Scale-Invariant Diffusions: Transience and Non-Polar Points

Consider a diffusion in Rd (d ≥2) whose generator has coefficients independent of the distance to the origin. Then there is a parameter α so that the origin is almost surely hit when α< 1 and almost surely not hit when α> 1. Moreover, the process is transient to ∞ for α> 1. We identify α in terms of the diffusion coefficients and a certain invariant measure. In some special two-dimensional cases we explicitly compute the invariant measure and resolve the critical case α= 1. This work complements and extends certain results of Pinsky (1995) and Williams (1985).