Network Consensus in the Wasserstein Space of Probability Measures Defined on Multi-Dimensional Euclidean Spaces

The consensus problem -- achieving agreement among a network of agents -- is a central theme in both theory and applications. Recently, this problem has been extended from Euclidean spaces to the space of probability measures, where the natural notion of averaging is given by the Wasserstein barycenter. While prior work established convergence in one dimension, the case of higher dimensions poses additional challenges due to the curved geometry of Wasserstein space. In this paper, we develop a framework for analyzing such consensus algorithms by employing a Wasserstein version of Jensen's inequality. This tool provides convexity-type estimates that allow us to prove convergence of nonlinear consensus dynamics in the Wasserstein space of probability measures on $\mathbb{R}^d$.