Results 1  10
of
90
On the relation between elliptic and parabolic Harnack inequalities
, 2001
"... We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in que ..."
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Cited by 66 (6 self)
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We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for on M , (i.e., for @ t + ) and elliptic Harnack inequality for @ 2 t + on R M . 1
Dirichlet forms and stochastic completeness of graphs and subgraphs, to appear
 J. Reine Angew. Math. (Crelle’s Journal
"... Abstract. We characterize stochastic completeness for regular Dirichlet forms on discrete sets. We then study how stochastic completeness of a subgraph is related to stochastic completeness of the whole graph. We show that any graph is a subgraph of a stochastically complete graph and that stochasti ..."
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Cited by 50 (21 self)
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Abstract. We characterize stochastic completeness for regular Dirichlet forms on discrete sets. We then study how stochastic completeness of a subgraph is related to stochastic completeness of the whole graph. We show that any graph is a subgraph of a stochastically complete graph and that stochastic incompleteness of a suitably modified subgraph implies stochastic incompleteness of the whole graph. Along our way we give a sufficient condition for essential selfadjointness of generators of Dirichlet forms on discrete sets and explicitely determine the generators on all ℓ p, 1 ≤ p < ∞, in this case.
Sobolev Spaces, Laplacian, And Heat Kernel On Alexandrov Spaces
, 1998
"... We prove the compactness of the imbedding of the Sobolev space W 1;2 0 (\Omega\Gamma into L 2(\Omega\Gamma for any relatively compact open subset\Omega of an Alexandrov space. As a corollary, the generator induced from the Dirichlet (energy) form has discrete spectrum. The generator can be approx ..."
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Cited by 44 (7 self)
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We prove the compactness of the imbedding of the Sobolev space W 1;2 0 (\Omega\Gamma into L 2(\Omega\Gamma for any relatively compact open subset\Omega of an Alexandrov space. As a corollary, the generator induced from the Dirichlet (energy) form has discrete spectrum. The generator can be approximated by the Laplacian induced from the DCstructure on the Alexandrov space. We also prove the existence of the locally Holder continuous Dirichlet heat kernel.
Unbounded Laplacians on graphs: basic spectral properties and the heat equation
"... Abstract. We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focus on phenomena related to unboundedness of the Laplacians. This includes (failure of) essential selfadjointness, absence of essential spectrum and stochastic incompleteness. ..."
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Cited by 35 (14 self)
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Abstract. We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focus on phenomena related to unboundedness of the Laplacians. This includes (failure of) essential selfadjointness, absence of essential spectrum and stochastic incompleteness.
Intrinsic metrics for nonlocal symmetric Dirichlet forms and applications to spectral theory. to appear
, 2010
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A dual characterization of length spaces with application to Dirichlet metric spaces
, 2009
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Stochastically Incomplete Manifolds and Graphs
"... Abstract. We survey geometric properties which imply the stochastic incompleteness of the minimal diffusion process associated to the Laplacian on manifolds and graphs. In particular, we completely characterize stochastic incompleteness for spherically symmetric graphs and show that, in contrast to ..."
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Cited by 17 (1 self)
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Abstract. We survey geometric properties which imply the stochastic incompleteness of the minimal diffusion process associated to the Laplacian on manifolds and graphs. In particular, we completely characterize stochastic incompleteness for spherically symmetric graphs and show that, in contrast to the case of Riemannian manifolds, there exist examples of stochastically incomplete graphs of polynomial volume growth.
Sch’nol’s theorem for strongly local forms
, 2009
"... We prove a variant of Sch’nol’s theorem in a general setting: for generators of strongly local Dirichlet forms perturbed by measures. As an application, we discuss quantum graphs with δ or Kirchhoff boundary conditions. ..."
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Cited by 17 (9 self)
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We prove a variant of Sch’nol’s theorem in a general setting: for generators of strongly local Dirichlet forms perturbed by measures. As an application, we discuss quantum graphs with δ or Kirchhoff boundary conditions.
Volume growth and stochastic completeness of graphs, Trans
 Amer. Math. Soc
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