LARGE DEVIATIONS FOR MARKOV JUMP PROCESSES IN PERIODIC AND LOCALLY PERIODIC ENVIRONMENTS

• 成果类型: Article
• 署名作者: Piatnitski, Andrey; Pirogov, Sergey; Zhizhina, Elena
• 署名单位: UiT The Arctic University of Tromso; Kharkevich Institute for Information Transmission Problems of the RAS; Russian Academy of Sciences
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1797
• 发表日期: 2022
• 页码: 4611-4641
• 关键词: homogenization

摘要

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 corre-sponding 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 con-vex, 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.