Parabolicity and stochastic completeness of manifolds in terms of the Green formula

Grigor’yan, Alexander; Masamune, Jun

doi:10.1016/j.matpur.2013.01.015

Cited by 42 publications

(66 citation statements)

References 30 publications

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“…We provide the arguments for the special case 2α = d in Corollary 4. By Giraud's lemma, [4,Chapter 4,Proposition 4.12], together with (14) and (16), the densities u Consequently, using (14) and (16), and with positive constants c independent of x but possibly varying from line to line,…”

mentioning

confidence: 99%

Probabilistic Characterizations of Essential Self-Adjointness and Removability of Singularities

Hinz¹,

Kang²,

Masamune³

2017

Vestn. Volgogr. gos. univ., Ser. 1. Math. Phy. and Comp. Model.

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Abstract. We consider the Laplacian and its fractional powers of order less than one on the complement R ∖ Σ of a given compact set Σ ⊂ R of zero Lebesgue measure. Depending on the size of Σ, the operator under consideration, equipped with the smooth compactly supported functions on R ∖ Σ, may or may not be essentially self-ajoint. We survey well-known descriptions for the critical size of Σ in terms of capacities and Hausdorff measures. In addition, we collect some known results for certain two-parameter stochastic processes. What we finally want to point out is, that, although a priori essential self-adjointness is not a notion directly related to classical probability, it admits a characterization via Kakutani-type theorems for such processes.

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confidence: 99%

Probabilistic Characterizations of Essential Self-Adjointness and Removability of Singularities

Hinz¹,

Kang²,

Masamune³

2017

Vestn. Volgogr. gos. univ., Ser. 1. Math. Phy. and Comp. Model.

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“…This L 1 -Stokes theorem implies the L 2 -Stokes theorem for all p-forms, but the inverse does not hold (see Grigor'yan and Masamune [19] p. 614, Proposition 2.4).…”

Section: -Stokes Theoremmentioning

confidence: 99%

L2-Harmonic Forms on Incomplete Riemannian Manifolds with Positive Ricci Curvature

Takahashi¹

2018

Mathematics

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“…In general, this is not the case. But if the boundary ∂B is a polar set in the sense of Grigor'yan and Masamune (see [29]), it is also true. In the weighted situation that we are considering the boundary is a polar set in this sense.…”

Section: Weighted Riemannian Manifoldsmentioning

confidence: 99%

Cones over metric measure spaces and the maximal diameter theorem

Ketterer¹

2015

Journal de Mathématiques Pures et Appliquées

113

112

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The main result of this article states that the (K, N )-cone over some metric measure space satisfies the reduced Riemannian curvature-dimension condition RCD * (KN, N +1) if and only if the underlying space satisfies RCD * (N −1, N ). The proof uses a characterization of reduced Riemannian curvature-dimension bounds by Bochner's inequality that was established for general metric measure spaces by Erbar, Kuwada and Sturm in [21] (independently, the same result has been announced by Ambrosio, Mondino and Savaré). As a corollary of our result and the Gigli-Cheeger-Gromoll splitting theorem [25] we obtain a maximal diameter theorem in the context of metric measure spaces that satisfy the condition RCD * .

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Parabolicity and stochastic completeness of manifolds in terms of the Green formula

Cited by 42 publications

References 30 publications

Probabilistic Characterizations of Essential Self-Adjointness and Removability of Singularities

Probabilistic Characterizations of Essential Self-Adjointness and Removability of Singularities

L2-Harmonic Forms on Incomplete Riemannian Manifolds with Positive Ricci Curvature

Cones over metric measure spaces and the maximal diameter theorem

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