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 BT 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.