All-time minimum of a transient Bessel process
= All-time minimum of a transient Bessel process
{title2=$\mathbb P_x(\inf_{t\geq0}X_t\leq r)=(r/x)^{\delta-2}$}
For a <Bessel process> of dimension $\delta>2$ started at $x>0$, its <scale function of a one-dimensional diffusion> gives $\mathbb P_x(\tau_r<\infty)=(r/x)^{\delta-2}$ for $0<r<x$. In particular, $\inf_{t\geq0}X_t>0$ almost surely. The minimum is attained because the process tends to infinity; the hitting formula alone already implies positivity of the infimum.