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The paper deals with a family of jump Markov process defined in a medium witha periodic or locally periodic microstructure. We assume that the generator ofthe process is a zero order convolution type operator with rapidly oscillatinglocally periodic coefficient and, under natural ellipticity and localizationconditions, show that the family satisfies the large deviation principle in thepath space equipped with Skorokhod topology. The corresponding rate function isdefined in terms of a family of auxiliary periodic spectral problems.

Large Deviations for Markov jump processes in periodic and locally periodic environments