Brownian motion with polar drift

Source abstract

Consider a strong Markov process X 0 {X^0} that has continuous sample paths in R d ( d ⩾ 2 ) {R^d}(d \geqslant 2) and the following two properties. (1) Away from the origin X 0 {X^0} behaves like Brownian motion with a polar drift given in spherical polar coordinates by μ ( θ ) / 2 r \mu (\theta )/2r . Here μ \mu is a bounded Borel measurable function on the unit sphere in R d {R^d} , with average value μ ¯ \overline \mu . (2) X 0 {X^0} is absorbed at the origin. It is shown that X 0 {X^0} reaches the origin with probability zero or one as μ ¯ ⩾ 2 − d \overline \mu \geqslant 2 - d or > 2 − d > 2 - d . Indeed, X 0 {X^0} is transient to + ∞ + \infty if μ ¯ > 2 − d \overline \mu > 2 - d and null recurrent if μ ¯ = 2 − d \bar \mu = 2 - d . Furthermore, if μ ¯ > 2 − d \bar \mu > 2 - d (i.e., X 0 {X^0} reaches the origin), then X 0 {X^0} does not approach the origin in any particular direction. Indeed, there is a single Martin boundary point for X 0 {X^0} at the origin. The question of the existence and uniqueness of a strong Markov process with continuous sample paths in R d {R^d} that behaves like X 0 {X^0} away from the origin, but spends zero time there (in the sense of Lebesgue measure), is also resolved here.