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

Daniel Lacker; Lane Chun Yeung

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arXiv · 2012.02304

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

Daniel Lacker·Lane Chun Yeung

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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.

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Daniel Lacker, Lane Chun Yeung. 2020-12-03. A characterization of transportation-information inequalities for Markov processes in terms of dimension-free concentration. https://arxiv.org/abs/2012.02304

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