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

Alexander Grigor'yan; Jun Masamune

doi:10.1016/j.matpur.2013.01.015

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

Alexander Grigor'yan, Jun Masamune

SCI - Mathematics

Bielefeld University

Research output: Contribution to journal › Article › peer-review

27 Citations (Scopus)

Abstract

We present and prove new characterizations of parabolicity and stochastic completeness for a general weighted manifold M as well as the uniqueness of the Markov extensions of the Laplacian in terms of Green's formula. Moreover, we study the relationship between those properties and the singularity of M in terms of a fractal dimension and capacity.

Original language	English
Pages (from-to)	607-632
Number of pages	26
Journal	Journal des Mathematiques Pures et Appliquees
Volume	100
Issue number	5
DOIs	https://doi.org/10.1016/j.matpur.2013.01.015
Publication status	Published - 2013 Nov

Keywords

Conservation property
Green's formula
Heat kernel
Markov extensions
Non-explosion
Parabolicity
Primary
Recurrence
Riemannian manifold
Secondary
Self-adjoint extensions
Stochastic completeness
Weighted manifold

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10.1016/j.matpur.2013.01.015

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title = "Parabolicity and stochastic completeness of manifolds in terms of the Green formula",

abstract = "We present and prove new characterizations of parabolicity and stochastic completeness for a general weighted manifold M as well as the uniqueness of the Markov extensions of the Laplacian in terms of Green's formula. Moreover, we study the relationship between those properties and the singularity of M in terms of a fractal dimension and capacity.",

keywords = "Conservation property, Green's formula, Heat kernel, Markov extensions, Non-explosion, Parabolicity, Primary, Recurrence, Riemannian manifold, Secondary, Self-adjoint extensions, Stochastic completeness, Weighted manifold",

author = "Alexander Grigor'yan and Jun Masamune",

note = "Funding Information: This research was conducted when J.M. visited Worcester Polytechnic Institute and Bielefeld University. He wishes to express his gratitude towards these institutes for their professional and warm hospitality as well as for the financial support. The National Science Foundation (Grant No. 1109356) is greatly acknowledged. He thanks X.-P. Huang and Umberto Mosco for several stimulating discussions during the visits.",

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doi = "10.1016/j.matpur.2013.01.015",

language = "English",

volume = "100",

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T1 - Parabolicity and stochastic completeness of manifolds in terms of the Green formula

AU - Grigor'yan, Alexander

AU - Masamune, Jun

N1 - Funding Information: This research was conducted when J.M. visited Worcester Polytechnic Institute and Bielefeld University. He wishes to express his gratitude towards these institutes for their professional and warm hospitality as well as for the financial support. The National Science Foundation (Grant No. 1109356) is greatly acknowledged. He thanks X.-P. Huang and Umberto Mosco for several stimulating discussions during the visits.

PY - 2013/11

Y1 - 2013/11

N2 - We present and prove new characterizations of parabolicity and stochastic completeness for a general weighted manifold M as well as the uniqueness of the Markov extensions of the Laplacian in terms of Green's formula. Moreover, we study the relationship between those properties and the singularity of M in terms of a fractal dimension and capacity.

AB - We present and prove new characterizations of parabolicity and stochastic completeness for a general weighted manifold M as well as the uniqueness of the Markov extensions of the Laplacian in terms of Green's formula. Moreover, we study the relationship between those properties and the singularity of M in terms of a fractal dimension and capacity.

KW - Conservation property

KW - Green's formula

KW - Heat kernel

KW - Markov extensions

KW - Non-explosion

KW - Parabolicity

KW - Primary

KW - Recurrence

KW - Riemannian manifold

KW - Secondary

KW - Self-adjoint extensions

KW - Stochastic completeness

KW - Weighted manifold

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U2 - 10.1016/j.matpur.2013.01.015

DO - 10.1016/j.matpur.2013.01.015

M3 - Article

AN - SCOPUS:84885381196

SN - 0021-7824

VL - 100

SP - 607

EP - 632

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

IS - 5

ER -

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

Abstract

Keywords

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