Stability of Martin boundary under non-local Feynman-Kac perturbations

Abstract

Recently the authors showed that the Martin boundary and the minimal Martin boundary for a censored (or resurrected) α-stable process Y in a bounded C 1,1 -open set D with α∈(1,2) can all be identified with the Euclidean boundary ∂D of D. Under the gaugeability assumption, we show that the Martin boundary and the minimal Martin boundary for the Schrödinger operator obtained from Y through a non-local Feynman-Kac transform can all be identified with ∂D. In other words, the Martin boundary and the minimal Martin boundary are stable under non-local Feynman-Kac perturbations. Moreover, an integral representation of nonnegative excessive functions for the Schrödinger operator is explicitly given. These results in fact hold for a large class of strong Markov processes, as are illustrated in the last section of this paper. As an application, the Martin boundary for censored relativistic stable processes in bounded C 1,1 -smooth open sets is studied in detail.