= Solution
For $0<\alpha<1$, use the power transformation $Y=X^{1-\alpha}$ on the positive <stochastic interval>. The <Itô formula>, with $d[X]=X^{2\alpha}dt$, gives
$$
dY_t=(1-\alpha)\,dB_t-\frac{\alpha(1-\alpha)}{2Y_t}\,dt,\qquad Y_0=x_0^{1-\alpha}.
$$
Thus $Y_t\leq x_0^{1-\alpha}+(1-\alpha)B_t$ simultaneously for $t<T$. The right side reaches zero <almost surely> by <recurrence of one-dimensional Brownian motion>, and the same positive-path contradiction proves $T$ cannot exceed that <Brownian first-passage time>. This establishes the <finite lifetime threshold for a power diffusion> throughout $0<\alpha<1$.
At $\alpha=1$, the solution is the <geometric Brownian motion>
$$
X_t=x_0\exp(B_t-t/2).
$$
It is finite and positive at every finite time. On every compact time interval it has a positive minimum, so the hitting times of $1/n$ tend to infinity. Although the <strong law for Brownian motion> implies $X_t\to0$ as $t\to\infty$, this is not a finite lifetime. Therefore
$$
\boxed{\mathbb P(T<\infty)=\begin{cases}1,&0<\alpha<1,\\0,&\alpha=1.\end{cases}}
$$
The power transformation is a rescaled <Lamperti transform>; at $\alpha=1$ the corresponding transformation is the <logarithm>. No claim about <pathwise uniqueness> after adjoining the boundary zero is needed: the coefficients are locally <Lipschitz continuous> inside the positive domain, which is the domain of the given <maximal local solution of a stochastic differential equation>.
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