Periodic homogenization of non-symmetric Lévy-type processes

Abstract

In this paper, we study homogenization problem for strong Markov processes on R d having infinitesimal generatorsin periodic media, where Π is a non-negative measure on R d that does not charge the origin 0, satisfiesand 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 [6,7] 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 non-negative bounded continuous function that is multivariate 1-periodic in both x and z variables, and, in spherical coordinate z = (r, θ) ∈ R+ × S d−1 ,with α ∈ (0, ∞) and ̺0 being any finite measure on the unit sphere S d−1 in R d . 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, ∞).