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arXiv · 2405.06503

Optimal transport of measures via autonomous vector fields

Nicola De Nitti·Xavier Fernández-Real
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Abstract

We study the problem of transporting one probability measure to another via an autonomous velocity field. We rely on tools from the theory of optimal transport. In one space-dimension, we solve a linear homogeneous functional equation to construct a suitable autonomous vector field that realizes the (unique) monotone transport map as the time-$1$ map of its flow. Generically, this vector field can be chosen to be Lipschitz continuous. We then use Sudakov's disintegration approach to deal with the multi-dimensional case by reducing it to a family of one-dimensional problems.

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Nicola De Nitti, Xavier Fernández-Real. 2024-05-10. Optimal transport of measures via autonomous vector fields. https://arxiv.org/abs/2405.06503

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