PERIODIC HOMOGENIZATION OF NONSYMMETRIC LÉVY-TYPE PROCESSES

Xin Chen*, Zhen Qing Chen, Takashi Kumagai, Jian Wang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper we study homogenization problem for strong Markov processes on ℝd having infinitesimal gener (formula presented) in periodic media, where Π is a nonnegative measure on d that does not charge the origin 0, satisfies (formula presented) and can be singular with respect to the Lebesgue measure on ℝd. Under a proper scaling we show the scaled processes converge weakly to Lévy processes on ℝd. 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 b(x) is a bounded continuous multivariate 1-periodic ℝd -valued function, k(x,z) is a nonnegative bounded continuous function that is multivariate 1-periodic in both x and z variables and, in spherical coordinate (formula presented) (formula presented) with (formula presented) and e0 being any finite measure on the unit sphere (formula presented) in Rd. Different phenomena occur depending on the values of α; there are five cases: α ∈(0, 1), α = 1, α ∈ (1, 2), α = 2 and α ∈ (2,∞).

Original languageEnglish
Pages (from-to)2874-2921
Number of pages48
JournalAnnals of Probability
Volume49
Issue number6
DOIs
Publication statusPublished - 2021
Externally publishedYes

Keywords

  • Corrector
  • Homogenization
  • Lévy-type process
  • Martingale problem

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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