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

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...

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Bibliographic Details
Published in:The Annals of applied probability 2022-12, Vol.32 (6), p.4611
Main Authors: Piatnitski, Andrey, Pirogov, Sergey, Zhizhina, Elena
Format: Article
Language:English
Subjects:
Boundary conditions
Convex analysis
Convexity
Deviation
Ellipticity
Generators
Infinity
Markov analysis
Markov processes
Stochastic models
Topology
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Summary: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.
ISSN:1050-5164
2168-8737
2168-8737
DOI:10.1214/22-AAP1797