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Hopf-Cole transformation via generalized Schr"odinger bridge problem

arXiv 1901.09051 ↗First posted 25 Jan 2019math.DS math.AP math.OCcited by 8

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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.

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

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 2; 2023: 1; 2024: 2; 2025: 1; 2026: 2.

8 papers.

  1. 2022An optimal control perspective on diffusion-based generative modelingcited by 147
  2. 2023A Computational Framework for Solving Wasserstein Lagrangian Flowscited by 46
  3. 2022Maximum Likelihood Training of Implicit Nonlinear Diffusion Modelscited by 42
  4. 2024Learning stochastic dynamics from snapshots through regularized unbalanced optimal transportcited by 33
  5. 2024HJ-sampler: A Bayesian sampler for inverse problems of a stochastic process by leveraging Hamilton-Jacobi PDEs and score-based generative modelscited by 2
  6. 2026Stochastic Control Methods for Optimizationcited by 1
  7. 2026Nonlocal Mean Field Schr"odinger Bridge with Learned Interactionscited by 1
  8. 2025Contact Wasserstein Geodesics for Non-Conservative Schr"odinger Bridgescited by 0