In this article, we consider the problem of periodic homogenization of a Feller process generated by a pseudo-differential operator, the so-called Lévy-type process. Under the assumptions that the generator has rapidly periodically oscillating coefficients, and that it admits “small jumps” only (that is, the jump kernel has finite second moment), we prove that the appropriately centered and scaled process converges weakly to a Brownian motion with covariance matrix given in terms of the coefficients of the generator. The presented results generalize the classical and well-known results related to periodic homogenization of a diffusion process.

中文翻译：

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Lévy型过程的周期性均质化，跳跃很小

在本文中，我们考虑了由伪微分算子（即所谓的Lévy型过程）生成的Feller过程的周期性均质化问题。假设发生器具有快速的周期性振荡系数，并且仅允许“小跳动”（即，跳动核具有有限的第二矩），我们证明了适当居中和按比例缩放的过程弱收敛为具有根据发生器的系数给出的协方差矩阵。提出的结果概括了与扩散过程的周期性均质化有关的经典和众所周知的结果。