, does not converge.