In [3], Kotani proved analytically that expectations for
additive functionals of Brownian motion $\{ B_t, t \ge 0 \}$ of the form
$$E_0 \bigg[ f(B_t)g \bigg( \int_0^t \varphi (B_s)ds \bigg) \bigg]$$
have the asymptotics $t^{-3/2}$ as $t\to \infty$ for some suitable non-negative functions $\varphi$ , $f$
and $g$ . This generalizes, in the asymptotic form, Yor's explicit formula [10] for exponential Brownian functionals.
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In the present paper, we discuss this generalization probabilistically, by using a time-change argument. We may easily see from our argument that this asymptotics $t^{-3/2}$ comes from the transition probability of 3-dimensional Bessel process.