On the Occupation Time on the Half Line of Pinned Diffusion Processes

Abstract

The aim of the present paper is to generalize Lévy’s result of the occupation e time on the half line of pinned Brownian motion for pinned diffusion processes. An asymptotic behavior of the distribution function at the origin of the occupation time${\Gamma }_{+}(t)$ and limit theorem for the law of the fraction ${\Gamma }_{+}(t)/t$ when $t\to \infty$ are studied. An expression of the distribution function by the Riemann–Liouville fractional integral for pinned skew Bessel diffusion processes is also obtained. Krein’s spectral theory and Tauberian theorem play important roles in the proofs.