Unbiased shifts of Brownian motion

Let B = (B(t) :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 (B(T +t) − B(T) :t∈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 ν on R we construct a stopping time T ≥ 0 with the above properties such that B(T) has distribution ν. 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.