TY - JOUR
T1 - Global properties of Dirichlet forms in terms of Green’s formula
AU - Haeseler, Sebastian
AU - Keller, Matthias
AU - Lenz, Daniel
AU - Masamune, Jun
AU - Schmidt, Marcel
N1 - Funding Information:
M.K. and D.L. gratefully acknowledge partial support from German Research Foundation (DFG). J.M. gratefully acknowledges partial support from the Japan–Korea Basic Scientific Cooperation Program “Non-commutative Stochastic Analysis: New Prospects of Quantum White Noise and Quantum Walk” (2015–2016) and Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 26400062 (2014–2016), “Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers”. M.S. has been financially supported by the Graduiertenkolleg 1523/2: Quantum and gravitational fields and by the European Science Foundation (ESF) within the project Random Geometry of Large Interacting Systems and Statistical Physics. The authors gratefully acknowledge the constructive comments on the paper by the anonymous referee, which improved the quality of the paper significantly.
Publisher Copyright:
© 2017, Springer-Verlag GmbH Germany.
PY - 2017/10/1
Y1 - 2017/10/1
N2 - We study global properties of Dirichlet forms such as uniqueness of the Dirichlet extension, stochastic completeness and recurrence. We characterize these properties by means of vanishing of a boundary term in Green’s formula for functions from suitable function spaces and suitable operators arising from extensions of the underlying form. We first present results in the framework of general Dirichlet forms on σ-finite measure spaces. For regular Dirichlet forms our results can be strengthened as all operators from the previous considerations turn out to be restrictions of a single operator. Finally, the results are applied to graphs, weighted manifolds, and metric graphs, where the operators under investigation can be determined rather explicitly, and certain volume growth criteria can be (re)derived.
AB - We study global properties of Dirichlet forms such as uniqueness of the Dirichlet extension, stochastic completeness and recurrence. We characterize these properties by means of vanishing of a boundary term in Green’s formula for functions from suitable function spaces and suitable operators arising from extensions of the underlying form. We first present results in the framework of general Dirichlet forms on σ-finite measure spaces. For regular Dirichlet forms our results can be strengthened as all operators from the previous considerations turn out to be restrictions of a single operator. Finally, the results are applied to graphs, weighted manifolds, and metric graphs, where the operators under investigation can be determined rather explicitly, and certain volume growth criteria can be (re)derived.
KW - 31C25
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U2 - 10.1007/s00526-017-1216-7
DO - 10.1007/s00526-017-1216-7
M3 - Article
AN - SCOPUS:85026873724
SN - 0944-2669
VL - 56
JO - Calculus of Variations and Partial Differential Equations
JF - Calculus of Variations and Partial Differential Equations
IS - 5
M1 - 124
ER -