Unbiased shifts of Brownian motion

Let B = (B-t)(t epsilon 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 - B-T)(t epsilon R) is a Brownian motion independent of B-T. We characterise unbiased shifts in terms of allocation rules balancing mixtures of local times of B. For any probability distribution nu on R we construct a stopping time T >= 0 with the above properties such that B-T has distribution nu. 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.