A weak transport approach to the Schr\"odinger-Bass bridge

Abstract

We study the Schr\"odinger-Bass problem, a one-parameter family of semimartingale optimal transport problems indexed by $\beta>0$, whose limiting regimes interpolate between the classical Schr\"odinger bridge, the Brenier-Strassen problem, and, after rescaling, the martingale Benamou-Brenier (Bass) problem. For each $\beta>0$, we prove that the dynamic Schr\"odinger-Bass problem is equivalent to a static weak optimal transport (WOT) problem with explicit cost $C_{\mathrm{SB}}^\beta$. This yields primal and dual attainment, as well as a structural characterization of the optimal semimartingales. The cost $C_{\mathrm{SB}}^\beta$ is constructed via infimal convolution and deconvolution of the Schr\"odinger cost with the Wasserstein distance. In a broader setting, we show that such infimal convolutions preserve the WOT structure and inherit continuity, coercivity, and stability of both values and optimizers with respect to the marginals. Building on this formulation, we propose a dual ascent algorithm for numerical computation. We establish monotone improvement of the dual objective and convergence of the iteration to the unique optimizer under suitable integrability assumptions.