Random walk on graphs with regular resistance and volume growth
Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 1, pp. 143-169.

Dans cet article, nous caractérisons des graphes qui satisfont des estimées du noyau de la chaleur pour un large ensemble de fonctions d'echelles spaciaux-temporelles. L'équivalence entre l'estimée du noyau de la chaleur et l'inégalité parabolique de Harnack est également démontrée par l'équivalence de l'estimée haute (basse) du noyau de la chaleur et l'inégalité parabolique de la valeur moyenne (et de la valeur moyenne supérieure).

In this paper characterizations of graphs satisfying heat kernel estimates for a wide class of space-time scaling functions are given. The equivalence of the two-sided heat kernel estimate and the parabolic Harnack inequality is also shown via the equivalence of the upper (lower) heat kernel estimate to the parabolic mean value (and super mean value) inequality.

DOI : 10.1214/AIHP114
Classification : 60J10, 60J45
Mots-clés : random walk, heat kernel, parabolic inequalities 
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Telcs, András. Random walk on graphs with regular resistance and volume growth. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 1, pp. 143-169. doi : 10.1214/AIHP114. http://archive.numdam.org/articles/10.1214/AIHP114/

D. G. Aronson. Non-negative solutions of linear parabolic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci (3) 22 (1968) 607-694; Addendum 25 (1971) 221-228. | EuDML | Numdam | MR | Zbl

M. T. Barlow. Diffusion on fractals. Lectures on Probability Theory and Statistics, Ecole d'été de probabilités de Saint-Flour XXV -1995, pp. 1-121. Lecture Notes in Math. 1690. Springer, Berlin, 1998. | MR | Zbl

M. T. Barlow. Some remarks on the elliptic Harnack inequality. Bull. Lond. Math. Soc. 37 (2005) 200-208. | MR | Zbl

M. T. Barlow and R. Bass. Stability of the parabolic Harnack inequalities. Trans. Amer. Math. Soc. 356 (2003) 1501-1533. | MR | Zbl

M. T. Barlow, R. Bass and T. Kumagai. Stability of parabolic Harnack inequalities on metric measure spaces. J. Math. Soc. Japan. 58 (2006) 485-519. | MR | Zbl

M. Barlow, T. Coulhon and A. Grigor'Yan. Manifolds and graphs with slow heat kernel decay. Invent. Math. 144 (2001) 609-649. | MR | Zbl

M. T. Barlow, T. Coulhon and T. Kumagai. Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs. Comm. Pure Appl. Math. 58 (2005) 1642-1677. | MR | Zbl

T. Coulhon and A. Grigor'Yan. Random walks on graphs with regular volume growth. Geom. Funct. Anal. 8 (1998) 656-701. | MR | Zbl

E. B. Davies. Heat Kernels and Spectral Theory. Cambridge University Press, 1989. | MR | Zbl

T. Delmotte. Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoamericana 1 (1999) 181-232. | MR | Zbl

T. Delmotte. Graphs between the elliptic and parabolic Harnack inequalities. Potential Anal. 16 (2002) 151-168. | MR | Zbl

E. B. Fabes and D. W. Stroock. A new proof of Moser's parabolic Harnack inequality using an old idea of Nash. Arch. Rational Mech. Anal. 96 (1986) 327-338. | MR | Zbl

A. Grigor'Yan. Heat equation on a non-compact Riemannian manifold. Math. USSR Sb. 72 (1992) 47-77. | Zbl

A. Grigor'Yan. Heat kernel upper bounds on fractal spaces. Unpublished manuscript.

A. Grigor'Yan and A. Telcs. Sub-Gaussian estimates of heat kernels on infinite graphs. Duke Math. J. 109 (2001) 452-510. | MR | Zbl

A. Grigor'Yan and A. Telcs. Harnack inequalities and sub-Gaussian estimates for random walks. Math. Ann. 324 (2002) 521-556. | MR | Zbl

A. Grigor'Yan and A. Telcs. Heat kernel estimates in measure metric spaces. Unpublished manuscript.

M. Gromov. Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. 53 (1981) 57-73. | Numdam | MR | Zbl

B. Hambly and T. Kumagai. Heat kernel estimates for symmetric random walks on a class of fractal graphs and stability under rough isometries. In Proc. of Symposia in Pure Math. 72. Part 2, pp. 233-260. Amer. Math. Soc., Providence, RI, 2004. | MR | Zbl

M. Hino and J. A. Ramírez. Small-time Gaussian behavior of symmetric diffusion semigroups. Ann. Probab. 31 (2003) 1254-1295. | MR | Zbl

W. Hebisch and L. Saloff-Coste. On the relation between elliptic and parabolic Harnack inequalities. Ann. Inst. Fourier 51 (2001) 1437-1481. | Numdam | MR | Zbl

O. D. Jones. Transition probabilities for the simple random walk on the Sierpinski graph. Stochastic Process Appl. 61 (1996) 45-69. | MR | Zbl

T. Kumagai and K.-T. Sturm. Construction of diffusion processes on fractals, d-sets, and general metric measure spaces, J. Math. Kyoto Univ. 45 (2005) 307-327. | MR | Zbl

P. Li and J. Wang. Mean value inequalities. Indiana Univ. Math. J. 48 (1999) 1257-1283. | MR | Zbl

P. Li and S.-T. Yau. On the parabolic kernel of the Schrödinger operator. Acta Math. 156 (1986) 153-201. | MR | Zbl

J. Moser. On Harnack's theorem for elliptic differential equations. Comm. Pure Appl. Math. 16 (1964) 101-134. | MR | Zbl

J. Moser. On Harnack's theorem for parabolic differential equations. Comm. Pure Appl. Math. 24 (1971) 727-740. | MR | Zbl

J. R. Norris. Heat kernel asymptotics and the distance function in Lipschitz Riemannian manifolds. Acta Math. 179 (1997) 79-103. | MR | Zbl

C. Sabot. Existence and uniqueness of diffusions on finitely ramified self-similar fractals. Ann. Sci. École Norm. Sup (4) 30 (1997) 605-673. | Numdam | MR | Zbl

L. Saloff-Coste. A note on Poincaré, Sobolev and Harnack inequalities. Internat. Math. Res. 2 (1992) 27-38. | MR | Zbl

C.-J. Sung. Parabolic super mean value inequality. Proc. Amer. Math. Soc. 130 (2002) 3401-3408. | MR | Zbl

R. S. Strichartz. Fractafolds based on the Sierpinski gasket and their spectra. Trans. Amer. Math. Soc. 355 (2003) 4019-4043. | MR | Zbl

K.-T. Sturm. Analysis on local Dirichlet spaces III. The parabolic Harnack inequality. J. Math. Pures Appl. (9) 75 (1996) 273-297. | MR | Zbl

A. Telcs. Volume and time doubling of graphs and random walk, the strongly recurrent case. Comm. Pure Appl. Math. (2001) 975-1018. | MR | Zbl

A. Telcs. Some notes on the Einstein relation. J. Stat. Phys. 122 (2006) 617-645. | MR | Zbl

A. Telcs. Random walks on graphs with volume and time doubling. Revista Mat. Iberoamericana 22 (2006) 17-54. | MR | Zbl

A. Telcs. The Art of Random Walks. Springer, Berlin, 2006. | MR | Zbl

S. R. S. Varadhan. On the behavior of the fundamental solution of the heat equation with variable coefficients. Comm. Pure Appl. Math. 20 (1967) 431-455. | MR | Zbl

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