A time-change approach to Kotani's extension of Yor's formula

In [3], Kotani proved analytically that expectations for additive functional of Brownian motion {Bt,t ≥ 0} of the form E₀[f(B _t)_g(∫₀φ(B_s)ds)] have the asymptotics t^-3/2 as t → ∞ for some suitable non-negative functions φ f and g. This generalizes, in the asymptotic form, Yor's explicit formula [10] for exponential Brownian functionals. 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.