Uniformly integrable operators and large deviations for Markov processes
type: Item
instance of: scholarly article
title: Uniformly integrable operators and large deviations for Markov processes
zbMATH Open document ID: 0957.60032
published in: Journal of Functional Analysis
publication date: 2001-03-18T00:00:00Z
review text: The notion of uniformly integrable operators on \(L^p\) spaces was introduced by the author [C. R. Acad. Sci., Paris, Sér. I 321, No. 6, 777-782 (1995; Zbl 0837.60025)]. Unlike compactness, this property is satisfied by certain infinite-dimensional Markov semigroups such as the Ornstein-Uhlenbeck semigroup, and it is also weaker than hypercontractivity. The main goal of this work is to show that uniform integrability, as hypercontractivity, implies large deviation principles. The paper starts with a study of the relations of uniform integrability with compactness and with a discussion of spectral properties of uniformly integrable operators. Sufficient conditions are given for the uniform integrability of positive, essentially irreducible semigroups of operators and their resolvents. Large deviation results for occupation functionals of Markov processes with transition semigroup \(P_t(x,dy)\) are obtained under uniform integrability assumptions on \(Q(T,\lambda) = \int_T^\infty e^{-\lambda t} P_t dt\). Other applications of uniform integrability are given to Feynman-Kac semigroups, thermodynamical limits of grand ensembles, and to semigroups perturbed by a Girsanov formula.
reviewed by: Nicolas Privault
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 47B15
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 47D07
Mathematics Subject Classification ID: 47N30
zbMATH DE Number: 1455751
zbMATH Keywords: large deviations
zbMATH Keywords: uniformly integrable operators
zbMATH Keywords: compact operators
zbMATH Keywords: hypercontractivity
zbMATH Keywords: Perron-Frobenius theorems
zbMATH Keywords: spectral gap
zbMATH Keywords: Feynman-Kac formula
zbMATH Keywords: Girsanov formula
author: Liming Wu
MaRDI profile type: Publication
full work available at URL: jfan.1999.3544
OpenAlex ID: W2088811292
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DOI: 10.1006/JFAN.1999.3544
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description: scientific article; zbMATH DE number 1455751