Transporting random measures on the line and embedding excursions into Brownian motion

Abstract

We consider two jointly stationary and ergodic random measures $\xi$ and $\eta$ on the real line $\mathbb{R}$ with equal intensities. An allocation is an equivariant random mapping from $\mathbb{R}$ to $\mathbb{R}$. We give sufficient and partially necessary conditions for the existence of allocations transporting $\xi$ to $\eta$. An important ingredient of our approach is to introduce a transport kernel balancing $\xi$ and $\eta$, provided these random measures are mutually singular. In the second part of the paper, we apply this result to the path decomposition of a two-sided Brownian motion into three independent pieces: a time reversed Brownian motion on $(-\infty ,0]$, an excursion distributed according to a conditional Itô's law and a Brownian motion starting after this excursion. An analogous result holds for Bismut's excursion law.

22 pages, 2 figures. This paper is published by https://projecteuclid.org/euclid.aihp/1539849799

References in corpus (4)

• Extra heads and invariant allocations
• Invariant transports of stationary random measures and mass-stationarity
• The Slepian zero set, and Brownian bridge embedded in Brownian motion by a spacetime shift
• Optimal embeddings by unbiased shifts of Brownian motion