Factorising Brownian motion at two boundaries; an example (Q910829)

The author studies in details the processes \(B^+\) and \(B^-\) obtained by time-changing a Brownian motion B by the respective inverses \(\tau^+_ t=\inf \{s:\) \(A_ s>t\}\) and \(\tau^-_ t=\inf \{s:\) \(A_ s<-t\}\) of the fluctuating additive functional \[ A_ t=\int^{t}_{0}1_{\{| B_ s| >p\}}ds-\delta^ 2\int^{t}_{0}1_{\{| B_ s| <p\}}ds. \] The kernel \(\Pi (x,dy)=P_ x(B_{\tau}\in dy)\), where \(\tau =\inf \{t>0:\) \(A_ t=0\}\), is computed using a complex martingale. See also \textit{N. Baker}, Stochastic analysis and applications, Proc. int. Conf., Swansea 1983, Lect. Notes Math. 1095, 169-186 (1984; Zbl 0557.60071).