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Uniformly Integrable Operators and Large Deviations for Markov Processes

Uniformly integrable operators and large deviations for Markov processes
Authors: Wu, Liming;

Uniformly Integrable Operators and Large Deviations for Markov Processes

Abstract

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.

Keywords

Perron-Frobenius theorems, Girsanov formula, uniformly integrable operators, Transition functions, generators and resolvents, Feynman-Kac formula, large deviations, Large deviations, Feynman–Kac formula, spectral gap, Hermitian and normal operators (spectral measures, functional calculus, etc.), Applications of operator theory in probability theory and statistics, Perron–Frobenius theorems, Markov semigroups and applications to diffusion processes, hypercontractivity, Analysis, compact operators

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
75
Top 10%
Top 10%
Top 10%
hybrid
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