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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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: Consider a strong Markov process $ {X^0}$ that has continuous sample paths in $ {R^d}(d \geqslant 2)$ and the following two properties.

(1) Away from the origin $ {X^0}$ behaves like Brownian motion with a polar drift given in spherical polar coordinates by $ \mu (\theta )/2r$. Here $ \mu $ is a bounded Borel measurable function on the unit sphere in $ {R^d}$, with average value $ \overline \mu $.

(2) $ {X^0}$ is absorbed at the origin. It is shown that $ {X^0}$ reaches the origin with probability zero or one as $ \overline \mu \geqslant 2 - d$ or $ < 2 - d$. Indeed, $ {X^0}$ is transient to $ + \infty $ if $ \overline \mu > 2 - d$ and null recurrent if $ \bar \mu = 2 - d$. Furthermore, if $ \bar \mu < 2 - d$ (i.e., $ {X^0}$ reaches the origin), then $ {X^0}$ does not approach the origin in any particular direction. Indeed, there is a single Martin boundary point for $ {X^0}$ at the origin. The question of the existence and uniqueness of a strong Markov process with continuous sample paths in $ {R^d}$ that behaves like $ {X^0}$ away from the origin, but spends zero time there (in the sense of Lebesgue measure), is also resolved here.


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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