# On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint

arXiv: https://arxiv.org/abs/1412.4430 · HTML: https://mlsanity.com/paper/1412.4430/

## Key facts

- Cited by 99 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Dec 2014.

## Built on by (most cited first, 30 of 99)

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- [Convergence Rates for Regularized Optimal Transport via Quantization](https://mlsanity.com/paper/2208.14391/) (2022)
- [Schr"odinger Bridge with Quadratic State Cost is Exactly Solvable](https://mlsanity.com/paper/2406.00503/) (2024)
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## Abstract (arXiv)

We take a new look at the relation between the optimal transport problem and the Schr"odinger bridge problem from the stochastic control perspective. We show that the connections are richer and deeper than described in existing literature. In particular: a) We give an elementary derivation of the Benamou-Brenier fluid dynamics version of the optimal transport problem; b) We provide a new fluid dynamics version of the Schr"odinger bridge problem; c) We observe that the latter provides an important connection with optimal transport without zero noise limits; d) We propose and solve a fluid dynamic version of optimal transport with prior; e) We can then view optimal transport with prior as the zero noise limit of Schr"odinger bridges when the prior is any Markovian evolution. In particular, we work out the Gaussian case. A numerical example of the latter convergence involving Brownian particles is also provided.