Boundary Harnack principle for diffusion with jumps

Consider the operator Lb=L0+b1⋅∇+Sb2 on Rd, where L0 is a second order differential operator of non-divergence form, the drift function b1 belongs to some Kato class and Sb2f(x)≔∫Rdf(x+z)−f(x)−∇f(x)⋅z1{|z|≤1}b2(x,z)j0(z)dz,f∈Cb2(Rd).Here j0(z) is a nonnegative locally bounded function on Rd∖{0} satisfying that ∫Rd(1∧|z|2)j0(z)dz<∞ and that there are constants β∈(1,2) and c0>0 so that j0(z)≤c0|z|d+βfor|z|≤1,and b2(x,z) is a real-valued bounded function on Rd×Rd. There is conservative Feller process Xb associated with the non-local operator Lb. We derive sharp two-sided Green function estimates of Lb on bounded C1,1 domains, identify the Martin and minimal Martin boundary, and establish the Martin integral representation of Lb-harmonic functions on these domains. The latter in particular reveals how the process Xb exits a bounded C1,1 domain D, or equivalently, the structure of the harmonic measure of Lb on D, which consists of the continuously exiting term and the jump-off term. These results are then used to establish, under some mild conditions, Harnack principle and the boundary Harnack principle with explicit boundary decay rate for the operator Lb on C1,1 open sets.