Unbiased shifts of Brownian motion

Günter Last, Peter Mörters, Hermann Thorisson
2014 Annals of Probability  
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
more » ... ties 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.
doi:10.1214/13-aop832 fatcat:v2d5qadu45ecbfwnajbscij5gq