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

TL;DR

This paper introduces a static weak optimal transport formulation for the Schr"odinger-Bass problem, providing theoretical insights and a numerical Sinkhorn-type algorithm, with analysis of asymptotic regimes.

Contribution

It establishes a static weak optimal transport framework for the Schr"odinger-Bass problem, including explicit cost construction, structural characterization, and convergence analysis.

Findings

Static formulation of Schr"odinger-Bass problem as weak optimal transport.

Development of a Sinkhorn-type algorithm with proven convergence.

Asymptotic analysis showing convergence to classical costs and problems.

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. Our first main result is a static formulation. 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, through the general WOT framework. The cost $C_{\mathrm{SB}}^\beta$ is constructed via an 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 Sinkhorn-type algorithm for numerical computation. We establish monotone improvement of the dual objective and, under suitable integrability assumptions on the marginals, convergence of the iteration to the unique optimizer. We then study the asymptotic regimes $\beta\uparrow\infty$ and $\beta\downarrow0$. We prove that the costs $C_{\mathrm{SB}}^\beta$ converge pointwise to the Schr\"odinger cost and, after natural rescaling, to the Brenier-Strassen and Bass costs. The associated values and optimal solutions are shown to converge to those of the corresponding limiting problems.