Uniformly integrable operators and large deviations for Markov processes
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.
scientific article; zbMATH DE number 1455751
scientific article