Uniform in time propagation of chaos for noisy mean-field coupled maps
We study discrete-time $N$-dimensional mean-field systems on the torus subject to additive noise whose probability density is bounded away from zero. Given a Lipschitz one-particle map and interaction term, we prove that, if the lower bound on the noise density is sufficiently large, the $N$-dimensional transfer operator $\mathcal P_N$ preserves a dimension-independent class of sub-Gaussian probability measures. Moreover, $\mathcal P_N$ is a one-step contraction in the Dobrushin-Wasserstein distance and therefore admits a unique invariant measure $\rho_N$, which is itself sub-Gaussian. Under the same condition on the noise strength, we show that the associated self-consistent transfer operator is a one-step contraction in Wasserstein distance and hence admits a unique fixed point $\rho$. We further prove, using these contraction estimates, that $\mathcal P_N$ preserves an $O(N^{-1/2})$ neighbourhood of the product measure $\rho^{\otimes N}$ in the Dobrushin-Wasserstein metric, and in particular that $\rho_N$ belongs to this neighbourhood. Finally, we establish uniform-in-time propagation of chaos in the Dobrushin-Wasserstein metric and, under the additional assumption that the noise density is of bounded variation, in total variation for every fixed-dimensional marginal.