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- W2912309118 abstract "It is known that the Poincar'e inequality is equivalent to the quadratic transportation-variance inequality (namely $W_2^2(fmu,mu) leqslant C_V mathrm{Var}_mu(f)$), see Jourdain cite{Jourdain} and most recently Ledoux cite{Ledoux18}. We give two alternative proofs to this fact. In particular, we achieve a smaller $C_V$ than before, which equals the double of Poincar'e constant. Applying the same argument leads to more characterizations of the Poincar'e inequality. Our method also yields a by-product as the equivalence between the logarithmic Sobolev inequality and strict contraction of heat flow in Wasserstein space provided that the Bakry-'Emery curvature has a lower bound (here the control constants may depend on the curvature bound). Next, we present a comparison inequality between $W_2^2(fmu,mu)$ and its centralization $W_2^2(f_cmu,mu)$ for $f_c = frac{|sqrt{f} - mu(sqrt{f})|^2}{mathrm{Var}_mu (sqrt{f})}$, which may be viewed as some special counterpart of the Rothaus' lemma for relative entropy. Then it yields some new bound of $W_2^2(fmu,mu)$ associated to the variance of $sqrt{f}$ rather than $f$. As a by-product, we have another proof to derive the quadratic transportation-information inequality from Lyapunov condition, avoiding the Bobkov-Gotze's characterization of the Talagrand's inequality." @default.
- W2912309118 created "2019-02-21" @default.
- W2912309118 creator A5046214153 @default.
- W2912309118 date "2019-02-11" @default.
- W2912309118 modified "2023-10-14" @default.
- W2912309118 title "The Poincar'e inequality and quadratic transportation-variance inequalities" @default.
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- W2912309118 doi "https://doi.org/10.48550/arxiv.1902.04196" @default.
- W2912309118 hasPublicationYear "2019" @default.
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