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Heat Kernel and Monotonicity Inequalities on the Graph

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Abstract

Let \(G=(V,E)\) be a connected locally finite graph with the \(\mathrm {CD}^{\psi }\) condition for some \(\mathrm {C}^1\), concave function \(\psi : (0,+\infty )\rightarrow \mathbf {R}\). In this paper, based on the gradient estimate and Harnack inequality for the positive solution to the heat equation on G established by Münch, we get the heat kernel estimate. We also derive a logarithmic inequality of the heat kernel by an observation of the short time behavior of the heat kernel. We also define the Fisher information, the Shannon entropy and the \(\mathcal {W}\)-functional, and establish monotone inequalities for these functionals along the heat equation on G.

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References

  1. Bauer, F., Horn, P., Lin, Y., Lippner, G., Mangoubi, D., Yau, S.T.: Li-Yau inequality on graphs. J. Differ. Geom. 99, 359–405 (2015)

    MathSciNet  MATH  Google Scholar 

  2. Cao, H.D., Zhu, X.P.: A complete proof of the Poincaré and geometrization conjectures-Application of the Hamilton–Perelman theory of the Ricci flow. Asian J. Math. 10(2), 165–492 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Colding, T.H.: New monotonicity formulas for Ricci curvature and applications I. Acta Math. 209, 229–263 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Colding, T.H., Minicozzi II, W.P.: Monotonicity-analytic and geometric implications. arXiv:1205.6768v2

  5. Gong, C., Lin, Y., Liu, S., Yau, S.T.: Li-Yau inequality for unbounded Laplacian on graphs. Adv. Math. 357(1), 106822 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  6. Horn, P., Lin, Y., Liu, S., Yau, S.T.: Volume doubling, Poincaré inequality and Gaussian heat kernel estimate for non-negatively curved graphs. J. Reine Angew. Math. (Crelles Journal) 757, 89–130 (2019)

    Article  MATH  Google Scholar 

  7. Hua, B.B., Lin, Y.: Stochastic completeness for graphs with curvature dimension conditions. Adv. Math. 306(14), 279–302 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kleiner, B., Lott, J.: Notes on Perelman’s papers. Geom. Topol. 12(5), 2587–2855 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, P., Yau, S.T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156(3–4), 153–201 (1986)

    Article  MathSciNet  Google Scholar 

  10. Liu, S.: Buser’s inequality on infinite graphs. J. Math. Anal. Appl. 475(2), 1416–1426 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  11. Liu, S.P., Münch, F., Peyerimhoff, N.: Bakry–Émery curvature and diameter bounds on graphs. Calc. Var. Partial Differ. Equ. 57, 67 (2018)

    Article  MATH  Google Scholar 

  12. Lv, Y., Wang, L.F.: Gradient estimates on connected graphs with the \(CD\psi (m, K)\) condition. Ann. Mater. Pura Appl. 198, 2207–2225 (2019)

    Article  MATH  Google Scholar 

  13. Morgan, J., Tian, G.: Ricci Flow and the Poincaré Conjecture. American Mathematical Society, Providence (2007)

    MATH  Google Scholar 

  14. Münch, F.: Remarks on curvature dimension conditions on graphs. Calc. Var. Partial Differ. Equ. 56, 11 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Münch, F.: Li-Yau inequality on finite graphs via non-linear curvature dimension conditions. J. Math. Pures Appl. 120, 130–164 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ni, L.: The entropy formula for linear heat equation. J. Geom. Anal. 14(1), 87–100 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ni, L.: Addenda to ‘The entropy formula for linear heat equation’. J. Geom. Anal. 14(2), 329–335 (2004)

    Article  MATH  Google Scholar 

  18. Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv: math.DG/0211159

  19. Vladimir, I.A.: Ordinary differential equations. Springer, 1992-4. ISBN: 9780387548135

  20. Wang, L.F.: A new entropy formula for the linear heat equation. Arch. Math. 96, 473–481 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wang, L.F.: Monotonicity formulas via the Bakry-Émery curvature. Nonlinear Anal. 89, 230–241 (2013)

  22. Wang, L.F., Zhang, Z.Y.: Gradient estimates for the heat equation on graphs. Commun. Anal. Geom. 27(4), 965–989 (2019)

    Article  MathSciNet  Google Scholar 

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  1. School of Science, Nantong University, Nantong, 226007, Jiangsu, China

    Lin Feng Wang

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  1. Lin Feng Wang
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Correspondence to Lin Feng Wang.

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Wang, L.F. Heat Kernel and Monotonicity Inequalities on the Graph. J Geom Anal 33, 38 (2023). https://doi.org/10.1007/s12220-022-01093-w

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  • DOI: https://doi.org/10.1007/s12220-022-01093-w

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