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For the positive-domain power diffusion with $0<\alpha \le 1$, 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 givesThe 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 $\alpha =1$, $X_t=X_0{e}^{B_t-t/2}$ 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.

On an interval where $\sigma >0$ is continuously differentiable, define $F(x)={\int }^x\sigma (y{)}^{-1}dy$. The Itô formula transforms $dX=b(X)dt+\sigma (X)dB$ intoThe transformed Itô diffusion has constant noise coefficient. For a power diffusion $\sigma (x)=x^{\alpha }$, $F(x)=x^{1-\alpha }/(1-\alpha )$ when $\alpha \ne 1$, and $F(x)=\log x$ when $\alpha =1$. This simplifies comparison with Brownian motion and exposes the boundary drift.