Existence of optimal transport maps in very strict $CD(K,\infty)$ -spaces

TL;DR

This paper introduces a more restrictive version of the strict $CD(K, inity)$ condition, called very strict $CD(K, inity)$, and proves the existence of optimal transport maps in these spaces even when optimal plans are not unique.

Contribution

It defines the very strict $CD(K, inity)$ condition and establishes the existence of optimal maps under this new framework, advancing the understanding of optimal transport in non-unique plan settings.

Findings

Existence of optimal maps in very strict $CD(K, inity)$ spaces.

The new condition ensures optimal map existence despite non-uniqueness.

Provides a framework for further analysis of transport in complex metric measure spaces.

Abstract

We introduce a more restrictive version of the strict $CD(K,\infty)$ -condition, the so-called very strict $CD(K,\infty)$ -condition, and show the existence of optimal maps in very strict $CD(K,\infty)$ -spaces despite the possible lack of uniqueness of optimal plans.