A characterization of Brownian motion in a Lipschitz domain by its killing distributions
The Blumenthal-Getoor-McKean theorem states that if X and Y are two standard Markov processes with identical hitting distributions for the compact sets, then there is a strictly increasing continuous additive functional A of X such that \(X_ t=Y_{A(t)}\) for all t. This paper presents the following refinement of the B-G-MK theorem in the case when X is a Brownian motion in a bounded Lipschitz domain D in \({\mathbb{R}}^ n\), and Y a normal strong Markov process in D with continuous path up to its lifetime \({\tilde \zeta}\): if \(X_{\zeta -}\) and \(Y_{{\tilde \zeta}-}\) have the same law for every starting point, then the conclusion of the B- G-MK theorem holds. The idea for the proof is to show that hitting distributions for compact sets are the same for both processes, and then apply the B-G-MK theorem.
scientific article; zbMATH DE number 4192833