Abstract
In this paper we study homogenization problem for strong Markov processes on having infinitesimal generators
in periodic media, where Π is a nonnegative measure on that does not charge the origin 0, satisfies and can be singular with respect to the Lebesgue measure on . Under a proper scaling we show the scaled processes converge weakly to Lévy processes on . The results are a counterpart of the celebrated work (Asymptotic Analysis for Periodic Structures (1978) North-Holland; Ann. Probab. 13 (1985) 385–396) in the jump-diffusion setting. In particular, we completely characterize the homogenized limiting processes when is a bounded continuous multivariate 1-periodic -valued function, is a nonnegative bounded continuous function that is multivariate 1-periodic in both x and z variables and, in spherical coordinate ,
with and being any finite measure on the unit sphere in . Different phenomena occur depending on the values of α; there are five cases: , , , and .
Funding Statement
The research of Xin Chen is supported by the National Natural Science Foundation of China (No. 11871338). The research of Zhen-Qing Chen is partially supported by Simons Foundation grant 520542 and a Victor Klee Faculty Fellowship at UW. The research of Takashi Kumagai is supported by JSPS KAKENHI Grant Number JP17H01093 and by the Alexander von Humboldt Foundation. The research of Jian Wang is supported by the National Natural Science Foundation of China (Nos. 11831014 and 12071076), the Program for Probability and Statistics: Theory and Application (No. IRTL1704) and the Program for Innovative Research Team in Science and Technology in Fujian Province University (IRTSTFJ).
Acknowledgments
The authors thank Boris Solomyak for the reference [8] on almost periodic functions.
Citation
Xin Chen. Zhen-Qing Chen. Takashi Kumagai. Jian Wang. "Periodic homogenization of nonsymmetric Lévy-type processes." Ann. Probab. 49 (6) 2874 - 2921, November 2021. https://doi.org/10.1214/21-AOP1518
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