We study the mean field Langevin dynamics and the associated particle system. By assuming the functional convexity of the energy, we obtain the $L^p$-convergence of the marginal distributions towards the unique invariant measure for the mean field dynamics. Furthermore, we prove the uniform-in-time propagation of chaos in both the $L^2$-Wasserstein metric and relative entropy.

翻译：我们研究平均场朗之万动力学及其相关粒子系统。通过假设能量的函数凸性，我们得到了边际分布关于平均场动力学唯一不变测度的$L^p$收敛性。此外，我们证明了在$L^2$-Wasserstein度量与相对熵中时间一致的混沌传播性质。