Dirichlet forms and stochastic completeness of graphs and subgraphs

By Matthias Keller, Daniel Lenz. 2009

252 citations recorded by the source corpus.

Abstract We study Laplacians on graphs and networks via regular Dirichlet forms. We give a sufficient geometric condition for essential selfadjointness and explicitly determine the generators of the associated semigroups on all ℓp, 1 ≦ p < ∞, in this case. We characterize stochastic completeness thereby generalizing all earlier corresponding results for graph Laplacians. Finally, we study how stochastic completeness of a subgraph is related to stochastic completeness of the whole graph.

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

View Paper: https://www.semanticscholar.org/paper/7c5e0d2b3e54a40607f09a168de67089645efe48

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Citations in the corpus

Time regularity and long-time behavior of parabolic p-Laplace equations on infinite graphs (2014, background): “Alternatively, it has been proved in [12, 24, 14] that the Laplacian is essentially self-adjoint, hence L D 2 = L N 2 if • p = 2 and ν is nondegenerate.”
Volume growth and stochastic completeness of graphs (2012, background): “Stochastic completeness of subgraphs is discussed in [21], and [19] establishes analytic criteria which are equivalent to stochastic completeness of graphs and which have analogues in the manifold setting.”
Graphs of finite measure (2013, background): “These forms are naturally in one-to-one correspondence with graphs (b, c) as discussed in [33].”
Generalized Solutions and Spectrum for Dirichlet Forms on Graphs (2010, background): “ y∈V b(x,y)|w(y)| < ∞ for all x ∈ V } and Lwe (x) := 1 m(x) X y∈V b(x,y)(w(x) − w(y)) + c(x) m(x) u(x), where, for each x ∈ V , the sum exists by assumption on u. Then L is a restriction of Le(see [KL]). In [KL] the question whether the operator L is essentiay self-adjoint is addressed. Essential”
Analysis of the physical Laplacian and the heat flow on a locally finite graph (2008, background): “3,24]). Furthermore, the essentail self-adjointness of the Laplacian was independently proved by Palle Jorgensen in [9]. See also the preprint [10] by Palle Jorgensen and Erin Pearse. In the preprint [12], Matthias Keller and Daniel Lenz extended some of the mentioned results to the more general con”
A note on self-adjoint extensions of the Laplacian on weighted graphs (2012, methodology): “We generally follow the setting of [30].”
Volume growth, spectrum and stochastic completeness of infinite graphs (2011, background): “Such forms are regular Dirichlet forms on the measure space (V,m), see [37].”
On the conservativeness and the recurrence of symmetric jump-diffusions (2012, background): “Recently, there has been a tremendous amount of work devoted to the conservation property of a non-local Dirichelt form; for instance, the physical Laplacian on an infinite graph [7, 6, 39, 40, 41, 24, 18, 19, 20] and non-local Dirichlet forms [26, 15, 31]; however, as far as the authors know, there”
Gaussian Upper Bounds for Heat Kernels of Continuous Time Simple Random Walks (2011, background): “cs are closely related to the intrinsic metric associated with a given Dirichlet form; some details on the latter may be found in [18]. Recent work using similar metrics includes [4], [12], [15], and [20]. We will need the following condition: Deﬁnition: A monotonically increasing function g: (a,b) ”
Cheeger inequalities for unbounded graph Laplacians (2012, background): “The formQ defines a regular Dirichlet form on `2(X,m) (see [20, 36]).”
Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation (2011, background): “As discussed in [22] (see [15] as well) the following holds.”
Stochastic completeness for graphs with curvature dimension conditions (2015, background): “e denote by P(D) t = etL (D) the C 0-semigroup associated to the Dirichlet form Q(D) on ℓ2 m. It extrapolates to C0-semigroups on ℓ p m for all p ∈ [1,∞], for simplicity still denoted by P(D) t , see [KL12]. Deﬁnition 3.1. A weighted graph (V,E,m,µ) is called stochastically complete if P(D) t 1= 1, ”