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December 2022 Large deviations for Markov jump processes in periodic and locally periodic environments
Andrey Piatnitski, Sergey Pirogov, Elena Zhizhina
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Ann. Appl. Probab. 32(6): 4611-4641 (December 2022). DOI: 10.1214/22-AAP1797


The paper deals with a family of jump Markov process defined in a medium with a periodic or locally periodic microstructure. We assume that the generator of the process is a zero order convolution type operator with rapidly oscillating locally periodic coefficient and, under natural ellipticity and localization conditions, show that the family satisfies the large deviation principle in the path space equipped with Skorokhod topology. The corresponding rate function is defined in terms of a family of auxiliary periodic spectral problems. It is shown that the corresponding Lagrangian is a convex function of velocity that has a superlinear growth at infinity. However, neither the Lagrangian nor the corresponding Hamiltonian need not be strictly convex, we only claim their strict convexity in some neighbourhood of infinity. It then depends on the profile of the generator kernel whether the Lagrangian is strictly convex everywhere or not.


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Andrey Piatnitski. Sergey Pirogov. Elena Zhizhina. "Large deviations for Markov jump processes in periodic and locally periodic environments." Ann. Appl. Probab. 32 (6) 4611 - 4641, December 2022.


Received: 1 September 2021; Published: December 2022
First available in Project Euclid: 6 December 2022

Digital Object Identifier: 10.1214/22-AAP1797

Primary: 60F10 , 60J76
Secondary: 47A10

Keywords: jump Markov processes , large deviations , locally periodic microstructure , spectral properties of convolution type operators

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.32 • No. 6 • December 2022
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