Periodic homogenization of nonsymmetric Lévy-type processes
Xin Chen, Zhen-Qing Chen, Takashi Kumagai, Jian Wang · The Annals of Probability · 2021
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)Π(dz)+⟨b(x),∇f(x)⟩,f∈Cb2(Rd) in periodic media, where Π is a nonnegative measure on Rd that does not charge the origin 0, satisfies ∫Rd(1∧|z|2)Π(dz)1}Π(dz)=1{r>1}ϱ0(dθ)dr r1+α with α∈(0,∞) and ϱ0 being any finite measure on the unit sphere Sd−1 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,∞).