Stochastic Completeness of Graphs

By Radoslaw K. Wojciechowski. 2007

117 citations recorded by the source corpus.

In this thesis, we analyze the stochastic completeness of a heat kernel on graphs which is a function of three variables: a pair of vertices and a continuous time, for infinite, locally finite, connected graphs. For general graphs, a sufficient condition for stochastic completeness is given in terms of the maximum valence on spheres about a fixed vertex. That this result is optimal is shown by studying a particular family of trees. We also prove a lower bound on the bottom of the spectrum for the discrete Laplacian and use this lower bound to show that in certain cases the Laplacian has empty essential spectrum.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/05425efd9eec6259c430b7d530bb3a9d02c1867a

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In Computational Theory and Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation (2011, methodology): “This has been shown in [22] in our setting after earlier results in [8, 32, 33] for locally finite graphs.”
Cheeger inequalities for unbounded graph Laplacians (2012, background): “Moreover, Cheeger estimates for the bottom of the essential spectrum and criteria for discreteness of spectrum are given in [19, 34, 49, 50].”
Dirichlet forms and stochastic completeness of graphs and subgraphs (2009, unknown): “Our setting generalizes the setting of [9], [11], [17], [26], [28] to Dirichlet forms on countable sets.”
Hardy inequality and asymptotic eigenvalue distribution for discrete Laplacians (2011, background): “ences therein). In [Jor], he says that every discrete Laplacian is essentially self-adjoint on simple graphs. The proof of [Jor] was incomplete (see [JP]). It was ﬁxed later in [JP]. In the meantime, [Woj] proves this fact. An alternative proof can be found in [Web] where one uses the maximum princi”
Heat Kernel and Essential Spectrum of Infinite Graphs (2008, unknown): “27 45 v4 [ m at h. SP ] 2 4 A pr 2”
A note on self-adjoint extensions of the Laplacian on weighted graphs (2012, background): “More specifically, this problem is studied for adjacency, (magnetic) Laplacian, and Schrödinger-type operators on graphs in [4, 5, 6, 16, 17, 18, 23, 27, 28, 30, 31, 34, 37, 39, 40, 42, 43, 44, 45] among others.”
Analysis of the physical Laplacian and the heat flow on a locally finite graph (2008, unknown): “MSC 2000: 05C50, 47B39.”