Weak Poincar\'e inequalities in the absence of spectral gaps
Functional Analysis
2019-11-01 v5 Probability
Spectral Theory
Abstract
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 Poincar\'e inequality" (WPI), originally introduced by Liggett [Ann. Probab., 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.
Keywords
Cite
@article{arxiv.1805.08557,
title = {Weak Poincar\'e inequalities in the absence of spectral gaps},
author = {Jonathan Ben-Artzi and Amit Einav},
journal= {arXiv preprint arXiv:1805.08557},
year = {2019}
}
Comments
15 pages, improved results including precise constants in the inequalities (including Nash), expanded class of examples