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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.

Because $v\in L^1(0,1)$, its indefinite integral $V$ is bounded, so $e^{-V}$ is bounded above and away from zero. Hence $\psi$ extends continuously and strictly increasingly to $[0,1]$. The quadratic-variation clock of $\psi (X)$ iswhose rate is bounded above and away from zero before exit. The Dambis-Dubins-Schwarz theorem therefore identifies $\psi (X)$, up to an equivalent time change, with Brownian motion in the bounded interval $(0,\psi (1))$; in particular, $\mathcal{T}<\infty$ almost surely.

The bounded stopped local martingale $\psi (X_{t\wedge \mathcal{T}})$ is a martingale. If $q={\mathbb{P}}_x(X_{\mathcal{T}}=0)$, the optional sampling theorem givesThereforeThis is the boundary hitting probability from a diffusion scale function.

For the Bessel processthe scale function of a one-dimensional diffusion is $s(x)=x^{2-d}$ when $d\ne 2$. If $0<ϵ<x<R$, the boundary hitting probability from a diffusion scale function gives

If $d<2$, then $s(0)=0$ and $s(R)\to \infty$. Letting $ϵ\downarrow 0$ and then $R\to \infty$ shows that the process cannot escape to infinity before reaching zero. The exit time from each bounded interval is finite almost surely, so $T_0<\infty$ almost surely.

If $d>2$, then $s(ϵ)\to \infty$ in absolute value as $ϵ\downarrow 0$. Equivalently,Thus the process does not hit zero. The borderline case $d=2$ has scale function $\log x$ and also does not hit zero. This is the Hitting-zero classification for a Bessel process.