# On the Hopf-Cole Transform for Control-affine Schr"odinger Bridge

arXiv: https://arxiv.org/abs/2503.17640 · HTML: https://mlsanity.com/paper/2503.17640/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 60 references that are in the map.
- First posted on arXiv 22 Mar 2025.

## Builds on (most cited first, 21 of 21)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint](https://mlsanity.com/paper/1412.4430/) (2014)
- [Wasserstein Proximal Algorithms for the Schr"odinger Bridge Problem: Density Control with Nonlinear Drift](https://mlsanity.com/paper/1912.01244/) (2019)
- [An Optimal Transport approach for the Schr"odinger bridge problem and convergence of Sinkhorn algorithm](https://mlsanity.com/paper/1911.06850/) (2019)
- [Entropic and displacement interpolation: a computational approach using the Hilbert metric](https://mlsanity.com/paper/1506.04255/) (2015)
- [Reflected Schr"odinger Bridge: Density Control with Path Constraints](https://mlsanity.com/paper/2003.13895/) (2020)
- [The most likely evolution of diffusing and vanishing particles: Schrodinger Bridges with unbalanced marginals](https://mlsanity.com/paper/2108.02879/) (2021)
- [Schr"odinger Bridge with Quadratic State Cost is Exactly Solvable](https://mlsanity.com/paper/2406.00503/) (2024)
- [Finite Horizon Density Steering for Multi-input State Feedback Linearizable Systems](https://mlsanity.com/paper/1909.12511/) (2019)
- [Harmonic Path Integral Diffusion](https://mlsanity.com/paper/2409.15166/) (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)
- [Prediction and Optimal Feedback Steering of Probability Density Functions for Safe Automated Driving](https://mlsanity.com/paper/2009.09055/) (2020)
- [A Physics-informed Deep Learning Approach for Minimum Effort Stochastic Control of Colloidal Self-Assembly](https://mlsanity.com/paper/2208.09182/) (2022)
- [Fast cooling for a system of stochastic oscillators](https://mlsanity.com/paper/1411.1323/) (2014)
- [Stochastic Uncertainty Propagation in Power System Dynamics using Measure-valued Proximal Recursions](https://mlsanity.com/paper/2108.13405/) (2021)
- [Weyl Calculus and Exactly Solvable Schr"odinger Bridges with Quadratic State Cost](https://mlsanity.com/paper/2407.15245/) (2024)
- [A geometric perspective on regularized optimal transport](https://mlsanity.com/paper/1703.10243/) (2017)
- [Linear PDEs and eigenvalue problems corresponding to ergodic stochastic optimization problems on compact manifolds](https://mlsanity.com/paper/1303.0126/) (2013)

## Built on by (most cited first, 5 of 5)

- [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)
- [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)
- [Schr"odinger Bridges over Kinetic Swarming Models](https://mlsanity.com/paper/2608.25281/) (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)

## Abstract (arXiv)

The purpose of this note is to clarify the importance of the relation boldsymbolggtoppropto boldsymbolσσtop in solving control-affine Schr"odinger bridge problems via the Hopf-Cole transform, where boldsymbolg,boldsymbolσ are the control and noise coefficients, respectively. We show that the Hopf-Cole transform applied to the conditions of optimality for generic control-affine Schr"odinger bridge problems, i.e., without the assumption boldsymbolggtopproptoboldsymbolσσtop, gives a pair of forward-backward PDEs that are neither linear nor equation-level decoupled. We explain how the resulting PDEs can be interpreted as nonlinear forward-backward advection-diffusion-reaction equations, where the nonlinearity stem from additional drift and reaction terms involving the gradient of the log-likelihood a.k.a. the score. These additional drift and reaction vanish when boldsymbolggtopproptoboldsymbolσσtop, and the resulting boundary-coupled system of linear PDEs can then be solved by dynamic Sinkhorn recursions. A key takeaway of our work is that the numerical solution of the generic control-affine Schr"odinger bridge requires further algorithmic development, possibly generalizing the dynamic Sinkhorn recursion or otherwise.