# Stochastic bridges of linear systems

arXiv: https://arxiv.org/abs/1407.3421 · HTML: https://mlsanity.com/paper/1407.3421/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jul 2014.

## Built on by (most cited first, 14 of 14)

- [Generalized Schr"odinger Bridge Matching](https://mlsanity.com/paper/2310.02233/) (2023)
- [Rethinking Score Distillation as a Bridge Between Image Distributions](https://mlsanity.com/paper/2406.09417/) (2024)
- [Provably Convergent Schr"odinger Bridge with Applications to Probabilistic Time Series Imputation](https://mlsanity.com/paper/2305.07247/) (2023)
- [Generative Modeling with Phase Stochastic Bridges](https://mlsanity.com/paper/2310.07805/) (2023)
- [Adjoint Schr"odinger Bridge Sampler](https://mlsanity.com/paper/2506.22565/) (2025)
- [Reflected Schr"odinger Bridge for Constrained Generative Modeling](https://mlsanity.com/paper/2401.03228/) (2024)
- [Momentum Multi-Marginal Schr"odinger Bridge Matching](https://mlsanity.com/paper/2506.10168/) (2025)
- [Variational Schr"odinger Diffusion Models](https://mlsanity.com/paper/2405.04795/) (2024)
- [Diffusion Methods for Generating Transition Paths](https://mlsanity.com/paper/2309.10276/) (2023)
- [Maximum entropy optimal density control of discrete-time linear systems and Schr"odinger bridges](https://mlsanity.com/paper/2204.05263/) (2022)
- [Branched Schr"odinger Bridge Matching](https://mlsanity.com/paper/2506.09007/) (2025)
- [Learning non-equilibrium diffusions with Schr"odinger bridges: from exactly solvable to simulation-free](https://mlsanity.com/paper/2505.16644/) (2025)
- [BridgeShape: Latent Diffusion Schr"odinger Bridge for 3D Shape Completion](https://mlsanity.com/paper/2506.23205/) (2025)
- [Bridging Vision and Language Concepts through Optimal Transport Semantic Flow](https://mlsanity.com/paper/2606.26891/) (2026)

## Abstract (arXiv)

We study a generalization of the Brownian bridge as a stochastic process that models the position and velocity of inertial particles between the two end-points of a time interval. The particles experience random acceleration and are assumed to have known states at the boundary. Thus, the movement of the particles can be modeled as an Ornstein-Uhlenbeck process conditioned on position and velocity measurements at the two end-points. It is shown that optimal stochastic control provides a stochastic differential equation (SDE) that generates such a bridge as a degenerate diffusion process. Generalizations to higher order linear diffusions are considered.