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Blaise Pascal
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Satoshi Ishiwata
Discrete version of Dungey’s proof for the gradient heat kernel estimate on coverings
(Version discrète d’une preuve de Dungey pour les estimations du gradient du noyau de la chaleur sur les revêtements)
Annales mathématiques Blaise Pascal, 14 no. 1 (2007), p. 93-102, doi: 10.5802/ambp.229
Article PDF | Reviews Zbl 1137.60033
Class. Math.: 60J10, 58J35, 58J37
Keywords: Gradient estimates, Random walks, Gaussian estimates for the heat kernel

Résumé - Abstract

We obtain another proof of a Gaussian upper estimate for a gradient of the heat kernel on cofinite covering graphs whose covering transformation group has a polynomial volume growth. It is proved by using the temporal regularity of the discrete heat kernel obtained by Blunck [2] and Christ [3] along with the arguments of Dungey [7] on covering manifolds.

Bibliography

[1] P. Auscher, T. Coulhon, X. T. Duong and S. Hofmann. Riesz transform on manifolds and heat kernel regurality. Ann. Scient. Éc. Norm. Sup., 37:911-957, 2004. Numdam |  MR 2119242 |  Zbl 02174958
[2] S. Blunck. Perturbation of analytic operators and temporal regularity. Colloq. Math., 86:189-201, 2000.  MR 1808675 |  Zbl 0961.47005
[3] M. Christ. Temporal regularity for random walk on discrete nilpotent groups. Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993). J. Fourier Anal. Appl., Special Issue:141-151, 1995.  MR 1364882 |  Zbl 0889.60007
[4] T. Coulhon and X. T. Duong. Riesz transforms for $1\le p\le 2$. Trans. Amer. Math. Soc., 351:1151-1169, 1999. Article |  MR 1458299 |  Zbl 0973.58018
[5] E. B. Davies. Non-gaussian aspects of heat kernel behaviour. J. London Math. Soc., 55:105-125, 1997. Article |  MR 1423289 |  Zbl 0879.35064
[6] N. Dungey. Heat kernel estimates and Riesz transforms on some Riemannian covering manifolds. Math. Z., 247:765-794, 2004. Article |  MR 2077420 |  Zbl 1080.58022
[7] N. Dungey. Some gradient estimates on covering manifolds. Bull. Pol. Acad. Sci. Math., 52:437-443, 2004. Article |  MR 2128280 |  Zbl 02170013
[8] N. Dungey. A note on time regularity for discrete time heat kernel. Semigroup Forum, 72:404-410, 2006. Article |  MR 2228535 |  Zbl 1102.47016
[9] M. Gromov. Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math., 53:53-73, 1981. Numdam |  MR 623534 |  Zbl 0474.20018
[10] W. Hebisch and L. Saloff-Coste. Gaussian estimates for Markov chains and random walks on groups. Ann. Probab., 21:673-709, 1993. Article |  MR 1217561 |  Zbl 0776.60086
[11] S. Ishiwata. Asymptotic behavior of a transition probability for a random walk on a nilpotent covering graph. Contemp. Math., 347:57-68, 2004.  MR 2077030 |  Zbl 1061.22009
[12] S. Ishiwata. A Berry-Esseen type theorem on nilpotent covering graphs. Canad. J. Math., 56:963-982, 2004. Article |  MR 2085630 |  Zbl 1062.22018
[13] E. Russ. Riesz transform on graphs for $p>2$. unpublished manuscript