Brownian motion with polar drift

Author:
R. J. Williams

Journal:
Trans. Amer. Math. Soc. **292** (1985), 225-246

MSC:
Primary 60J60; Secondary 60J65

DOI:
https://doi.org/10.1090/S0002-9947-1985-0805961-0

MathSciNet review:
805961

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Abstract | References | Similar Articles | Additional Information

Abstract: Consider a strong Markov process that has continuous sample paths in and the following two properties.

(1) Away from the origin behaves like Brownian motion with a *polar* drift given in spherical polar coordinates by . Here is a bounded Borel measurable function on the unit sphere in , with average value .

(2) is absorbed at the origin. It is shown that reaches the origin with probability zero or one as or . Indeed, is transient to if and null recurrent if . Furthermore, if (i.e., reaches the origin), then does not approach the origin in any particular direction. Indeed, there is a single Martin boundary point for at the origin. The question of the existence and uniqueness of a strong Markov process with continuous sample paths in that behaves like away from the origin, but spends zero time there (in the sense of Lebesgue measure), is also resolved here.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1985-0805961-0

Keywords:
Brownian motion,
pole,
drift,
diffusion,
Martin boundary,
martingale,
twisted product

Article copyright:
© Copyright 1985
American Mathematical Society