Wiener-Hopf factorisation of Brownian motion (Q1113212)

We study how Brownian motion behaves under time change by a fluctuating additive functional \(A_ t\), in particular letting \(\tau\) be the first passage time of \(A_ t\) to zero we compute \(P_{-x}[B_{\tau}\in dy]\) explicitly in certain cases. The calculation is not an easy one, our method uses the Désiré André relation for the overshoot of a Lévy process and depends on some elliptic function identities. This paper only considers the one-boundary case where \(A_ t\) is increasing (resp. decreasing) on the positive (resp. negative) half line.