Finite-time access to zero for Bessel dimensions below two ($0<a<\frac{1}{2},{\mathbb{P}}_x(T_0<\infty )=1$)

For $dX=dB+a{X}^{-1}dt$, put $\gamma =1-2a\in (0,1)$ and $v_R(x)=(R^{2-\gamma }{x}^{\gamma }-x^2)/(1+2a)$. It is nonnegative on $[0,R]$, vanishes at both endpoints and solves $\frac{1}{2}{v}_R^{\prime \prime }+(a/x)v_R^{\prime }=-1$ on $(0,R)$. Optional stopping at $T_{\epsilon }\wedge T_R$ gives $\mathbb{E}(T_{\epsilon }\wedge T_R)\le v_R(x)$ uniformly as $\epsilon \downarrow 0$. Thus the limiting exit to zero or $R$ occurs in finite time, not merely as an asymptotic approach to zero. Scale probabilities give ${\mathbb{P}}_x(T_0<T_R)=1-(x/R{)}^{\gamma }$; letting $R\to \infty$ gives almost sure finite hitting of zero.