Random walk on graphs with regular resistance and volume growth
Telcs, András
Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008), p. 143-169 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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.

Publié le : 2008-01-01
DOI : https://doi.org/10.1214/AIHP114
Classification:  60J10,  60J45 
@article{AIHPB_2008__44_1_143_0,
     author = {Telcs, Andr\'as},
     title = {Random walk on graphs with regular resistance and volume growth},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {44},
     year = {2008},
     pages = {143-169},
     doi = {10.1214/AIHP114},
     mrnumber = {2451575},
     zbl = {1177.60047},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2008__44_1_143_0}
}

Telcs, András. Random walk on graphs with regular resistance and volume growth. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) pp. 143-169. doi : 10.1214/AIHP114. http://gdmltest.u-ga.fr/item/AIHPB_2008__44_1_143_0/

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. | Numdam | Numdam | MR 435594 | Zbl 0223.35046

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 1668115 | Zbl 0916.60069

M. T. Barlow. Some remarks on the elliptic Harnack inequality. Bull. Lond. Math. Soc. 37 (2005) 200-208. | MR 2119019 | Zbl 1067.31002

M. T. Barlow and R. Bass. Stability of the parabolic Harnack inequalities. Trans. Amer. Math. Soc. 356 (2003) 1501-1533. | MR 2034316 | Zbl 1034.60070

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 2228569 | Zbl 1102.60064

M. Barlow, T. Coulhon and A. Grigor'Yan. Manifolds and graphs with slow heat kernel decay. Invent. Math. 144 (2001) 609-649. | MR 1833895 | Zbl 1003.58025

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 2177164 | Zbl 1083.60060

T. Coulhon and A. Grigor'Yan. Random walks on graphs with regular volume growth. Geom. Funct. Anal. 8 (1998) 656-701. | MR 1633979 | Zbl 0918.60053

E. B. Davies. Heat Kernels and Spectral Theory. Cambridge University Press, 1989. | MR 990239 | Zbl 0699.35006

T. Delmotte. Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoamericana 1 (1999) 181-232. | MR 1681641 | Zbl 0922.60060

T. Delmotte. Graphs between the elliptic and parabolic Harnack inequalities. Potential Anal. 16 (2002) 151-168. | MR 1881595 | Zbl 1081.39012

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 855753 | Zbl 0652.35052

A. Grigor'Yan. Heat equation on a non-compact Riemannian manifold. Math. USSR Sb. 72 (1992) 47-77. | Zbl 0776.58035

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 1853353 | Zbl 1010.35016

A. Grigor'Yan and A. Telcs. Harnack inequalities and sub-Gaussian estimates for random walks. Math. Ann. 324 (2002) 521-556. | MR 1938457 | Zbl 1011.60021

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 623534 | Zbl 0474.20018

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 2112125 | Zbl 1065.60041

M. Hino and J. A. Ramírez. Small-time Gaussian behavior of symmetric diffusion semigroups. Ann. Probab. 31 (2003) 1254-1295. | MR 1988472 | Zbl 1085.31008

W. Hebisch and L. Saloff-Coste. On the relation between elliptic and parabolic Harnack inequalities. Ann. Inst. Fourier 51 (2001) 1437-1481. | Numdam | MR 1860672 | Zbl 0988.58007

O. D. Jones. Transition probabilities for the simple random walk on the Sierpinski graph. Stochastic Process Appl. 61 (1996) 45-69. | MR 1378848 | Zbl 0853.60058

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 2161694 | Zbl 1086.60052

P. Li and J. Wang. Mean value inequalities. Indiana Univ. Math. J. 48 (1999) 1257-1283. | MR 1757075 | Zbl 1003.58026

P. Li and S.-T. Yau. On the parabolic kernel of the Schrödinger operator. Acta Math. 156 (1986) 153-201. | MR 834612 | Zbl 0611.58045

J. Moser. On Harnack's theorem for elliptic differential equations. Comm. Pure Appl. Math. 16 (1964) 101-134. | MR 159139 | Zbl 0149.06902

J. Moser. On Harnack's theorem for parabolic differential equations. Comm. Pure Appl. Math. 24 (1971) 727-740. | MR 288405 | Zbl 0227.35016

J. R. Norris. Heat kernel asymptotics and the distance function in Lipschitz Riemannian manifolds. Acta Math. 179 (1997) 79-103. | MR 1484769 | Zbl 0912.58041

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 1474807 | Zbl 0924.60064

L. Saloff-Coste. A note on Poincaré, Sobolev and Harnack inequalities. Internat. Math. Res. 2 (1992) 27-38. | MR 1150597 | Zbl 0769.58054

C.-J. Sung. Parabolic super mean value inequality. Proc. Amer. Math. Soc. 130 (2002) 3401-3408. | MR 1913020 | Zbl 1010.58021

R. S. Strichartz. Fractafolds based on the Sierpinski gasket and their spectra. Trans. Amer. Math. Soc. 355 (2003) 4019-4043. | MR 1990573 | Zbl 1041.28006

K.-T. Sturm. Analysis on local Dirichlet spaces III. The parabolic Harnack inequality. J. Math. Pures Appl. (9) 75 (1996) 273-297. | MR 1387522 | Zbl 0854.35016

A. Telcs. Volume and time doubling of graphs and random walk, the strongly recurrent case. Comm. Pure Appl. Math. (2001) 975-1018. | MR 1829530 | Zbl 1021.60037

A. Telcs. Some notes on the Einstein relation. J. Stat. Phys. 122 (2006) 617-645. | MR 2213945 | Zbl 1149.82025

A. Telcs. Random walks on graphs with volume and time doubling. Revista Mat. Iberoamericana 22 (2006) 17-54. | MR 2267312 | Zbl 1118.60062

A. Telcs. The Art of Random Walks. Springer, Berlin, 2006. | MR 2240535 | Zbl 1104.60003

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 208191 | Zbl 0155.16503