*Accepted Paper*

**Inserted:** 23 may 2019

**Last Updated:** 30 apr 2021

**Journal:** CPAM

**Year:** 2019

**Notes:**

This paper replaces and supersedes the first (deterministic) part of the preprint 'A large-scale regularity theory for the Monge-Ampère equation with rough data and application to the optimal matching problem' (which will therefore not be submitted). The first main difference is that we are able here to directly relate the displacements to the flux given by the Poisson equation and thus confirming the linearization ansatz by Caracciolo and al. The second main difference is in the statement (which is more general) and the proofs of the harmonic approximation result.

**Abstract:**

This paper is about quantitative linearization results for the Monge-Ampère equation with rough data. We develop a large-scale regularity theory and prove that if a measure $\mu$ is close to the Lebesgue measure in Wasserstein distance at all scales, then the displacement of the macroscopic optimal coupling is quantitatively close at all scales to the gradient of the solution of the corresponding Poisson equation. The main ingredient we use is a harmonic approximation result for the optimal transport plan between arbitrary measures. This is used in a Campanato iteration which transfers the information through the scales.

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