Curvature bounds for the cost of coherence in conditional optimal transport

Xiaozhen Wang; Xiang Zhou; Zihan Zhang

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arXiv · 2610.09738

Curvature bounds for the cost of coherence in conditional optimal transport

Xiaozhen Wang·Xiang Zhou·Zihan Zhang

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Abstract

We study the gap between the integrated direction-wise kinetic minima and the minimum expected Dirichlet energy of a joint particle realization of a prescribed two-parameter family of conditional probability measures. Under regular coarea assumptions with compact connected fibres and a star-shaped parameter domain, we derive a curvature lower bound valid for every admissible law of Sobolev maps realizing the prescribed smooth positive conditional measures. Our method combines the weighted-Poisson characterization of local optimal-transport velocities with an exact energy-residual identity, a weak Stokes formula, and explicit transport constructions. The minimum excess vanishes exactly when the Lie-bracket curvature vanishes throughout the parameter domain, whereas at a nonzero-curvature centre it is of order $r^4$ as $r\to0$ on squares of half-side length $r$ with unnormalized area measure. In a two-phase torus model fitted to electricity-demand data, numerical tests illustrate this quartic order for explicit couplings, with radial transport approximately halving the ordered construction's excess on the tested small squares.

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Xiaozhen Wang, Xiang Zhou, Zihan Zhang. 2026-10-07. Curvature bounds for the cost of coherence in conditional optimal transport. https://arxiv.org/abs/2610.09738

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