Marginal flows of non-entropic weak Schr\"odinger bridges

TL;DR

This paper develops a new dynamic framework for divergence-regularized optimal transport with weak terminal constraints, generalizing Schrödinger bridges beyond entropic cases.

Contribution

It introduces a convex divergence-based formulation, establishes well-posedness, duality, and explicit optimizer characterizations, extending classical Schrödinger systems.

Findings

Explicit density formulas for optimal path measures.

Duality and structural characterizations of solutions.

New descriptions for non-entropic divergences like $oldsymbol{ extchi^2}$-divergence.

Abstract

This paper introduces a dynamic formulation of divergence-regularized optimal transport with weak targets on the path space. In our formulation, the classical relative entropy penalty is replaced by a general convex divergence, and terminal constraints are imposed in a weak sense. We establish well-posedness and a convex dual formulation, together with a dual existence result and explicit structural characterizations of primal and dual optimizers. Specifically, the optimal path measure admits an explicit density relative to a reference diffusion, generalizing the classical Schr{\"o}dinger system. In the case of zero transport cost, which corresponds to a non-entropic dynamic Schr{\"o}dinger problem, we further characterize the flow of time marginals of the optimal bridge, recovering known results in the entropic setting and providing new descriptions for non-entropic divergences, including the $\chi^2$-divergence