Gradient estimate of the heat kernel on modified graphs. (English) Zbl 1196.58012

The author obtains a condition on the modification of graphs (in the sense of a finite combinations of an edge modification and a vertex modification) which guarantees the preservation of the Gaussian upper bound for the gradient of the heat kernel, more precisely we have \[ \nabla k_n(x,y)\leq \frac{C}{\sqrt{n}V(x,\sqrt{n})}e^{-cd(x,y)^2/n} \] for all \(x,y\in V\) and \(n\in \mathbb{N}^{\ast}\) where \(V\) is the modified graph, \(V(x,y)\) is the volume of the ball centered at \(x\) with radius \(r\) for the combinatorial distance \(d\), and \(C,c\) are some positive constant.


58J37 Perturbations of PDEs on manifolds; asymptotics
60G50 Sums of independent random variables; random walks
58J65 Diffusion processes and stochastic analysis on manifolds
Full Text: DOI Link


[1] Auscher, P., Coulhon, T., Duong, X.T., Hofmann, S.: Riesz transform on manifolds and heat kernel regularity. Ann. Sci. École Norm. Sup. 37(6), 911–957 (2004) · Zbl 1086.58013
[2] Coulhon, T., Grigor’yan, A.: Random walks on graphs with regular volume growth. GAFA 8, 656–701 (1998) · Zbl 0918.60053 · doi:10.1007/s000390050070
[3] Coulhon, T., Dungey, N.: Riesz transforms and perturbations. J. Geom. Anal. 17(2), 213–226 (2007) · Zbl 1122.58014
[4] Dungey, N.: Heat kernel estimates and Riesz transforms on some Riemannian covering manifolds. Math. Z. 247(4), 765–794 (2004) · Zbl 1080.58022 · doi:10.1007/s00209-003-0646-4
[5] Dungey, N.: Some gradient estimates on covering manifolds. Bull. Polish Acad. Sci. Math. 52(4), 437–443 (2004) · Zbl 1112.58027 · doi:10.4064/ba52-4-10
[6] Hebisch, W., Saloff-Coste, L.: Gaussian estimates for Markov chains and random walks on groups. Ann. Probab. 21(2), 673–709 (1993) · Zbl 0776.60086 · doi:10.1214/aop/1176989263
[7] Ishiwata, S.: A Berry-Esseen type theorem on nilpotent covering graphs. Canad. J. Math. 56(5), 963–982 (2004) · Zbl 1062.22018 · doi:10.4153/CJM-2004-044-4
[8] Ishiwata, S.: Discrete version of Dungey’s proof for the gradient heat kernel estimate on coverings. Ann. Math. Blaise Pascal 14(1), 93–102 (2007) · Zbl 1137.60033
[9] Ishiwata, S.: Connectivity on discrete nilpotent groups (in preparation)
[10] Kanai, M.: Rough isometries, and combinatorial approximations of geometries of non-compact riemannian manifolds. J. Math. Soc. Japan 37, 391–413 (1985) · Zbl 0554.53030 · doi:10.2969/jmsj/03730391
[11] Russ, E.: Riesz transforms on graphs for p (preprint) · Zbl 1008.60085
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.