Unbiased shifts of Brownian motion

Abstract

Let $B=(B_t)_{t\in {\mathbb{R}}}$ 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 $(B_{T+t}-B_T)_{t\in {\mathbb{R}}}$ 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 ${\mathbb{R}}$ we construct a stopping time $T\ge0$ 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 ${\mathbb{R}}$. Another new result is an analogue for diffuse random measures on ${\mathbb{R}}$ of the cycle-stationarity characterisation of Palm versions of stationary simple point processes.