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On-diagonal lower bounds for heat kernels and Markov chains. (English) Zbl 0920.58064

The authors establish estimates for on-diagonal lower bound for heat kernels on Riemannian manifolds of two kinds: (i) a suplower bound (i.e., a lower bound for \(\sup_{x\in M}p_t(x,x)\) where \(p_t(x,y)\) is the heat kernel on a manifold \(M\)); (ii) a pointwise lower bound (i.e., a lower bound for \(p_t(x,x)\) for a fixed given point \(x\in M\)). There are also two kinds of assumptions about the geometry of the manifold (considered alternatively): (i) an anti-isoperimetric (or anti–Faber-Krahn-type) inequality; (ii) an upper bound for the volume growth function or a doubling volume property.
The authors’ approach to lower bounds of heat kernels is independent of known ones for upper bounds and it works even if the Harnack principle is unknown or false. The approach involves, in particular, random walks on graphs and other probabilistic ideas.

MSC:

58J65 Diffusion processes and stochastic analysis on manifolds
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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[1] G. Alexopoulos, A lower estimate for central probabilities on polycyclic groups , Canad. J. Math. 44 (1992), no. 5, 897-910. · Zbl 0762.31003
[2] 1 D. G. Aronson, Non-negative solutions of linear parabolic equations , Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 607-694. · Zbl 0182.13802
[3] 2 D. G. Aronson, Addendum: “Non-negative solutions of linear parabolic equations” , Ann. Scuola Norm. Sup. Pisa (3) 25 (1971), 221-228. · Zbl 0223.35046
[4] R. Azencott, Behavior of diffusion semi-groups at infinity , Bull. Soc. Math. France 102 (1974), 193-240. · Zbl 0293.60071
[5] I. Benjamini, I. Chavel, and E. A. Feldman, Heat kernel lower bounds on Riemannian manifolds using the old ideas of Nash , Proc. London Math. Soc. (3) 72 (1996), no. 1, 215-240. · Zbl 0853.58098
[6] A. Carey, T. Coulhon, V. Mathai, and J. Phillips, Von Neumann spectra near the mass gap , to appear in Bull. Sci. Math. · Zbl 0911.58034
[7] E. A. Carlen, S. Kusuoka, and D. W. Stroock, Upper bounds for symmetric Markov transition functions , Ann. Inst. H. Poincaré Probab. Statist. 23 (1987), no. 2, suppl., 245-287. · Zbl 0634.60066
[8] T. K. Carne, A transmutation formula for Markov chains , Bull. Sci. Math. (2) 109 (1985), no. 4, 399-405. · Zbl 0584.60078
[9] G. Carron, Inégalités isopérimétriques de Faber-Krahn et conséquences , Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), Sémin. Congr., vol. 1, Soc. Math. France, Paris, 1996, pp. 205-232. · Zbl 0884.58088
[10] J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian , Problems in analysis (Papers dedicated to Salomon Bochner, 1969), Princeton Univ. Press, Princeton, N. J., 1970, pp. 195-199. · Zbl 0212.44903
[11] J. Cheeger and S.-T. Yau, A lower bound for the heat kernel , Comm. Pure Appl. Math. 34 (1981), no. 4, 465-480. · Zbl 0481.35003
[12] T. Coulhon, Inégalités de Gagliardo-Nirenberg pour les semi-groupes d’opérateurs et applications , Potential Anal. 1 (1992), no. 4, 343-353. · Zbl 0768.47018
[13] T. Coulhon, Ultracontractivity and Nash type inequalities , J. Funct. Anal. 141 (1996), no. 2, 510-539. · Zbl 0887.58009
[14] T. Coulhon, Dimensions at infinity for Riemannian manifolds , Potential Anal. 4 (1995), no. 4, 335-344. · Zbl 0847.53022
[15] T. Coulhon and M. Ledoux, Isopérimétrie, décroissance du noyau de la chaleur et transformations de Riesz: un contre-exemple , Ark. Mat. 32 (1994), no. 1, 63-77. · Zbl 0826.53035
[16] T. Coulhon and L. Saloff-Coste, Isopérimétrie pour les groupes et les variétés , Rev. Mat. Iberoamericana 9 (1993), no. 2, 293-314. · Zbl 0782.53066
[17] T. Coulhon and L. Saloff-Coste, Minorations pour les chaînes de Markov unidimensionnelles , Probab. Theory Related Fields 97 (1993), no. 3, 423-431. · Zbl 0792.60063
[18] E. B. Davies, Heat kernels and spectral theory , Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1990. · Zbl 0699.35006
[19] E. B. Davies, Non-Gaussian aspects of heat kernel behaviour , J. London Math. Soc. (2) 55 (1997), no. 1, 105-125. · Zbl 0879.35064
[20] A. Debiard, B. Gaveau, and E. Mazet, Théorèmes de comparaison en géométrie riemannienne , Publ. Res. Inst. Math. Sci. 12 (1976/77), no. 2, 391-425. · Zbl 0382.31007
[21] E. B. Fabes and D. W. Stroock, A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash , Arch. Rational Mech. Anal. 96 (1986), no. 4, 327-338. · Zbl 0652.35052
[22] A. A. Grigor’yan, Stochastically complete manifolds , Dokl. Akad. Nauk SSSR 290 (1986), no. 3, 534-537, (in Russian); Engl. transl. in Soviet Math. Dokl. 34 (1987), 310-313. · Zbl 0632.58041
[23] A. A. Grigor’yan, Heat kernel upper bounds on a complete non-compact manifold , Rev. Mat. Iberoamericana 10 (1994), no. 2, 395-452. · Zbl 0810.58040
[24] A. A. Grigor’yan, Integral maximum principle and its applications , Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), no. 2, 353-362. · Zbl 0812.58082
[25] A. A. Grigor’yan, Heat kernel of a noncompact Riemannian manifold , Stochastic analysis (Ithaca, NY, 1993) ed. M. Pinsky, Proc. Sympos. Pure Math., vol. 57, Amer. Math. Soc., Providence, RI, 1995, pp. 239-263. · Zbl 0829.58041
[26] A. A. Grigor’yan, Stochastically complete manifolds and summable harmonic functions , Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 5, 1102-1108, 1120, (in Russian); Engl. transl. in Math. USSR-Izv. 33 (1989), 425-432. · Zbl 0661.60090
[27] A. A. Grigor’yan, The heat equation on noncompact Riemannian manifolds , Mat. Sb. 182 (1991), no. 1, 55-87, Engl. transl. in Math. USSR Sb. 72 (1992), 47-77. · Zbl 0743.58031
[28] A. A. Grigor’yan, Gaussian upper bounds for the heat kernel and for its derivatives on a Riemannian manifold , Classical and modern potential theory and applications (Chateau de Bonas, 1993) ed. K. Gowri Sankaran, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 430, Kluwer Acad. Publ., Dordrecht, 1994, pp. 237-252. · Zbl 0885.58088
[29] Y. Guivarc’h, Sur la loi des grands nombres et le rayon spectral d’une marche aléatoire , Conference on Random Walks (Kleebach, 1979) (French), Astérisque, vol. 74, Soc. Math. France, Paris, 1980, 47-98, 3. · Zbl 0448.60007
[30] D. Hoffman and J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds , Comm. Pure Appl. Math. 27 (1974), 715-727. · Zbl 0295.53025
[31] L. Karp and P. Li, The heat equation on complete Riemannian manifolds , unpublished. · Zbl 0776.58035
[32] S. P. Lalley, Finite range random walk on free groups and homogeneous trees , Ann. Probab. 21 (1993), no. 4, 2087-2130. · Zbl 0804.60006
[33] P. Li and S.-T. Yau, On the parabolic kernel of the Schrödinger operator , Acta Math. 156 (1986), no. 3-4, 153-201. · Zbl 0611.58045
[34] F. Lust-Piquard, Lower bounds on \(\| K^ n\| _ 1\to\infty\) for some contractions \(K\) of \(L^ 2(\mu)\), with applications to Markov operators , Math. Ann. 303 (1995), no. 4, 699-712. · Zbl 0836.47021
[35] T. Lyons, Random thoughts on reversible potential theory , Summer School in Potential Theory (Joensuu, 1990) ed. Ilpo Laine, Joensuun Yliop. Luonnont. Julk., vol. 26, Univ. Joensuu, Joensuu, 1992, pp. 71-114. · Zbl 0757.31007
[36] J. Nash, Continuity of solutions of parabolic and elliptic equations , Amer. J. Math. 80 (1958), 931-954. JSTOR: · Zbl 0096.06902
[37] C. Pittet, Følner sequences in polycyclic groups , Rev. Mat. Iberoamericana 11 (1995), no. 3, 675-685. · Zbl 0842.20035
[38] F. O. Porper and S. D. Èĭ del’man, Two-sided estimates of the fundamental solutions of second-order parabolic equations and some applications of them , Uspekhi Mat. Nauk 39 (1984), no. 3(237), 107-156, (in Russian); Engl. transl. in Russian Math. Surveys 39 (1984), 119-178. · Zbl 0582.35052
[39] L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities , Internat. Math. Res. Notices (1992), no. 2, 27-38. · Zbl 0769.58054
[40] D. W. Stroock, Estimates on the heat kernel for second order divergence form operators , Probability theory (Singapore, 1989) ed. Y. Chen, et al., de Gruyter, Berlin, 1992, pp. 29-44. · Zbl 0779.60065
[41] K.-Th. Sturm, Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and \(L^ p\)-Liouville properties , J. Reine Angew. Math. 456 (1994), 173-196. · Zbl 0806.53041
[42] M. Takeda, On a martingale method for symmetric diffusion processes and its applications , Osaka J. Math. 26 (1989), no. 3, 605-623. · Zbl 0717.60090
[43] N. Th. Varopoulos, Long range estimates for Markov chains , Bull. Sci. Math. (2) 109 (1985), no. 3, 225-252. · Zbl 0583.60063
[44] N. Th. Varopoulos, Hardy-Littlewood theory for semigroups , J. Funct. Anal. 63 (1985), no. 2, 240-260. · Zbl 0608.47047
[45] N. Th. Varopoulos, Small time Gaussian estimates of heat diffusion kernels. I. The semigroup technique , Bull. Sci. Math. 113 (1989), no. 3, 253-277. · Zbl 0703.58052
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