Weak Optimal Entropy Transport Problems
Nhan-Phu Chung, Thanh-Son Trinh
TL;DR
The paper develops Weak Optimal Entropy Transport (WOET), a unifying framework that extends both Optimal Entropy Transport (OET) and Weak Optimal Transport (WOT) to include disintegration-based costs and entropy-penalized marginal relaxations. It establishes a Kantorovich-type duality for WOET, framing the problem through dual sets \(\Lambda\) and \(\Lambda_R\) and via a RC-transform representation \(F_1^\circ(R_C\varphi)\) and \(F_2^\circ(-\varphi)\); the duality holds under mild assumptions with entropy functions that are superlinear. The work proves existence of minimizers under coercivity and discusses feasibility conditions, illustrating that classical OET, WOT, and martingale transport are recovered as special cases; a MOET variant with martingale constraints is also introduced. Examples including WLET and χ^2-type divergences demonstrate the framework’s breadth and potential applications in probability, optimization, and related PDE contexts. A companion paper further addresses non-superlinear entropy scenarios, highlighting the robustness and extensibility of WOET.
Abstract
In this paper, we introduce weak optimal entropy transport problems that cover both optimal entropy transport problems and weak optimal transport problems introduced by Liero, Mielke, and Savaré [27]; and Gozlan, Roberto, Samson and Tetali [20], respectively. Under some mild assumptions of entropy functionals, we establish a Kantorovich type duality for our weak optimal entropy transport problem. We also introduce martingale optimal entropy transport problems, and express them in terms of duality, homogeneous marginal perspective functionals and homogeneous constraints.