Transportation-information inequalities for Markov processes

In this paper, one investigates the transportation-information TcI inequalities: α(Tc(ν, μ)) ≤ I (ν|μ) for all probability measures ν on a metric space $${(\mathcal{X}, d)}$$, where μ is a given probability measure, Tc(ν, μ) is the transportation cost from ν to μ with respect to the cost function c(x, y) on $${\mathcal{X}^2}$$, I(ν|μ) is the Fisher–Donsker–Varadhan information of ν with respect to μ and α : [0, ∞) → [0, ∞] is a left continuous increasing function. Using large deviation techniques, it is shown that TcI is equivalent to some concentration inequality for the occupation measure of a μ-reversible ergodic Markov process related to I(·|μ). The tensorization property of TcI and comparisons of TcI with Poincaré and log-Sobolev inequalities are investigated. Several easy-to-check sufficient conditions are provided for special important cases of TcI and several examples are worked out.