Finite-dimensional reduction of a Wasserstein gradient flow and sharp decay rates

Abstract

We study the Wasserstein gradient flow generated by a family of extended generalized variance functionals, defined as the expected squared $n$-dimensional volume of a simplex, which includes the classical variance-type interaction and generalized variance as special cases. The key structural observation is that this functional depends only on the covariance matrix. Consequently, the Wasserstein gradient flow reduces to a finite-dimensional system for the eigenvalues of the covariance matrix, and the full measure-valued solution can be recovered through an explicit linear pushforward representation. Using this representation, we establish global well-posedness for arbitrary initial data in ${\mathcal{P}}_2({\mathbb{R}}^d)$ without assuming compact support. We also study the long-time behavior of the flow. For every initial datum in ${\mathcal{P}}_2({\mathbb{R}}^d)$, the solution converges to a limiting equilibrium measure whose covariance has rank strictly less than $n$. Moreover, we obtain sharp convergence rates in all spectral regimes: exponential in the non-degenerate case and algebraic with the optimal exponent in the degenerate case.