# Schr"odinger Bridge with Quadratic State Cost is Exactly Solvable

arXiv: https://arxiv.org/abs/2406.00503 · HTML: https://mlsanity.com/paper/2406.00503/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 68 references that are in the map.
- First posted on arXiv 1 Jun 2024.

## Builds on (most cited first, 27 of 27)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [I2SB: Image-to-Image Schr"odinger Bridge](https://mlsanity.com/paper/2302.05872/) (2023)
- [Denoising Diffusion Bridge Models](https://mlsanity.com/paper/2309.16948/) (2023)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [Deep Generative Learning via Schr"odinger Bridge](https://mlsanity.com/paper/2106.10410/) (2021)
- [On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint](https://mlsanity.com/paper/1412.4430/) (2014)
- [Diffusion Bridge Mixture Transports, Schr"odinger Bridge Problems and Generative Modeling](https://mlsanity.com/paper/2304.00917/) (2023)
- [The Schr"odinger Bridge between Gaussian Measures has a Closed Form](https://mlsanity.com/paper/2202.05722/) (2022)
- [Bayesian Learning via Neural Schr"odinger-F"ollmer Flows](https://mlsanity.com/paper/2111.10510/) (2021)
- [Wasserstein Proximal Algorithms for the Schr"odinger Bridge Problem: Density Control with Nonlinear Drift](https://mlsanity.com/paper/1912.01244/) (2019)
- [Entropic Neural Optimal Transport via Diffusion Processes](https://mlsanity.com/paper/2211.01156/) (2022)
- [Deep Generalized Schr"odinger Bridge](https://mlsanity.com/paper/2209.09893/) (2022)
- [Entropic and displacement interpolation: a computational approach using the Hilbert metric](https://mlsanity.com/paper/1506.04255/) (2015)
- [Optimal steering of a linear stochastic system to a final probability distribution](https://mlsanity.com/paper/1408.2222/) (2014)
- [Reflected Schr"odinger Bridge: Density Control with Path Constraints](https://mlsanity.com/paper/2003.13895/) (2020)
- [Building the Bridge of Schr"odinger: A Continuous Entropic Optimal Transport Benchmark](https://mlsanity.com/paper/2306.10161/) (2023)
- [Optimal steering of a linear stochastic system to a final probability distribution, Part III](https://mlsanity.com/paper/1608.03622/) (2016)
- [Reflected Schr"odinger Bridge for Constrained Generative Modeling](https://mlsanity.com/paper/2401.03228/) (2024)
- [Neural Schr"odinger Bridge with Sinkhorn Losses: Application to Data-driven Minimum Effort Control of Colloidal Self-assembly](https://mlsanity.com/paper/2307.14442/) (2023)
- [A Physics-informed Deep Learning Approach for Minimum Effort Stochastic Control of Colloidal Self-Assembly](https://mlsanity.com/paper/2208.09182/) (2022)
- [Schr"odinger Meets Kuramoto via Feynman-Kac: Minimum Effort Distribution Steering for Noisy Nonuniform Kuramoto Oscillators](https://mlsanity.com/paper/2202.09734/) (2022)
- [Hopfield Neural Network Flow: A Geometric Viewpoint](https://mlsanity.com/paper/1908.01270/) (2019)
- [On the Contraction Coefficient of the Schr"odinger Bridge for Stochastic Linear Systems](https://mlsanity.com/paper/2309.06622/) (2023)
- [Solution of the Probabilistic Lambert Problem: Connections with Optimal Mass Transport, Schr"odinger Bridge and Reaction-Diffusion PDEs](https://mlsanity.com/paper/2401.07961/) (2024)

## Built on by (most cited first, 12 of 12)

- [Harmonic Path Integral Diffusion](https://mlsanity.com/paper/2409.15166/) (2024)
- [On the Hopf-Cole Transform for Control-affine Schr"odinger Bridge](https://mlsanity.com/paper/2503.17640/) (2025)
- [Weyl Calculus and Exactly Solvable Schr"odinger Bridges with Quadratic State Cost](https://mlsanity.com/paper/2407.15245/) (2024)
- [A Generalized Sinkhorn Algorithm for Mean-Field Schr"odinger Bridge](https://mlsanity.com/paper/2604.06531/) (2026)
- [Schr"odinger Bridge Over A Compact Connected Lie Group](https://mlsanity.com/paper/2603.14049/) (2026)
- [Adaptive Path Integral Diffusion: AdaPID](https://mlsanity.com/paper/2512.11858/) (2025)
- [Nonlinear Non-Gaussian Density Steering with Input and Noise Channel Mismatch: Sinkhorn with Memory for Solving the Control-affine Schr"odinger Bridge Problem](https://mlsanity.com/paper/2604.23370/) (2026)
- [Twisted Schr"odinger Bridge Matching](https://mlsanity.com/paper/2607.16987/) (2026)
- [Lifted Schr"odinger Bridges for Gaussian Mixture Endpoints: Projection Gaps and Path-Space Obstructions](https://mlsanity.com/paper/2605.24795/) (2026)
- [Linear Exponential Quadratic Gaussian Covariance Steering](https://mlsanity.com/paper/2609.12463/) (2026)
- [On the Equivalence of Stochastic Control and Path Space Formulations for Schr"odinger Bridges over Compact Connected Lie Groups](https://mlsanity.com/paper/2609.13758/) (2026)
- [Cost-augmented Schr"odinger bridges on graphs are exactly solvable: a Feynman-Kac tilt replaces learned control](https://mlsanity.com/paper/2610.02195/) (2026)

## Abstract (arXiv)

Schr"odinger bridge is a diffusion process that steers a given distribution to another in a prescribed time while minimizing the effort to do so. It can be seen as the stochastic dynamical version of the optimal mass transport, and has growing applications in generative diffusion models and stochastic optimal control. blackWe say a Schr"odinger bridge is ``exactly solvable'' if the associated uncontrolled Markov kernel is available in closed form, since then the bridge can be numerically computed using dynamic Sinkhorn recursion for arbitrary endpoint distributions with finite second moments. In this work, we propose a regularized variant of the Schr"odinger bridge with a quadratic state cost-to-go that incentivizes the optimal sample paths to stay close to a nominal level. Unlike the conventional Schr"odinger bridge, the regularization induces a state-dependent rate of killing and creation of probability mass, and its solution requires determining the Markov kernel of a reaction-diffusion partial differential equation. We derive this Markov kernel in closed form, blackshowing that the regularized Schr"odinger bridge is exactly solvable, even for non-Gaussian endpoints. This advances the state-of-the-art because closed form Markov kernel for the regularized Schr"odinger bridge is available in existing literature only for Gaussian endpoints. Our solution recovers the heat kernel in the vanishing regularization (i.e., diffusion without reaction) limit, thereby recovering the solution of the conventional Schr"odinger bridge blackas a special case. We deduce properties of the new kernel and explain its connections with certain exactly solvable models in quantum mechanics.