Uniformly Integrable Operators and Large Deviations for Markov Processes

Abstract

In this paper we introduce and investigate the notion of uniformly integrable operators on Lp(E, μ). Its relations to classical compactness and hypercontractivity are exhibited. Several consequences of this notion are established, such as Perron– Frobenius type theorems, independence on p of the spectral radius in Lpp, continuity of spectral radius, and especially the existence of spectral gap in the irreducible case. We also present some infinitesimal criteria ensuring the uniform integrability of a positive semigroup or of its resolvent. For a μ-essentially irreducible Markov process, we show that the uniform integrability in Lp(μ) of some type of resolvent associated with the transition semigroup implies the large deviation principle of level-3 with some rate function given by a modified Donsker–Varadhan entropy functional. We also prove that the uniform integrability condition becomes even necessary in the symmetric case. Finally, we present several applications of our results to Feynman–Kac semigroups, to the thermodynamical limits of grand ensembles, and to (non-symmetric) Markov processes given by Girsanov's formula.