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Brownian motion with polar drift. (English) Zbl 0573.60072

Consider a strong Markov process \(X^ 0\) that has continuous sample paths in \(R^ d\) (d\(\geq 2)\) 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 \({\bar \mu}\).
(2) \(X^ 0\) is absorbed at the origin.
It is shown that \(X^ 0\) reaches the origin with probability zero or one as \({\bar \mu}\geq 2-d\) or \(<2-d\). Indeed, \(X^ 0\) is transient to \(+\infty\) if \({\bar \mu}>2-d\) 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.

MSC:

60J65 Brownian motion
60J60 Diffusion processes
35J15 Second-order elliptic equations
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