On the Poincaré inequality for infinitely divisible measures
type: Item
instance of: scholarly article
title: On the Poincaré inequality for infinitely divisible measures
zbMATH Open document ID: 1071.31005
author: Wilhelm Stannat
published in: Potential Analysis
publication date: 2005-06-09T00:00:00Z
review text: Let \(S\) be a locally compact separable metric space and let \(M\) denote the set of all nonnegative Radon measures on \(S\), with the \(\sigma \)-algebra generated by the vague topology. With each infinitely divisible measure \(m\) on \(M\), one associates canonically a nonnegative measure \(\Pi \) on \(M\setminus \{0\}\) such that \[ -\log \int e^{-} m(d\mu ) = +\int 1 - e^{-} \Pi (d\mu ) \] A one parameter convolution semigroup \((m_{t})_{t\geq 0}\) can be associated. A gradient form \[ {\mathcal E}_{m} (F,G) := \int _ {M} \left < \mu , \Gamma \left ( \frac{\partial F}{\partial \delta } (\mu ), \frac{\partial G}{\partial \delta } (\mu )\right ) \right > m(d\mu ) \] is well-defined. The fact that \({\mathcal E}_m\) satisfies a Poincaré inequality means that: \[ \int \left (F(\mu ) - _{m} \right )^{2} m(d\mu ) \leq \kappa {\mathcal E}_{m} (F, F) \] The main result of the paper gives a necessary and sufficient condition for Poincaré inequality to hold for bilinear forms of gradient type on \(L^ {2}\), with respect to an infinitely divisible measure. As an application, a strong law of large numbers for time-inhomogeneous one-dimensional subordinators is proved.
reviewed by: Liliana Popa
Mathematics Subject Classification ID: 31C25
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60E07
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60H07
zbMATH DE Number: 2174518
zbMATH Keywords: infinitely divisible
zbMATH Keywords: Poincaré inequality
zbMATH Keywords: convolution semigroup
zbMATH Keywords: branching process
zbMATH Keywords: subordinator
MaRDI profile type: Publication
full work available at URL: s11118-005-2610-x
OpenAlex ID: W2046060921
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DOI: 10.1007/S11118-005-2610-X
description: scientific article; zbMATH DE number 2174518