# Approximate Monge solutions continuously depending on the parameter

## TL;DR

This paper studies how approximate optimal transportation solutions vary continuously with a parameter, ensuring stability of solutions in parametric optimal transport problems.

## Contribution

It proves the existence of parameter-dependent approximate Monge mappings that are continuous, extending the understanding of stability in optimal transport.

## Key findings

- Existence of continuous approximate Monge mappings.
- Stability results for optimal transport solutions under parameter variation.
- Extension to costs and marginals depending continuously on a parameter.

## Abstract

We consider Kantorovich optimal transportation problem in the case where the cost function and marginal distributions continuously depend on a parameter with values in a metric space. We prove the existence of approximate optimal Monge mappings continuous with respect to the parameter.