PERIODIC HOMOGENIZATION OF NONSYMMETRIC LÉVY-TYPE PROCESSES

In this paper we study homogenization problem for strong Markov processes on ℝ^d 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 ℝ^d. Under a proper scaling we show the scaled processes converge weakly to Lévy processes on ℝ^d. 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 ℝ^d -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,∞).