On a T₁ Transport inequality for the adapted Wasserstein distance

2025/07/25 by Jonghwa Park, Park, Jonghwa
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Statistical Mechanics and Entropy

paper · pdf · doi:10.48550/arxiv.2507.19215

Abstract

The L¹ transport-entropy inequality (or T₁ inequality), which bounds the 1-Wasserstein distance in terms of the relative entropy, is known to characterize Gaussian concentration. To extend the T₁ inequality to laws of discrete-time processes while preserving their temporal structure, we investigate the adapted T₁ inequality which relates the 1-adapted Wasserstein distance to the relative entropy. Building on the Bolley--Villani inequality, we establish the adapted T₁ inequality under the same moment assumption as the classical T₁ inequality.

Citations

• Pinsker's inequality for adapted total variation
• Estimating causal distances with non-causal ones
• Computational methods for adapted optimal transport
• Designing Universal Causal Deep Learning Models: The Geometric (Hyper)Transformer
• Causal transport in discrete time and applications
• Transportation cost-information inequalities and applications to random dynamical systems and diffusions

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