On the (In)stability of the Identity Map in Optimal Transportation

TL;DR

This paper investigates the stability of the identity map in optimal transportation by examining how small perturbations in density and domain shape affect optimal transport plans, especially near irregular domain features.

Contribution

It provides new insights into the stability behavior of the identity map under various domain and density perturbations, focusing on two-dimensional cases with corners.

Findings

Stability of the identity map varies with domain regularity.

Perturbations near corners can significantly affect optimal transport.

Examples illustrate conditions leading to instability.

Abstract

We collect some examples of optimal transports in order to explore the (in)stability of the identity map as an optimal transport. First, we consider density and domain perturbations near regular portions of domains. Second, we investigate density and domains deformations to non-regular parts of domains. Here, we restrict our attention to two dimensions and focus near 90 degree corners.