Let E be a locally compact separable metric space and m be a positive Radon measure on it. Given a nonnegative function k defined on E × E off the diagonal whose anti-symmetric part is assumed to be less singular than the symmetric part, we construct an associated regular lower bounded semi-Dirichlet form η on L2(E; m) producing a Hunt process X0 on E whose jump behaviours are governed by k. For an arbitrary open subset D ⊂ E, we also construct a Hunt process XD,0 on D in an analogous manner. When D is relatively compact, we show that XD,0 is censored in the sense that it admits no killing inside D and killed only when the path approaches to the boundary. When E is a d-dimensional Euclidean space and m is the Lebesgue measure, a typical example of X0 is the stable-like process that will be also identified with the solution of a martingale problem up to an η-polar set of starting points. Approachability to the boundary ∂ D in finite time of its censored process XD,0 on a bounded open subset D will be examined in terms of the polarity of ∂ D for the symmetric stable processes with indices that bound the variable exponent α(x).
"Jump-type Hunt processes generated by lower bounded semi-Dirichlet forms." Ann. Probab. 40 (2) 858 - 889, March 2012. https://doi.org/10.1214/10-AOP633