Lifted Schrödinger Bridges for Gaussian Mixture Endpoints: Projection Gaps and Path-Space Obstructions

This research abstract introduces a lifted path-space construction to handle stochastic density control between Gaussian-mixture endpoint distributions under Brownian prior dynamics. By augmenting trajectories with source-target component labels, the problem decomposes into Gaussian component-to-component Schrödinger bridges, reducing mixture-level assignment to a finite-dimensional entropic coupling problem. The authors analyze the projection obtained by discarding labels, revealing a nonnegative conditional label-information gap between lifted and direct relative entropy, which indicates path-space obstructions. They derive posterior-averaged Markov drift, prove a kinetic-energy upper bound, and identify conditions under which the projection gap vanishes, supported by numerical illustrations for density and shape control.