= Solution
The lifetime of the time-changed process is
$$
\mathcal T=C_\infty=\int_0^\infty e^{2(B_t+at)}dt.
$$
By the <strong law for Brownian motion>, $B_t/t\to0$ almost surely. If $a<0$, the exponent is eventually at most $at<0$, so the integral is finite. If $a>0$, it is eventually positive and grows linearly; if $a=0$, the <recurrence of one-dimensional Brownian motion> makes $B$ spend infinite total time in, for example, $[-1,1]$, so the integral is infinite. Consequently
$$
\mathbb P(\mathcal T<\infty)=1
\quad\Longleftrightarrow\quad a<0
\quad\Longleftrightarrow\quad d<2.
$$
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