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.
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This research is supported in part by NSF Grant DMS-0071486 and a RRF Grant from University of Washington
Mathematics Subject Classification (2000): Primary 31C35, 60J45, 35J10; Secondary 60J50, 60J57
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Chen, ZQ., Kim, P. Stability of Martin boundary under non-local Feynman-Kac perturbations. Probab. Theory Relat. Fields 128, 525–564 (2004). https://doi.org/10.1007/s00440-003-0317-8
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DOI: https://doi.org/10.1007/s00440-003-0317-8
Keywords
- Green function
- Martin kernel
- Martin boundary
- Feynman-Kac transform
- Schrödinger semigroup
- Non-local perturbation
- Minimal harmonic function
- Excessive function
- Martin integral representation
- Stable process
- Resurrection
- h-transform
- Conditional Markov process
- Conditional gauge
- Kato class