Weak convergence of censored and reflected stable processes

It is shown that if a sequence of open nn-sets DkDk increases to an open nn-set DD then reflected stable processes in DkDk converge weakly to the reflected stable process in DD for every starting point xx in DD. The same result holds for censored αα-stable processes for every xx in DD if DD and DkDk satisfy the uniform Hardy inequality. Using the method in the proof of the above results, we also prove the weak convergence of reflected Brownian motions in unbounded domains.