On explicit occupation time distributions for Brownian processes (Q1613019)

Explicit expressions for the sojourn time distribution of Brownian bridge, excursion and meander are derived using Kac's formula and results of \textit{R. T. Durrett, D. L. Iglehart} and \textit{D. R. Miller} [Ann. Probab. 5, 117-129 (1977; Zbl 0356.60034)]. Such approach differs from the method used by \textit{L. Takács} [Methodol. Comput. Appl. Probab. 1, No. 1, 7-28 (1999; Zbl 0978.60094)], where the first two distributions were obtained using weak convergence of simple random walk. Also the author presents an explicit ~expression for sojourn time distribution of Brownian meander, which was not contained in Takács' above mentioned paper, but in [\textit{L. Takács}, J. Appl. Math. Stochastic Anal. 11, No. 3, 231-246 (1998; Zbl 0916.60073)]. All three distribution functions can be expressed in the terms of transition density of Brownian motion.