On Kelvin Transformation

K. Bogdan () and T. Żak ()
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K. Bogdan: Polish Academy of Sciences
T. Żak: Wrocław University of Technology

Journal of Theoretical Probability, 2006, vol. 19, issue 1, 89-120

Abstract: We prove that in the Euclidean space of arbitrary dimension the inversion of the isotropic stable Lévy process killed at the origin is, after an appropriate change of time, the same stable process conditioned in the sense of Doob by the Riesz kernel. Using this identification we derive and explain transformation rules for the Kelvin transform acting on the Green function and the Poisson kernel of the stable process and on solutions of Schrödinger equation based on the fractional Laplacian. The Brownian motion and the classical Laplacian are included as a special case.

Keywords: Inversion; Kelvin transform; isotropic stable Lévy process; Brownian motion; Doob conditional process; Riesz kernel; Green function; Schrödinger equation; Laplace transform; resolvent; Primary 31C05; 60J45; Secondary 60J65; 60J75 (search for similar items in EconPapers)
Date: 2006
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DOI: 10.1007/s10959-006-0003-8

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