# Minimizing relative entropy of path measures under marginal constraints

arXiv: https://arxiv.org/abs/2001.10920 · HTML: https://mlsanity.com/paper/2001.10920/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jan 2020.

## Built on by (most cited first, 4 of 4)

- [Towards a mathematical theory of trajectory inference](https://mlsanity.com/paper/2102.09204/) (2021)
- [Multi-marginal temporal Schr"odinger Bridge Matching from unpaired data](https://mlsanity.com/paper/2510.01894/) (2025)
- [Multi-Marginal Schr"odinger Bridge Matching](https://mlsanity.com/paper/2510.16587/) (2025)
- [Specificity-Aware Diffusion Steering via Variance-Reduced Sequential Monte Carlo](https://mlsanity.com/paper/2610.00395/) (2026)

## Abstract (arXiv)

We study generalizations of the Schr"odinger problem in statistical mechanics in two directions: when the density is constrained at more than two times, and when the joint law of the initial and final positions for the particles is prescribed. This is done in agreement with the so-called Br"odinger problem recently introduced to regularize Brenier's variational model for incompressible fluids. We recover generalizations of the standard factorization result for the Radon-Nikodym derivative of the solution P with respect to the reference measure R: this density can be written in terms of an additive functional on the set of constrained times. The specificity of this work is that we place ourselves in the case when R is Markov (or reciprocal), and that we use Markovian methods rather than classical convex analysis arguments. In this setting, it appears that a natural assumption to be made on the reference measure R is of irreducibility type.