# Hopf-Cole transformation via generalized Schr"odinger bridge problem

arXiv: https://arxiv.org/abs/1901.09051 · HTML: https://mlsanity.com/paper/1901.09051/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jan 2019.

## Built on by (most cited first, 8 of 8)

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- [A Computational Framework for Solving Wasserstein Lagrangian Flows](https://mlsanity.com/paper/2310.10649/) (2023)
- [Maximum Likelihood Training of Implicit Nonlinear Diffusion Models](https://mlsanity.com/paper/2205.13699/) (2022)
- [Learning stochastic dynamics from snapshots through regularized unbalanced optimal transport](https://mlsanity.com/paper/2410.00844/) (2024)
- [HJ-sampler: A Bayesian sampler for inverse problems of a stochastic process by leveraging Hamilton-Jacobi PDEs and score-based generative models](https://mlsanity.com/paper/2409.09614/) (2024)
- [Stochastic Control Methods for Optimization](https://mlsanity.com/paper/2601.01248/) (2026)
- [Nonlocal Mean Field Schr"odinger Bridge with Learned Interactions](https://mlsanity.com/paper/2606.04265/) (2026)
- [Contact Wasserstein Geodesics for Non-Conservative Schr"odinger Bridges](https://mlsanity.com/paper/2511.06856/) (2025)

## Abstract (arXiv)

We study generalized Hopf-Cole transformations motivated by the Schr"odinger bridge problem, which can be seen as a boundary value Hamiltonian system on the Wasserstein space. We prove that generalized Hopf-Cole transformations are symplectic submersions in the Wasserstein symplectic geometry. Many examples, including a Hopf-Cole transformation for the shallow water equations, are given. Based on this transformation, energy splitting inequalities are provided.