Stability of Martin boundary under non-local Feynman-Kac perturbations
The censored \(\alpha\)-stable processes (with \(\alpha\in(0,2)\)) on an open set from \( \mathbb R ^n \) were introduced by \textit{K. Bogdan, K. Burdzy} and \textit{Z.-Q. Chen} [Probab. Theory Relat. Fields 127, No. 1, 89--152 (2003; Zbl 1032.60047)]. Such symmetric Hunt processes with discontinuous sample paths can be constructed, for example, by means of the ``piecing together'' procedure of \textit{N. Ikeda, M. Nagasawa} and\ \textit{S. Watanabe} [Proc. Japan Acad. 42, 370--375 (1966; Zbl 0178.53401)], starting form a spherically symmetric \(\alpha \)-stable process. The processes studied in the present paper are (nonlocal) perturbations of censored \(\alpha\)-stable processes, whose infinitesimal generator is of the form: \[ \widetilde {\mathcal L} \varphi (x) = {\mathcal L}\varphi (x) + q(x) \varphi (x)+ \int _ {D} (e^ {F(x, y) -1} ) \varphi (y) N(x, d y) \mu _ {H} (d x) \] where \(\mathcal L\) is the infinitesimal generator of the original process; \(q\) is a Borel function on \(D\); \(F\) is a Borel function on \(D \times D\), vanishing on the diagonal; \((N, H)\) is a Levy system for the process; \(\mu_H\) is the Revuz measure of the additive functional \(H\). The \(C^ {1, 1}\) set \(D\) is an open, bounded subset of \(\mathbb R ^n\), \(n \geq 2\) satisfying a local regularity condition of the boundary (stronger than the Lipschitz property). The main result of the paper shows that the Martin boundary and the minimal Martin boundary for the above Schrödinger operator (obtained from the censored \(\alpha \)-stable process, with \(\alpha \in (1,2)\) through a nonlocal Feynmann-Kac transform), can be identified with the Euclidean boundary of \(D\). Moreover, the integral representation of nonnegative harmonic functions is proved. A similar result for censored \(\alpha \)-stable processes was established previously by the same authors \textit{Z.-Q. Chen} and \textit{P. Kim} [Probab. Theory Relat. Fields 124, No. 4, 595--610 (2002; Zbl 1016.60070)].
integral representation of nonnegative harmonic functions
scientific article; zbMATH DE number 2077978