Summary
LetX h be anh-Brownian motion in the unit ballD⊃R d withh harmonic, such that the representing measure ofh is not singular with respect to the surface measure on ∂D. IfY is a continuous strong Markov process inD with the same killing distributions asX h, thenY is a time change ofX h. Similar results hold in simply connected domains inC provided with either the Martin or the Euclidean boundary.
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Vondraček, Z. A characterization ofh-Brownian motion by its exit distributions. Probab. Th. Rel. Fields 92, 41–50 (1992). https://doi.org/10.1007/BF01205235
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DOI: https://doi.org/10.1007/BF01205235