Home A characterization of the rate of change ofФ-entropy via an integral form curvature-dimension condition
Article
Licensed
Unlicensed Requires Authentication

A characterization of the rate of change ofФ-entropy via an integral form curvature-dimension condition

  • Dejun Luo EMAIL logo
Published/Copyright: June 28, 2016
Become an author with De Gruyter Brill
Author Information Explore this Subject
Advances in Geometry
From the journal Advances in Geometry Volume 16 Issue 3

Abstract

Let M be a compact Riemannian manifold without boundary and V : M → ℝ a smooth function. Denote by Pt and dμ = eV dx the semigroup and symmetric measure of the second order differential operator L = ∆ + ∇V · ∇. For some suitable convex function Ф :𝓙 → ℝ defined on an interval 𝓙, we consider the Ф-entropy of Ptf (with respect to μ) for any f ∈ C∞(M, 𝓙). We show that an integral form curvature-dimension condition is equivalent to an estimate on the rate of change of the Ф-entropy. We also generalize this result to bounded smooth domains of a complete Riemannian manifold.

Keywords: Heat equation; Ф-entropy; curvature-dimension condition; second fundamental form; reflecting difusion semigroup
MSC 2010: 58J35

Communicated by: P. Eberlein

Acknowledgements

The author is very grateful to Professors Liming Wu for helpful discussions, and to Bin Qian for his suggestion of proving the necessity part of Theorem 1.1 by using the identity (2.1), which simplifies the argument.

Funding: Partly supported by the Key Laboratory of Random Complex Structures and Data Sciences, Chinese Academy of Sciences (2008DP173182), the Natural Science Foundation of China (11371099) and the Academy of Mathematics and Systems Science (Y129161ZZ1).

References

[1] L. Ambrosio, N. Gigli, G. Savaré, Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds. Ann. Probab. 43(2015), 339-404. MR3298475 Zbl 1307.4904410.1214/14-AOP907Search in Google Scholar

[2] B. Andrews, J. Clutterbuck, Proof of the fundamental gap conjecture. J.Amer. Math. Soc. 24(2011), 899-916. MR2784332 (2012d:35051) Zbl 1222.3513010.1090/S0894-0347-2011-00699-1Search in Google Scholar

[3] C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto, G. Scheffer, Surles inégalités de Sobolev logarithmiques, volume 10 of Panoramas et Synthèses. Société Mathématique de France, Paris 2000. MR1845806 (2002g:46132) Zbl 0982.46026Search in Google Scholar

[4] D. Bakry, Functional inequalities for Markov semigroups. In: Probability measures on groups: recent directions and trends, 91-147, Tata Inst. Fund. Res., Mumbai 2006. MR2213477 (2007g:60086) Zbl 1148.60057Search in Google Scholar

[5] D. Bakry, M. Emery, Diffusions hypercontractives. In: Séminaire de probabilités, XIX, 1983/84, volume 1123 of Lecture Notes in Math., 177-206, Springer 1985. MR889476 (88j:60131) Zbl 0561.6008010.1007/BFb0075847Search in Google Scholar

[6] F. Baudoin, N. Garofalo, Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries. J. Eur. Math. Soc., to appear.10.4171/JEMS/663Search in Google Scholar

[7] D. Chafaï, Entropies, convexity, and functional inequalities: on Ф-entropies and Ф-Sobolev inequalities. J. Math. Kyoto Univ. 44(2004), 325-363. MR2081075 (2005e:60170) Zbl 1079.2600910.1215/kjm/1250283556Search in Google Scholar

[8] F.-Z. Gong, H. Li, D. Luo, A probabilistic proof of the fundamental gap conjecture via the coupling by reflection. Potential Anal. 44(2016), 423-442. MR3489850 Zbl 0658053910.1007/s11118-015-9476-3Search in Google Scholar

[9] X.-D. Li, Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry-Emery Ricci curvature. Math. Ann. 353(2012), 403-437. MR2915542 Zbl 1280.5801610.1007/s00208-011-0691-ySearch in Google Scholar

[10] A. P. C. Lim, D. Luo, Asymptotic estimates on the time derivative of entropy on a Riemannian manifold. Adv. Geom. 13(2013), 97-115. MR3011536 Zbl 1267.5801710.1515/advgeom-2012-0022Search in Google Scholar

[11] L. Ni, The entropy formula for linear heat equation. J. Geom. Anal. 14(2004), 87-100. MR2030576 (2004m:53118a) Zbl 1044.5803010.1007/BF02921867Search in Google Scholar

[12] L. Ni, Addenda to: “The entropy formula for linear heat equation” J. Geom. Anal. 14(2 0 0 4), 369-374. MR2051693 (2004m:53118b) Zbl 1062.5802810.1007/BF02922078Search in Google Scholar

[13] B. Qian, Asymptotic estimates on the time derivative of P-entropy on Riemannian manifolds. Acta Math. Sin. (Engl. Ser.)30(2014), 609-618. MR3176915 Zbl 1304.6008510.1007/s10114-014-2720-8Search in Google Scholar

[14] F.-Y. Wang, Logarithmic Sobolev inequalities: conditions and counterexamples. J. Operator Theory46(2001), 183-197. MR1862186 (2003g:58060) Zbl 0993.58019Search in Google Scholar

[15] F.-Y. Wang, Equivalent semigroup properties for the curvature-dimension condition. Bull. Sci. Math. 135(2011), 803-815. MR2838102 (2012j:58052) Zbl 1230.6008310.1016/j.bulsci.2011.07.005Search in Google Scholar

[16] F.-Y. Wang, Analysis for diffusion processes on Riemannian manifolds. World Scientific Publ. Co., Hackensack, NJ 2014. MR3154951 Zbl 1296.58001Search in Google Scholar

[17] F.-Y. Wang, P-entropy inequality and application for SDEs with jumps. J. Math. Anal. Appl. 418(2014), 861-873. MR3206685 Zbl 0639053110.1016/j.jmaa.2014.03.086Search in Google Scholar

[18] F.-Y. Wang, Generalized curvature condition for subelliptic diffusion processes. arXiv:1202.0778v2 [math.PR]Search in Google Scholar

[19] L. Wu, A new modified logarithmic Sobolev inequality for Poisson point processes and several applications. Probab. Theory Related Fields118(2000), 427-438. MR1800540 (2002f:60109) Zbl 0970.6009310.1007/PL00008749Search in Google Scholar

[20] L. Yuan, Prescribing curvature problem of Bakry-Emery Ricci tensor. Sci. China Math. 56(2013), 1935-1944. MR3090864 Zbl 1280.5304310.1007/s11425-013-4595-zSearch in Google Scholar

Received: 2014-8-2
Published Online: 2016-6-28
Published in Print: 2016-7-1

© 2016 by Walter de Gruyter Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Research Article
  3. Framed curves in the Euclidean space
  4. Research Article
  5. A characterization of the rate of change ofФ-entropy via an integral form curvature-dimension condition
  6. Research Article
  7. On the density function on moduli spaces of toric 4-manifolds
  8. Research Article
  9. Conformal Ricci solitons and related integrability conditions
  10. Research Article
  11. Inequalities for casorati curvatures of submanifolds in real space forms
  12. Research Article
  13. Inequalities for hyperconvex sets
  14. Research Article
  15. Neighborly inscribed polytopes and delaunay triangulations
  16. Research Article
  17. Cellular homology of real maximal isotropic grassmannians
  18. Research Article
  19. Symmetric subspaces of SE(3)
  20. Research Article
  21. Clifford’s Theorem for graphs
https://doi.org/10.1515/advgeom-2015-0046
Keywords for this article
Heat equation; Ф-entropy; curvature-dimension condition; second fundamental form; reflecting difusion semigroup

Articles in the same Issue

  1. Frontmatter
  2. Research Article
  3. Framed curves in the Euclidean space
  4. Research Article
  5. A characterization of the rate of change ofФ-entropy via an integral form curvature-dimension condition
  6. Research Article
  7. On the density function on moduli spaces of toric 4-manifolds
  8. Research Article
  9. Conformal Ricci solitons and related integrability conditions
  10. Research Article
  11. Inequalities for casorati curvatures of submanifolds in real space forms
  12. Research Article
  13. Inequalities for hyperconvex sets
  14. Research Article
  15. Neighborly inscribed polytopes and delaunay triangulations
  16. Research Article
  17. Cellular homology of real maximal isotropic grassmannians
  18. Research Article
  19. Symmetric subspaces of SE(3)
  20. Research Article
  21. Clifford’s Theorem for graphs
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 27.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/advgeom-2015-0046/html
Scroll to top button