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

TL;DR

This paper derives a PDE-based formulation of the Schr"odinger--Bass Bridge problem, unifying classical Schr"odinger bridges and Bass martingale transport through explicit solutions and heat equation techniques.

Contribution

It provides the first direct PDE derivation of the SBB system in one dimension, linking optimal coupling problems with heat equations and Legendre transforms, and extends the framework to a semimartingale setting.

Findings

Explicit solution of the SBB system via Legendre transforms and heat equation

Unification of Schr"odinger bridge and Bass martingale transport models

Recovery of Sinkhorn algorithm and Bass construction as limits

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.