Marginal flows of non-entropic weak Schrödinger bridges
Camilo Hernández, Ludovic Tangpi
TL;DR
This work develops a dynamic, divergence-regularized OT framework on path space with weak terminal constraints, replacing the entropy penalty by a general convex divergence ${\cal I}_{\ell}$ and using a weak transport cost ${\cal W}_c$ to enforce terminal matching. It proves well-posedness and convex duality, and provides an explicit density representation for the primal optimizer with respect to a reference diffusion, linked through dual potentials via a generalized Schrödinger system. In the Schrödinger regime ($C=0$), the authors derive a formula for the time-marginals that unifies entropic and non-entropic divergences (including $\chi^2$ and Tsallis-type divergences) through forward–backward stochastic control and time-reversal arguments. The framework accommodates a variety of weak terminal costs, with concrete examples illustrating how different divergences and weak costs shape the optimal plan and the evolution of marginals. Overall, the paper blends stochastic control, convex duality, and time-reversal techniques to advance a flexible, computationally tractable theory of dynamic, divergence-regularized transport with weak targets.
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.