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

TL;DR

This paper studies conditions for transporting one random measure to another on the real line and applies these results to decompose Brownian motion into independent segments, including excursions.

Contribution

It introduces new sufficient and necessary conditions for measure transport and applies them to Brownian motion path decomposition involving excursions.

Findings

Established conditions for measure transport between random measures.

Applied transport results to decompose Brownian motion into independent parts.

Extended the approach to Bismut's excursion law.

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 It\^o's law and a Brownian motion starting after this excursion. An analogous result holds for Bismut's excursion law.