Title: Markov processes with jump kernels decaying at the boundary.
Abstract: In this talk, we discuss pure-jump Markov processes on smooth open sets whose jumping kernels vanish at the boundary and part processes obtained by killing at the boundary or (and) by killing via the killing potential. The killing potential may be subcritical or critical. This work can be viewed as developing a general theory for non-local singular operators whose kernel vanishes at the boundary. Due to the possible degeneracy at the boundary, such operators are, in a certain sense, not uniformly elliptic. These operators cover the restricted, censored, and spectral Laplacians in smooth open sets, and much more. The main results are the boundary Harnack principle, its possible failure, and sharp two-sided Green function estimates. This is a joint work with Soobin Cho (University of Illinois), Renming Song (University of Illinois), and Zoran Vondra\v{c}ek (University of Zagreb, Croatia).
