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

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.