= Finite-time access to zero for Bessel dimensions below two
{title2=$0<a<\tfrac12,\quad\mathbb P_x(T_0<\infty)=1$}
For $dX=dB+aX^{-1}dt$, 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 $\frac12v_R''+(a/x)v_R'=-1$ on $(0,R)$. Optional stopping at $T_\varepsilon\wedge T_R$ gives $\mathbb E(T_\varepsilon\wedge T_R)\leq v_R(x)$ uniformly as $\varepsilon\downarrow0$. 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.
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