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Curvature-dimension inequalities for non-local operators in the discrete setting

  • Adrian Spener
  • Frederic Weber
  • Rico ZacherEmail author
Article
  • 37 Downloads

Abstract

We study Bakry–Émery curvature-dimension inequalities for non-local operators on the one-dimensional lattice and prove that operators with finite second moment have finite dimension. Moreover, we show that a class of operators related to the fractional Laplacian fails to have finite dimension and establish both positive and negative results for operators with sparsely supported kernels. Furthermore, a large class of operators is shown to have no positive curvature. The results correspond to CD inequalities on locally infinite graphs.

Keywords

Gamma calculus Curvature-dimension inequality Bakry-Émery inequality Non-local operator Fractional Laplacian Markov chain Infinite graphs 

Mathematics Subject Classification

Primary 47D07 Secondary 05C63 60G22 26A33 

Notes

Acknowledgements

Adrian Spener and Rico Zacher are supported by the DFG (Project Number 355354916, GZ ZA 547/4-1). The authors thank the anonymous referee for the valuable comments, in particular the interesting observation described in Remark 5.1.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut für Angewandte AnalysisUniversität UlmUlmGermany

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