Summary
LetV=(V α)α≧0 be a (not necessarily sub-Markovian) resolvent such that the kernelV α for some α≧0 is compact and irreducible. We prove the following general gauge theorem: If there exists at least oneV-excessive function which is notV-inviriant, thenV 0 is bounded.
This result will be applied to resolventsU M arising from perturbation of sub-Markovian right resolventsU by multiplicative functionalsM (not necessarily supermartingale), for instance, by Feynman-Kac functionals. Among others, this leads to an extension of the gauge theorem of Chung/Rao and even of one direction of the conditional gauge theorem of Falkner and Zhao.
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Sturm, KT. Gauge theorems for resolvents with application to Markov processes. Probab. Th. Rel. Fields 89, 387–406 (1991). https://doi.org/10.1007/BF01199785
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DOI: https://doi.org/10.1007/BF01199785