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Stochastic incompleteness for graphs and weak Omori-Yau maximum principle

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Xueping Huang

Spectral Theory · Submitted 14 Sep 2010

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove an analogue of the weak Omori-Yau maximum principle and Khas'minskii's criterion for graphs in the general setting of Keller and Lenz.

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arXiv ID1009.2579 (version 4)
Pith IDSLVJXED6, Pith's permanent record of this paper and its review
CitationXueping Huang. "Stochastic incompleteness for graphs and weak Omori-Yau maximum principle." arXiv:1009.2579 (2010). https://pith.science/paper/SLVJXED6
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What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.49 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; .

We prove an analogue of the weak Omori-Yau maximum principle and Khas'minskii's criterion for graphs in the general setting of Keller and Lenz. Our approach naturally gives the stability of stochastic incompleteness under certain surgeries of graphs. It allows to develop a unified approach to all known criteria of stochastic completeness/incompleteness, as well as to obtain new criteria.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 We prove an analogue of the weak Omori-Yau maximum principle and Khas'minskii's criterion for graphs in the general setting of Keller and Lenz. contribution 0.57 0.96 0.74 asserted 1.00
s1 Our approach naturally gives the stability of stochastic incompleteness under certain surgeries of graphs. result 0.74 0.93 0.16 asserted 0.99
s2 It allows to develop a unified approach to all known criteria of stochastic completeness/incompleteness, as well as to obtain new criteria. interpretation 0.53 0.83 0.22 asserted 0.90

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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  • C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We prove an analogue of the weak Omori-Yau maximum principle and Khas'minskii's criterion for graphs in the general setting of Keller and Lenz. Our approach naturally gives the stability of stochastic incompleteness under certain surgeries of graphs. It allows to develop a unified approach to all known criteria of stochastic completeness/incompleteness, as well as to obtain new criteria.

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2 arXiv papers cite this in 3 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 2 sentences without a confident label

Cites it as background 1 paper

What it cites · 10 sentences without a confident label

Cites as background 4 papers
  • The explicit domains of these generators are irrelevant to the problem of stochastic completeness (for details see [ref] ).

    Dirichlet forms and stochastic completeness of graphs and subgraphs · Foundations · 0.93

  • Recently, Wojciechowski [ref] , [ref] , [ref] and Weber [ref] independently studied the problem of stochastic completeness for the following Laplace operator (Weber calls it “physical Laplacian”) on a locally finite, connected, undirected graph $(V, E)$ : [equation] where $V$ is the set of vertices and $E$ is the set of edges, and $y∼x$ means that $(x,y)∈E$.

    Stochastic Completeness of Graphs · Introduction · 0.94

  • Recently, Wojciechowski [ref] , [ref] , [ref] and Weber [ref] independently studied the problem of stochastic completeness for the following Laplace operator (Weber calls it “physical Laplacian”) on a locally finite, connected, undirected graph $(V, E)$ : [equation] where $V$ is the set of vertices and $E$ is the set of edges, and $y∼x$ means that $(x,y)∈E$.

    Stochastically Incomplete Manifolds and Graphs · Introduction · 0.94

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