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

TL;DR

This work investigates the Hopf–Cole transform for control-affine Schrödinger bridge problems and shows that, without identical input and noise channels, the transform yields a boundary-coupled pair of nonlinear forward–backward PDEs. The nonlinearities arise from additional drift and reaction terms $f_{\varphi}$ and $q_{\varphi}$, which depend on the gradient of the log-density through $f_{\varphi}=(\lambda gg^T-\Sigma)\nabla_x\log\varphi$ and $q_{\varphi}=\tfrac{1}{2}(\nabla_x\log\varphi)^T(\lambda gg^T-\Sigma)\nabla_x\log\varphi$, and these terms vanish when $gg^T$ is proportional to $\Sigma$, yielding decoupled linear PDEs solvable by dynamic Sinkhorn recursions. The paper thus identifies a fundamental limitation of the Hopf–Cole approach for generic problems and motivates developing new algorithms beyond Sinkhorn for the general case. The findings illuminate the role of channel mismatch in shaping the forward–backward dynamics and connect to interpretations via Doob transforms, current velocity, and Fisher-type terms, with potential implications for stochastic control and optimal transport methods.

Abstract

The purpose of this note is to clarify the importance of the relation $\boldsymbol{gg}^{\top}\propto \boldsymbol{σσ}^{\top}$ in solving control-affine Schrödinger bridge problems via the Hopf-Cole transform, where $\boldsymbol{g},\boldsymbolσ$ are the control and noise coefficients, respectively. We show that the Hopf-Cole transform applied to the conditions of optimality for generic control-affine Schrödinger bridge problems, i.e., without the assumption $\boldsymbol{gg}^{\top}\propto\boldsymbol{σσ}^{\top}$, 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 $\boldsymbol{gg}^{\top}\propto\boldsymbol{σσ}^{\top}$, 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.

Figures (1)

• Figure 1: When $m=p=1$, the vectors $\bm{g},\bm{\sigma}$ are shown as gray arrows. For fixed $\lambda>0$ and arbitrary nonzero vector $\bm{v}$, the dashed arrows are the projections $\lambda\bm{gg}^{\top}\bm{v}$ and $\bm{\sigma\sigma}^{\top}\bm{v}$ onto the span of $\bm{g}$ and $\bm{\sigma}$, respectively. When these projections become equal, their difference, shown in red, becomes zero.

Theorems & Definitions (12)

• Theorem 1: Coupled Nonlinear PDEs with linear boundary conditions
• proof
• Remark 1
• Remark 2
• Lemma 1: Weighted Laplacian of a product
• proof
• Theorem 2: Boundary-coupled nonlinear PDEs
• proof
• Remark 3
• Corollary 1: Boundary-coupled linear PDEs