Sojourn times for the Brownian motion
Let \(\omega(\alpha)\) be a sojourn time of a standard Brownian motion spent in the set \((\alpha,\infty)\), \(\alpha>0\) in the time interval \((0,1)\), and \(\omega^*(\alpha)\) be the corresponding sojourn time of the reflecting Brownian motion \(\xi(t)=| w(t)|\). Denote by \(\tau(\alpha)\) the Brownian local time at level \(\alpha\). The author uses a combinatorial approach to determine the moment of \(\tau(\alpha)\), \(\omega(\alpha)\) and \(\omega^*(\alpha)\). For instance, he proves that \[ M_r(\alpha) =E\bigl(\omega (\alpha)\bigr)^r=m_{2r} (\alpha)/(2^rr!) \] and \[ M^*_r(\alpha)= E\bigl(\omega^* (\alpha)\bigr)^r =(r-1)!2^{-(r-1)}\sum^m_{k=1}m_{ 2r}\bigl((2k-1)\alpha\bigr)/ \bigl \{ (r-k) !(r+k-1)!\bigr\}, \] where \(m_r(\alpha)=E(\tau(\alpha))^r\), \(r=1,2,\dots\). These relations enable one to find an explicit formula for the distribution function of \(\omega^*(\alpha)\) via Laplace-Stieltjes transform.
scientific article; zbMATH DE number 1221901