On a $T_1$ Transport inequality for the adapted Wasserstein distance

TL;DR

This paper extends the classical $T_1$ transport-entropy inequality to discrete-time stochastic processes by establishing an adapted $T_1$ inequality that relates the adapted Wasserstein distance to relative entropy, preserving temporal structure.

Contribution

It introduces an adapted $T_1$ inequality for stochastic processes, generalizing the classical inequality while maintaining the same moment assumptions.

Findings

Established the adapted $T_1$ inequality under classical moment conditions.

Extended the Bolley--Villani inequality to the adapted setting.

Provided a framework linking adapted Wasserstein distance and entropy for processes.

Abstract

The $L^1$ transport-entropy inequality (or $T_1$ inequality), which bounds the $1$-Wasserstein distance in terms of the relative entropy, is known to characterize Gaussian concentration. To extend the $T_1$ inequality to laws of discrete-time processes while preserving their temporal structure, we investigate the adapted $T_1$ inequality which relates the $1$-adapted Wasserstein distance to the relative entropy. Building on the Bolley--Villani inequality, we establish the adapted $T_1$ inequality under the same moment assumption as the classical $T_1$ inequality.