Large $N$ limit of the Langevin dynamics for the spin $O(N)$ model

In this paper, we prove that the large $N$ limit of the Langevin dynamics for the spin $O(N)$ model is given by a mean-field stochastic differential equation (SDE) in both finite and infinite volumes. We establish uniform in $N$ bounds for the dynamics, which enable us to demonstrate convergence to the mean-field SDE with polynomial interactions. Furthermore, the mean-field SDE is shown to be globally well-posed for suitable initial distributions. We also prove the existence of stationary measures for the mean-field SDE. For small inverse temperatures, we characterize the large $N$ limit of the spin $O(N)$ model through stationary coupling. Additionally, we establish the uniqueness of the stationary measure for the mean-field SDE.