A characterization of transportation-information inequalities for Markov processes in terms of dimension-free concentration

TL;DR

This paper characterizes the quadratic transportation-information inequality $W_2I$ for Markov processes using a dimension-free concentration property, extending known results and introducing a new Laplace-type principle for Feynman-Kac semigroups.

Contribution

It provides a novel characterization of $W_2I$ in terms of concentration, paralleling Gozlan's $W_2H$ characterization, and introduces a new Laplace-type principle for operator norms.

Findings

Characterization of $W_2I$ via dimension-free concentration.

Introduction of a new Laplace-type principle for Feynman-Kac semigroups.

Demonstration of a convex-analytic tensorization principle.

Abstract

Inequalities between transportation costs and Fisher information are known to characterize certain concentration properties of Markov processes around their invariant measures. This note provides a new characterization of the quadratic transportation-information inequality $W_2I$ in terms of a dimension-free concentration property for i.i.d. (conditionally on the initial positions) copies of the underlying Markov process. This parallels Gozlan's characterization of the quadratic transportation-entropy inequality $W_2H$. The proof is based on a new Laplace-type principle for the operator norms of Feynman-Kac semigroups, which is of independent interest. Lastly, we illustrate how both our theorem and (a form of) Gozlan's are instances of a general convex-analytic tensorization principle.