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 C1,1-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 cf this paper. As an application, the Martin boundary for censored relativistic stable processes in bounded C1,1-smooth open sets is studied in detail.