Unbiased shifts of Brownian motion

TL;DR

This paper characterizes unbiased shifts of Brownian motion, constructs such shifts for any given distribution, and introduces new theorems on allocation rules and stationarity for random measures.

Contribution

It provides a comprehensive characterization of unbiased shifts, constructs them for arbitrary distributions, and develops new theorems on allocation rules and stationarity of random measures.

Findings

Constructed unbiased shifts for any probability distribution.

Characterized unbiased shifts via allocation rules balancing local times.

Proved new theorems on allocation rules and stationarity of random measures.

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.