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Discrete versions of the Li-Yau gradient estimate. (English) Zbl 1476.35289

Summary: We study positive solutions to the heat equation on graphs. We prove variants of the Li-Yau gradient estimate and the differential Harnack inequality. For some graphs, we can show the estimates to be sharp. We establish new computation rules for differential operators on discrete spaces and introduce a relaxation function that governs the time dependency in the differential Harnack estimate.

MSC:

35R02 PDEs on graphs and networks (ramified or polygonal spaces)
35B45 A priori estimates in context of PDEs
35K05 Heat equation
35K10 Second-order parabolic equations
05C10 Planar graphs; geometric and topological aspects of graph theory
05C81 Random walks on graphs
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[1] D. BAKRY and M.ÉMERY, Diffusions hypercontractives, In: “Séminaire de Probabilités”, XIX, 1983/84, Lecture Notes in Math., Vol. 1123, 1985, 177-206.
[2] F. BAUER, P. HORN, Y. LIN, G. LIPPNER, D. MANGOUBI and S.-T. YAU, Li-Yau in-equality on graphs, J. Differential Geom. 99 (2015), 359-405. · Zbl 1323.35189
[3] A.-I. BONCIOCAT and K.-T. STURM, Mass transportation and rough curvature bounds for discrete spaces, J. Funct. Anal. 256 (2009), 2944-2966. · Zbl 1184.28015
[4] D. CUSHING, S. LIU and N. PEYERIMHOFF, Bakry-Émery curvature functions of graphs, Canad. J. Math. 72 (2020), 89-143. · Zbl 1430.05019
[5] F. R. K. CHUNG and S.-T. YAU, Logarithmic Harnack inequalities, Math. Res. Lett. 3 (1996), 793-812. · Zbl 0880.58026
[6] M. ERBAR and J. MAAS, Ricci curvature of finite Markov chains via convexity of the entropy, Arch. Ration. Mech. Anal. 206 (2012), 997-1038. · Zbl 1256.53028
[7] M. ERBAR, J. MAAS and P. TETALI, Discrete Ricci curvature bounds for Bernoulli-Laplace and random transposition models, Ann. Fac. Sci. Toulouse Math. (6) 24 (2016), 781-800. · Zbl 1333.60088
[8] B. HUA, J. JOST and S. LIU, Geometric analysis aspects of infinite semiplanar graphs with non-negative curvature, J. Reine Angew. Math. 700 (2015), 1-36. · Zbl 1308.05033
[9] P. HORN, Y. LIN, S. LIU and S.-T. YAU, Volume doubling, Poincaré inequality and Gaussian heat kernel estimate for non-negatively curved graphs, J. Reine Angew. Math. 757 (2019), 89-130. · Zbl 1432.35213
[10] J. JOST and S. LIU, Ollivier’s Ricci curvature, local clustering and curvature-dimension inequalities on graphs, Discrete Comput. Geom. 51 (2014), 300-322. · Zbl 1294.05061
[11] B. KLARTAG, G. KOZMA, P. RALLI and P. TETALI, Discrete curvature and Abelian groups, Canad. J. Math. 68 (2016), 655-674. · Zbl 1341.53068
[12] S. LAKZIAN and Z. MCGUIRK, A global Poincaré inequality on graphs via a conical curvature-dimension condition, Geom. Metr. Spaces 6 (2018), 32-47. · Zbl 1388.05183
[13] P. LI and S.-T. YAU, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), 153-201.
[14] Y. LIN and S.-T. YAU, Ricci curvature and eigenvalue estimate on locally finite graphs, Math. Res. Lett. 17 (2010), 343-356. · Zbl 1232.31003
[15] J. MAAS, Gradient flows of the entropy for finite Markov chains, J. Funct. Anal. 261 (2011), 2250-2292. · Zbl 1237.60058
[16] A. MIELKE, Geodesic convexity of the relative entropy in reversible Markov chains, Calc. Var. Partial Differential Equations 48 (2013), 1-31. · Zbl 1282.60072
[17] F. MÜNCH, Li-Yau inequality on finite graphs via non-linear curvature dimension condi-tions, J. Math. Pures Appl. (9) 120 (2018), 130-164. · Zbl 1400.05219
[18] F. MÜNCH, Remarks on curvature dimension conditions on graphs, Calc. Var. Partial Dif-ferential Equations 56 (2017), Art. 11, 8 pp.
[19] Y. OLLIVIER, Ricci curvature of Markov chains on metric spaces, J. Funct. Anal. 256 (2009), 810-864. · Zbl 1181.53015
[20] Y. OLLIVIER, A survey of Ricci curvature for metric spaces and Markov chains, In: “Prob-abilistic Approach to Geometry”, Adv. Stud. Pure Math., Vol. 57, Math. Soc. Japan, Tokyo, 2010, 343-381. · Zbl 1204.53035
[21] R. ZACHER, A weak Harnack inequality for fractional evolution equations with discontin-uous coefficients, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12 (2013), 903-940. Institut für Angewandte Analysis Universität Ulm Helmholtzstrasse 18 · Zbl 1285.35124
[22] D-33501 Bielefeld, Germany moritz.kassmann@uni-bielefeld.de
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