# SCS

## Seminar

### Unbiased shifts of Brownian Motion

Seminar Room 1, Newton Institute

#### Abstract

Let $B = (B_t)_{t\in\mathbb{R}}$ be a two-sided standard Brownian motion. Let $T$ be a measurable function of $B$. Call $T$ an \emph{unbiased shift} if $(B_{T+t}-B_T)_{t\in\mathbb{R}}$ is a Brownian motion independent of $B_T$. We characterize unbiased shifts in terms of allocation rules balancing local times of $B$. For any probability distribution $\nu$ on $\mathbb{R}$ we construct a nonnegative stopping time $T$ with the above properties such that $B_T$ has distribution $\nu$. In particular, we show that if we travel in time according to the clock of local time at zero we always see a two-sided Brownian motion. A crucial ingredient of our approach is a new theorem on the existence of allocation rules balancing jointly stationary diffuse random measures on $\mathbb{R}$. We also study moment and minimality properties of unbiased shifts.#### Video

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