Bridging Schrödinger and Bass

We introduce a novel  optimal coupling problem on path space that interpolates between the Schrödinger bridge and the Bass martingale transport. This yields a diffusion process matching prescribed marginal laws, with both controlled drift and volatility adapted to data. We establish a duality and characterize the solution via a triplet $(h,\nu,Y)$ solving a coupled Schrödinger Bass Bridge system:  $h$ solves a backward Kolmogorov equation, $\nu$ a Fokker-Planck equation, and $Y$ is a monotone transport map. The resulting process is a stretched Brownian motion under an entropic change of measure. This framework provides a principled foundation for data-driven diffusion models with generative applications.