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26
SubGaussian estimates of heat kernels on infinite graphs
 Duke Math. J
, 2000
"... We prove that a two sided subGaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay. ..."
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Cited by 30 (10 self)
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We prove that a two sided subGaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay.
Harnack inequalities and subGaussian estimates for random walks
 Math. Annalen
, 2002
"... We show that a fiparabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to so called fiGaussian estimates for the transition probability and, on the other hand, to the conjunction of the elliptic Harnack inequality, the doubling volume property, and the fact that the m ..."
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Cited by 29 (6 self)
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We show that a fiparabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to so called fiGaussian estimates for the transition probability and, on the other hand, to the conjunction of the elliptic Harnack inequality, the doubling volume property, and the fact that the mean exit time in any ball of radius R is of the order R . The latter condition can be replaced by a certain estimate of a resistance of annuli.
Manifolds and Graphs With Slow Heat Kernel Decay
 Invent. Math
, 1999
"... We give upper estimates on the long time behaviour of the heat kernel on a noncompact Riemannian manifold and infinite graphs, which only depend on a lower bound of the volume growth. We also show that these estimates are optimal. ..."
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Cited by 24 (2 self)
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We give upper estimates on the long time behaviour of the heat kernel on a noncompact Riemannian manifold and infinite graphs, which only depend on a lower bound of the volume growth. We also show that these estimates are optimal.
Growth of selfsimilar graphs
, 2001
"... Abstract. Locally finite selfsimilar graphs with bounded geometry and without bounded geometry as well as nonlocally finite selfsimilar graphs are characterized by the structure of their cell graphs. Geometric properties concerning the volume growth and distances in cell graphs are discussed. The ..."
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Cited by 8 (3 self)
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Abstract. Locally finite selfsimilar graphs with bounded geometry and without bounded geometry as well as nonlocally finite selfsimilar graphs are characterized by the structure of their cell graphs. Geometric properties concerning the volume growth and distances in cell graphs are discussed. The length scaling factor ν and the volume scaling factor µ can be defined similarly to the corresponding parameters of continuous selfsimilar sets. There are different notions of growth dimensions of graphs. For a rather general class of selfsimilar graphs it is proved that all these dimensions coincide and that they can be calculated in the same way as the Hausdorff dimension of continuous selfsimilar fractals: log µ dim X = log ν. 1.
Harnack inequality and hyperbolicity for subelliptic pLaplacians with applications to Picard type theorems
, 2000
"... Contents 1 Introduction 2 2 Preliminaries 2 2.1 The gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 The doubling property . . . . . . . . . . . . . . . . . . . ..."
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Cited by 5 (1 self)
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Contents 1 Introduction 2 2 Preliminaries 2 2.1 The gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 The doubling property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 The Poincar'e inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.5 The pLaplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.6 The nonsmooth case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 pparabolicity and phyperbolicity 10 3.1 An inequality for supersolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Volume growth and pparabolicity . . . . . . . . . . . . . . . . .
Interpolation of Sobolev spaces, LittlewoodPaley inequalities and Riesz transforms on graphs
 PUBLICACIONS MATEMATIQUES
"... Abstract. Let Γ be a graph endowed with a reversible Markov kernel p, and P the associated operator, defined by Pf(x) = ∑ y p(x, y)f(y). Denote by ∇ the discrete gradient. We give necessary and/or sufficient conditions on Γ in order to compare ‖∇f‖ p and ∥ ∥ (I − P) ..."
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Cited by 5 (2 self)
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Abstract. Let Γ be a graph endowed with a reversible Markov kernel p, and P the associated operator, defined by Pf(x) = ∑ y p(x, y)f(y). Denote by ∇ the discrete gradient. We give necessary and/or sufficient conditions on Γ in order to compare ‖∇f‖ p and ∥ ∥ (I − P)
Heat Kernel Estimates on Weighted Graphs
 PISA PI, ITALY FACHBEREICH MATHEMATIK, DER JOHANN WOLFGANG GOETHEUNIVERSITÄT, 60054 FRANKFURT AM MAIN, GERMANY EMAIL ADDRESS: ASCHMIDT@MATH.UNIFRANKFURT.DE
, 2000
"... We prove upper and lower heat kernel bounds for the Laplacian on weighted graphs which include the case that the weights have no strictly positive lower bound. Our estimates allow for a very explicit probabilistic interpretation and can be formulated in terms of a weighted metric. Interestingly, thi ..."
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Cited by 5 (1 self)
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We prove upper and lower heat kernel bounds for the Laplacian on weighted graphs which include the case that the weights have no strictly positive lower bound. Our estimates allow for a very explicit probabilistic interpretation and can be formulated in terms of a weighted metric. Interestingly, this metric is not equivalent to the intrinsic metric.
Eigenfunction expansions for generators of Dirichlet forms
, 2003
"... We present an eigenfunction expansion theorem for generators of strongly local, regular Dirichlet forms. Conditions are phrased in terms of the intrinsic metric. The result covers many cases of Hamiltonians which appear in Mathematical Physics and Geometry. ..."
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Cited by 5 (3 self)
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We present an eigenfunction expansion theorem for generators of strongly local, regular Dirichlet forms. Conditions are phrased in terms of the intrinsic metric. The result covers many cases of Hamiltonians which appear in Mathematical Physics and Geometry.
The Einstein relation for random walks on graphs
, 2008
"... This paper investigates the Einstein relation; the connection between the volume growth, the resistance growth and the expected time a random walk needs to leave a ball on a weighted graph. The Einstein relation is proved under different set of conditions. In the simplest case it is shown under the ..."
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Cited by 2 (1 self)
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This paper investigates the Einstein relation; the connection between the volume growth, the resistance growth and the expected time a random walk needs to leave a ball on a weighted graph. The Einstein relation is proved under different set of conditions. In the simplest case it is shown under the volume doubling and time comparison principles. This and the other set of conditions provide the basic vwork for the study of (sub) diffusive behavior of the random walks on weighted graphs. 1