On certain discounted arc-sine laws (Q1965871)

The joint distribution of the variable \((A_0,L_0)\) is characterized, where \[ A_0=\int _0^\infty d se^{-s}1_{[B_s>0]}=\int _0^\infty d yL_y\qquad \text{and}\qquad L_y=\int _0^\infty d_s(l^y_s)e^{-s}, \] \(\{l_t^y\); \(y\in R\), \(t\geq 0\}\) denoting the continuous family of local times of a one-dimensional Brownian motion \(B_t,t\geq 0\). The work extends earlier results by \textit{M. Baxter} and \textit{D. Williams} [Math. Proc. Camb. Philos. Soc. 111, No. 2, 387-397 (1992; Zbl 0756.60076) and ibid. 112, No. 3, 599-611 (1992; Zbl 0777.60013)].