A Liouville theorem for bounded empirical-harmonic functions on $\mathcal{P}_2(M)$

Abstract

The classical Liouville theorem states that every bounded harmonic function on Euclidean space is constant. On complete Rie mannian manifolds, analogous conclusions hold under geometric as sumptions such as nonnegative Ricci curvature. The quadratic Wasserstein space $\mathcal{P}_2(M)$ is often regarded (following Otto) as an infinite-dimensional Riemannian manifold; however, it has no canonical infinite-dimensional Riemannian volume and therefore no canonica Laplace--Beltrami operator. In this paper we introduce a canonical finite-dimensional trace notion of harmonicity: a continuous function $u:\mathcal{P}_2(M)\to\mathbb{R}$ is called empirically harmonic if, for every $N\geq 1$, its pullback to the $N$-particle empirical stratum, \begin{equation*} (x_1,\ldots,x_N)\longmapsto u\left(\frac1N\sum_{i=1}^Nδ_{x_i}\right), \end{equation*} is weakly harmonic on $M^N$. We prove that if $M$ has the finite-product Liouville property, then every bounded empirically harmonic function on $\mathcal{P}_2(M)$ is constant. In particular, this applies to $M=\mathbb{R}^d$ and, more generally, to every complete connected Riemannian manifold with nonnegative Ricci curvature. We also show sharpness: if $M$ admits a nonconstant bounded harmonic function, then $\mathcal{P}_2(M)$ admits a nonconstant bounded empirically harmonic function.