A weak transport approach to the Schrödinger-Bass bridge

We study the Schrödinger-Bass problem, a one-parameter family of semimartingale optimal transport problems indexed by $β>0$, whose limiting regimes interpolate between the classical Schrödinger bridge, the Brenier-Strassen problem, and, after rescaling, the martingale Benamou-Brenier (Bass) problem. Our first main result is a static formulation. For each $β>0$, we prove that the dynamic Schrödinger-Bass problem is equivalent to a static weak optimal transport (WOT) problem with explicit cost $C_{\mathrm{SB}}^β$. This yields primal and dual attainment, as well as a structural characterization of the optimal semimartingales, through the general WOT framework. The cost $C_{\mathrm{SB}}^β$ is constructed via an infimal convolution and deconvolution of the Schrödinger 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 $β\uparrow\infty$ and $β\downarrow0$. We prove that the costs $C_{\mathrm{SB}}^β$ converge pointwise to the Schrödinger 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.

Source: arXiv:2604.02312v1 - http://arxiv.org/abs/2604.02312v1 PDF: https://arxiv.org/pdf/2604.02312v1 Original Link: http://arxiv.org/abs/2604.02312v1