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On the Hopf-Cole Transform for Control-affine Schrödinger Bridge

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Abstract

The purpose of this note is to clarify the importance of the relation ggσσ\boldsymbol{gg}^{\top}\propto \boldsymbol{\sigma\sigma}^{\top} in solving control-affine Schrödinger bridge problems via the Hopf-Cole transform, where g,σ\boldsymbol{g},\boldsymbol{\sigma} are the control and noise coefficients, respectively. We show that the Hopf-Cole transform applied to the conditions of optimality for generic control-affine Schrödinger bridge problems, i.e., without the assumption ggσσ\boldsymbol{gg}^{\top}\propto\boldsymbol{\sigma\sigma}^{\top}, 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 ggσσ\boldsymbol{gg}^{\top}\propto\boldsymbol{\sigma\sigma}^{\top}, 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 Schrödinger bridge requires further algorithmic development, possibly generalizing the dynamic Sinkhorn recursion or otherwise.

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@article{teter2025_2503.17640,
  title={ On the Hopf-Cole Transform for Control-affine Schrödinger Bridge },
  author={ Alexis Teter and Abhishek Halder },
  journal={arXiv preprint arXiv:2503.17640},
  year={ 2025 }
}
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