Periodic homogenization of nonsymmetric Lévy-type processes

In this paper we study homogenization problem for strong Markov processes on Rd having infinitesimal generators Lf(x)=?Rd(f(x+z)?f(x)???f(x),z?1{|z|?1})k(x,z)?(dz)+?b(x),?f(x)?,f?Cb2(Rd) in periodic media, where ? is a nonnegative measure that does not charge the origin 0, satisfies ?Rd(1?|z|2)?(dz)1}?(dz)=1{r>1}?0(d?)dr r1+? ??(0,?) ?0 being any finite unit sphere Sd?1 Different phenomena occur depending values ?; there five cases: ??(0,1), ?=1, ??(1,2), ?=2 ??(2,?).