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In [3], Kotani proved analytically that expectations for additive functionals of Brownian motion $ {B_t, t geq 0} $ of the form $ E_0[f(B_t)g( int_0^t \varphi(B_s)ds)] $ have the asymptotics $ t^{-3}/...2 $ as $ t \rigtarrow infty $ for some suitable non-negative functions $ \varphi $, $ f $ and $ g $. This generalizes, in the asymptotic form, Yor’s explicit formula [9] 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.続きを見る
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