Unbiased shifts of Brownian motion

Let B=(Bt)t.R be a two-sided standard Brownian motion. An unbiased shift of B is a random time T, which is a measurable function of B, such that (BT+t.BT)t.R is a Brownian motion independent of BT. We characterise unbiased shifts in terms of allocation rules balancing mixtures of local times of B. For any probability distribution . on R we construct a stopping time T.0 with the above properties such that BT has distribution .. We also study moment and minimality properties of unbiased shifts. A crucial ingredient of our approach is a new theorem on the existence of allocation rules balancing stationary diffuse random measures on R. Another new result is an analogue for diffuse random measures on R of the cycle-stationarity characterisation of Palm versions of stationary simple point processes.