Unbiased shifts of Brownian motion

Details Unbiased shifts of Brownian motion Unbiased shifts of Brownian motion

2011-12-22

arxiv.org/abs/1112.5373v1

Mathematics

Probability

Scientific paper

Let $B=(B_t)_{t\in\R}$ be a two-sided standard Brownian motion. An \emph{unbiased shift} of $B$ is a random time $T$, which is a measurable function of $B$, such that $(B_{T+t}-B_T)_{t\in\R}$ is a Brownian motion independent of $B_T$. We characterise unbiased shifts in terms of allocation rules balancing additive functionals of $B$. For any probability distribution $\nu$ on $\R$ we construct a stopping time $T\ge 0$ 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 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 $\R$. We also study moment and minimality properties of unbiased shifts.

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