## The Annals of Probability

### 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.

#### Article information

Source
Ann. Probab., Volume 42, Number 2 (2014), 431-463.

Dates
First available in Project Euclid: 24 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1393251292

Digital Object Identifier
doi:10.1214/13-AOP832

Mathematical Reviews number (MathSciNet)
MR3178463

Zentralblatt MATH identifier
1295.60093

#### Citation

Last, Günter; Mörters, Peter; Thorisson, Hermann. Unbiased shifts of Brownian motion. Ann. Probab. 42 (2014), no. 2, 431--463. doi:10.1214/13-AOP832. https://projecteuclid.org/euclid.aop/1393251292

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