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Uniformly integrable operators and large deviations for Markov processes

机译：马尔可夫过程的一致可积算子和大偏差

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In this paper we introduce and investigate the notion of uniformly integrable operators on L-p(E, mu). 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 L-p, 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 mu-essentially irreducible Markov process, we show that the uniform integrability in L-p(mu) 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. (C) 2000 Academic Press. [References: 43]
机译：在本文中，我们介绍并研究了L-p（E，mu）上的一致可积算子的概念。它与经典的紧致性和超收缩性的关系得以展现。建立了该概念的若干结果，例如Perron-Frobenius型定理，L-p中光谱半径p的独立性，光谱半径的连续性，尤其是在不可约情况下光谱间隙的存在。我们还提出了一些无穷小的条件，以确保正半群或其分解体的一致可积性。对于一个本质上不可约的马尔可夫过程，我们表明，与过渡半群相关的某种类型的分解物在Lp（mu）中的均匀可积性暗示了带有修正的Donsker-Varadhan给出的某些速率函数的Level-3的大偏差原理熵函数。我们还证明了在对称情况下均匀可积性条件甚至变得必要。最后，我们将结果应用于Feynman-Kac半群，大合奏的热力学极限以及Girsanov公式给出的（非对称）Markov过程。 （C）2000学术出版社。 [参考：43]

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