Curvature, Mixing, and Entropic Interpolation
summary
Shows that a positive coarse Ricci curvature (one-step contracting coupling with rate κ) implies a quantitative W1-entropy transport inequality (transport cost ≤ const·Ent, with the constant behaving like ~1/κ as κ→0). Proof builds an entropic interpolation: the entropy-minimizing path measure is a time-inhomogeneous Markov chain, decomposes entropy by the chain rule, and is coupled to the stationary random walk using a total variation + contraction coupling; the argument uses Pinsker-type bounds to convert entropy spent into transport cost.
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