Weak Poincaré Inequalities in the Absence of Spectral Gaps

For generators of Markov semigroups which lack a spectral gap, it is shown how bounds on the density of states near zero lead to a so-called weak Poincaré inequality (WPI), originally introduced by Liggett (Ann Probab 19(3):935–959, 1991). Applications to general classes of constant coefficient pseu...

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Bibliographic Details
Published in:Annales Henri Poincaré 2020-02, Vol.21 (2), p.359-375
Main Authors: Ben-Artzi, Jonathan, Einav, Amit
Format: Article
Language:English
Subjects:
Classical and Quantum Gravitation
Decay rate
Dynamical Systems and Ergodic Theory
Elementary Particles
Mathematical and Computational Physics
Mathematical Methods in Physics
Physics
Physics and Astronomy
Quantum Field Theory
Quantum Physics
Relativity Theory
Theoretical
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Summary:For generators of Markov semigroups which lack a spectral gap, it is shown how bounds on the density of states near zero lead to a so-called weak Poincaré inequality (WPI), originally introduced by Liggett (Ann Probab 19(3):935–959, 1991). Applications to general classes of constant coefficient pseudodifferential operators are studied. Particular examples are the heat semigroup and the semigroup generated by the fractional Laplacian in the whole space, where the optimal decay rates are recovered. Moreover, the classical Nash inequality appears as a special case of the WPI for the heat semigroup.
ISSN:1424-0637
1424-0661
DOI:10.1007/s00023-019-00858-4