Exponential Brownian-to-Bessel time change

For $Z_t=e^{B_t+at}$, use the clock $C_t={\int }_0^t{Z}_s^{2}ds$. After the inverse time change, the Dambis-Dubins-Schwarz theorem givesso the time-changed process is a Bessel process of dimension $d=2a+2$. Its lifetime is $C_{\infty }$, which is finite almost surely exactly when $d<2$.

The strong law for Brownian motion gives $B_s/s\to 0$. For every $0<\epsilon <\mu$, eventually $e^{(2\mu -2\epsilon )s}\le e^{2B_s+2\mu s}\le e^{(2\mu +2\epsilon )s}$. Integrating these bounds, taking logarithms and dividing by $t$ squeezes the growth rate between $2\mu -2\epsilon$ and $2\mu +2\epsilon$. Let $\epsilon \downarrow 0$. The corresponding inverse clock grows like $(\log t)/(2\mu )$.