Marginal flows of non-entropic weak Schrödinger bridges

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 of the problem, together with explicit structural characterizations of primal and dual optimizers. Specifically, the optimal path measure is shown to admit an explicit density relative to a reference diffusion, generalizing the classical Schrödinger system. For the pure Schrödinger case, i.e., when the transport cost is zero, 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-divergence.