A time-change approach to Kotani’s extension of Yor’s formula. (English) Zbl 1093.60052

Author’s abstract: 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 \(t\to\infty\) for some suitable nonnegative functions \(\varphi,~f\) and \(g.\) This generalizes, in the asymptotic form, Yor’s explicit formula 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.


60J65 Brownian motion
60J35 Transition functions, generators and resolvents
60J55 Local time and additive functionals
60F99 Limit theorems in probability theory
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