= Divergence of a driftless Brownian exponential clock
{c}
{title2=$\int_0^\infty e^{cB_s}ds=\infty\quad\text{almost surely}$}
For any real constant $c$, one-dimensional <Brownian motion> repeatedly returns to zero. Consider successive segments from zero until exiting $(-1,1)$, separated by returns to zero. Their capped durations are independent identically distributed positive bounded random variables by the <Strong Markov property>, so their sum diverges by the <strong law of large numbers>. On each such segment $e^{cB_s}\geq e^{-|c|}$. Thus the exponential clock has infinite lifetime, despite arbitrarily deep negative excursions.
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