A PDE Derivation of the Schr\"odinger--Bass Bridge

Abstract

This short paper announces the main results of \cite{SBB2026}, where the Schr\"odinger--Bass Bridge (SBB) problem is introduced and studied in full generality. Here we provide a direct PDE derivation of the SBB system in dimension one, showing how the optimal coupling problem that interpolates between the classical Schr\"odinger bridge and the Bass martingale transport can be solved explicitly via Legendre transforms and the heat equation. A key insight is that the optimal SBB process is a Stretched Schr\"odinger Bridge: the composition of a monotone transport map with a Schr\"odinger bridge. This extends the stretched Brownian motion representation of Bass martingales to the semimartingale setting and provides a unified framework that recovers both the Sinkhorn algorithm (in the limit $\beta \to \infty$) and the Bass construction (as $\beta \to 0$). We refer to \cite{SBB2026} for complete proofs, the multidimensional setting, strong duality, dual attainment, and further developments.