  In this paper, we study the homogenization of a diffusion process with jumps,
that is, Feller process generated by an integro-differential operator. This
problem is closely related to the problem of homogenization of boundary value
problems arising in studying the behavior of heterogeneous media. Under the
assumptions that the corresponding generator has vanishing drift coefficient,
rapidly periodically oscillating diffusion and jump coefficients, that it
admits only "small jumps" (that is, the jump kernel has finite second moment)
and under certain additional regularity conditions, we prove that the
homogenized process is a Brownian motion. The presented results generalize the
classical and well-known results related to the homogenization of a diffusion
process.
