Brownian motion with polar drift

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.