Asymptotic estimates for the distribution of additive functionals of Brownian motion by the Wiener-Hopf factorization method
Let \((\tau_t,\xi_t)\) be a càdlàg process with stationary independent increments starting from \((0,0)\) with characteristic exponent \(\psi(\mu,\eta)\), The author proves a certain generalization of the Wiener-Hopf decomposition theorem for \(\psi(\mu,\eta)\), i.e. so called ``Spitzer-Rogozin identity in continuous time'', and uses it to investigate the asymptotics of the distribution of additive functionals of standard one-dimensional Brownian motion \(b(u)\). For instance, it is proved that \[ P\Biggl\{\sup_{0\leq t\leq A}\int_0^t \text{sgn}(b(u))|b(u)|^\alpha du\leq r\Biggr\} \asymp r^{1/2(\alpha+2)} A^{-1/4} \] as \(Ar^{-2/(\alpha+2)}\to \infty\). This result is compared with more precise results obtained by \textit{S. Watanabe} and \textit{Y. Isozaki} with the help of Kolmogorov diffusion and MeKean's formula in [Proc. Japan Acad., Ser. A 70, No. 9, 271-276 (1994; Zbl 0820.60066)].
scientific article; zbMATH DE number 950588