Poincaré Inequalities and Convergence to Equilibrium with Variable Curvature Bounds
summary
Shows how a uniform curvature lower bound is replaced by a variable or averaged curvature to get quantitative Poincaré constants and convergence rates: proves a pathwise gradient coupling bound (expectation of exp minus integrated curvature), uses transport–information inequalities to turn quantitative ergodic deviations into rates, and derives a self‑improving Bakry‑Émery estimate strictly better than 1/inf curvature except for Gaussian product cases. Gives W1 and L2 decay when ergodic control holds, and notes 1D counterexamples where average curvature alone fails.
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