# Gromov-Hausdorff limit of Wasserstein spaces on point clouds

## TL;DR

This paper proves that the discrete Wasserstein space on a random point cloud on a torus converges to the continuous Wasserstein space as the number of points increases, under certain conditions on the connection radius.

## Contribution

It establishes the Gromov-Hausdorff convergence of Wasserstein spaces on point clouds to the continuous space, providing a foundation for studying evolution equations on random data.

## Key findings

- Convergence of discrete to continuous Wasserstein spaces under specified conditions.
- Explicit rate for the decay of the connection radius $\
- First step towards stability analysis of evolution equations on random point clouds.

## Abstract

We consider a point cloud $X_n := \{ x_1, \dots, x_n \}$ uniformly distributed on the flat torus $\mathbb{T}^d : = \mathbb{R}^d / \mathbb{Z}^d $, and construct a geometric graph on the cloud by connecting points that are within distance $\varepsilon$ of each other. We let $\mathcal{P}(X_n)$ be the space of probability measures on $X_n$ and endow it with a discrete Wasserstein distance $W_n$ as introduced independently by Chow et al, Maas, and Mielke for general finite Markov chains. We show that as long as $\varepsilon= \varepsilon_n$ decays towards zero slower than an explicit rate depending on the level of uniformity of $X_n$, then the space $(\mathcal{P}(X_n), W_n)$ converges in the Gromov-Hausdorff sense towards the space of probability measures on $\mathbb{T}^d$ endowed with the Wasserstein distance. The analysis presented in this paper is a first step in the study of stability of evolution equations defined over random point clouds as the number of points grows to infinity.