 Hitting-zero classification for a Bessel process

A Bessel process of dimension $d>0$ started away from zero hits zero almost surely exactly when $d<2$. For $d\ne 2$, its scale function of a one-dimensional diffusion is $s(x)=x^{2-d}$; for $d=2$ it is $s(x)=\log x$. The boundary hitting probability from a diffusion scale function then gives the classification by taking the inner boundary to zero and the outer boundary to infinity.