[en] For a complete connected Riemannian manifold M let V∊ C^2(M) be such that µ(dx)=exp(-V(x))vol(dx) is a probability measure on M. Taking µ as reference measure, we derive inequalities for probability measures on M linking relative entropy, Fisher information, Stein discrepancy and Wasserstein distance. These inequalities strengthen in particular the famous log-Sobolev and transportation-cost inequality and extend the so-called Entropy/Stein-discrepancy/Information (HSI) inequality established by Ledoux, Nourdin and Peccati (2015) for the standard Gaussian measure on Euclidean space to the setting of Riemannian manifolds.
Disciplines :
Mathematics
Author, co-author :
Cheng, Li-Juan; Zhejiang University of Technology > Department of Applied Mathematics
Thalmaier, Anton ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)
Wang, Feng-Yu; Tianjin University > Center for Applied Mathematics
External co-authors :
yes
Language :
English
Title :
Some inequalities on Riemannian manifolds linking Entropy, Fisher information, Stein discrepancy and Wasserstein distance
Arnaudon, Marc, Plank, Holger, Thalmaier, Anton, A Bismut type formula for the Hessian of heat semigroups. C. R. Math. Acad. Sci. Paris 336:8 (2003), 661–666 MR 1988128.
Chen, Louis H.Y., Goldstein, Larry, Shao, Qi-Man, Normal Approximation by Stein's Method. Probability and Its Applications (New York), 2011, Springer, Heidelberg MR 2732624.
Courtade, Thomas A., Fathi, Max, Pananjady, Ashwin, Existence of Stein kernels under a spectral gap, and discrepancy bounds. Ann. Inst. Henri Poincaré Probab. Stat. 55:2 (2019), 777–790 MR 3949953.
Döbler, Christian, Peccati, Giovanni, The gamma Stein equation and noncentral de Jong theorems. Bernoulli 24:4B (2018), 3384–3421 MR 3788176.
Driver, Bruce K., Thalmaier, Anton, Heat equation derivative formulas for vector bundles. J. Funct. Anal. 183:1 (2001), 42–108 MR 1837533.
Elworthy, K. David, Li, Xue-Mei, Formulae for the derivatives of heat semigroups. J. Funct. Anal. 125:1 (1994), 252–286 MR 1297021.
Li, Xue-Mei, Hessian formulas and estimates for parabolic Schrödinger operators. J. Stoch. Anal., 2(3), 2021 Art. 7, 53. MR 4304478.
Milman, Emanuel, On the role of convexity in isoperimetry, spectral gap and concentration. Invent. Math. 177:1 (2009), 1–43 MR 2507637.
Nourdin, Ivan, Peccati, Giovanni, Normal Approximations with Malliavin Calculus. Cambridge Tracts in Mathematics, vol. 192, 2012, Cambridge University Press, Cambridge From Stein's method to universality. MR 2962301.
Nourdin, Ivan, Peccati, Giovanni, Swan, Yvik, Entropy and the fourth moment phenomenon. J. Funct. Anal. 266:5 (2014), 3170–3207 MR 3158721.
Otto, Felix, Villani, Cédric, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173:2 (2000), 361–400 MR 1760620.
Stein, Charles, A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. II: Probability Theory, Univ. California, Berkeley, Calif., 1970/1971, 1972, 583–602 MR 0402873.
Stein, Charles, Approximate Computation of Expectations. Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol. 7, 1986, Institute of Mathematical Statistics, Hayward, CA MR 882007.
Talagrand, Michel, Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6:3 (1996), 587–600 MR 1392331.
Tashiro, Yoshihiro, Complete Riemannian manifolds and some vector fields. Trans. Am. Math. Soc. 117 (1965), 251–275 MR 174022.
Thalmaier, Anton, On the differentiation of heat semigroups and Poisson integrals. Stoch. Stoch. Rep. 61:3–4 (1997), 297–321 MR 1488139.
Thompson, James, Derivatives of Feynman-Kac semigroups. J. Theor. Probab. 32:2 (2019), 950–973 MR 3959634.
Thompson, James, Approximation of Riemannian measures by Stein's method. arXiv:2001.09910, 2020.
Villani, Cédric, Optimal Transport, Old and New. Grundlehren der Mathematischen Wissenschaften, vol. 338, 2009, Springer.
von Renesse, Max-K., Sturm, Karl-Theodor, Transport inequalities, gradient estimates, entropy, and Ricci curvature. Commun. Pure Appl. Math. 58:7 (2005), 923–940 MR 2142879.
Wang, Feng-Yu, Analysis for Diffusion Processes on Riemannian Manifolds. Advanced Series on Statistical Science & Applied Probability, vol. 18, 2014, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ MR 3154951.
Wang, Feng-Yu, Identifying constant curvature manifolds, Einstein manifolds, and Ricci parallel manifolds. J. Geom. Anal. 29:3 (2019), 2374–2409 MR 3969430.
Wu, Guoqiang, Ye, Rugang, A note on Obata's rigidity theorem. Commun. Math. Stat. 2:3–4 (2014), 231–252 MR 3326231.