Dynamic characterization of barycentric optimal transport problems and their martingale relaxation

TL;DR

The paper develops a dynamic framework for barycentric weak optimal transport, extending the Benamou-Brenier formulation to weak and martingale settings. It introduces the two-parameter family $\overline{\mathcal{T}}^{\alpha,\beta}$ and proves a dynamic representation $\mathcal{BB}^{\alpha,\beta}$ in terms of controlled diffusions $dX_t = v_t dt + \sigma_t dB_t$, with objective $\mathbb{E}[\int_0^1 \alpha |v_t|^2 - \beta ( \langle B_t, v_t \rangle + \mathrm{Tr}(\sigma_t) ) dt]$. The results connect barycentric OT to convex order via a canonical $\bar{\mu} \preceq_c \nu$ and to stretched Brownian motions / Bass martingales, unifying dynamic OT with martingale OT. These dynamic characterizations illuminate how diffusion and martingale components interact within weak OT and extend the Benamou-Brenier framework beyond classical OT.

Abstract

We extend the Benamou-Brenier formula from classical optimal transport to weak optimal transport and show that the barycentric optimal transport problem studied by Gozlan and Juillet has a dynamic analogue. We also investigate a martingale relaxation of this problem, and relate it to the martingale Benamou-Brenier formula of Backhoff-Veraguas, Beiglböck, Huesmann and Källblad.

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Proposition 3: backhoffveraguas2019martingalebenamoubrenierprobabilisticperspective
• Proposition 4: Gozlan_2020
• Proposition 5: beiglböck2025fundamentaltheoremweakoptimal
• Lemma 6
• proof
• proof : Proof of Theorem \ref{['thm:intro1']}
• Corollary 7
• proof : Proof of Theorem \ref{['thm:intro2']}