= Finite lifetime threshold for a power diffusion
{title2=$\mathbb P(T<\infty)=\mathbf1_{\{\alpha<1\}}\quad(0<\alpha\leq1)$}
For the positive-domain <power diffusion> with $0<\alpha\leq1$, assuming the lifetime is the limit of the hitting times of $1/n$, it is finite <almost surely> exactly when $\alpha<1$. For $\alpha<1$, the <Itô formula> gives
$$
d(X^{1-\alpha})=(1-\alpha)dB-\frac{\alpha(1-\alpha)}{2X^{1-\alpha}}dt.
$$
The negative drift bounds this positive process above by $X_0^{1-\alpha}+(1-\alpha)B$, whose first hit of zero is finite by <recurrence of one-dimensional Brownian motion>. The lifetime must precede that hit. At $\alpha=1$, $X_t=X_0e^{B_t-t/2}$ is positive and finite on every compact time interval, so the lifetime is infinite. Approaching zero only as $t\to\infty$ is not a finite boundary lifetime.
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