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 Riemannian manifolds, analogous conclusions hold under geometric assumptions such as nonnegative Ricci curvature. The quadratic Wasserstein space ${\mathcal{P}}_2(M)$ has no canonical infinite-dimensional Riemannian volume and hence no canonical Laplace--Beltrami operator. We introduce a natural finite-particle notion of harmonicity: a continuous function $u:{\mathcal{P}}_2(M)\to \mathbb{R}$ is called empirically harmonic if, for every $N\ge 1$, its pullback under the empirical map $${\iota }_N(x_1,\dots ,x_N)=\frac{1}{N}\sum _{i=1}^N{\delta }_{x_i}$$ 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, the result applies to $M={\mathbb{R}}^d$ and to every complete connected Riemannian manifold with nonnegative Ricci curvature. We also derive a finite-particle chain rule for sufficiently regular functionals on ${\mathcal{P}}_2({\mathbb{R}}^d)$ and show that the empirical Laplacian is exactly the Hessian trace of a discrete $N$-particle lift. Finally, if $M$ admits a nonconstant bounded harmonic function, then ${\mathcal{P}}_2(M)$ admits a nonconstant bounded empirically harmonic linear statistic.

14 pages,modified some unclear expressions and added some details