Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Günter Last, Wenpin Tang, and Hermann Thorisson

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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 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 measure and a Brownian motion starting after this excursion. An analogous result holds for Bismut’s excursion measure.


On considère deux mesures aléatoires conjointement stationnaires et ergodiques $\xi$ et $\eta$ sur la droite réelle $\mathbb{R}$ et d’intensités égales. Une allocation est une carte aléatoire équivariante de $\mathbb{R}$ dans $\mathbb{R}$. On donne des conditions suffisantes et partiellement nécessaires pour l’existence d’allocations transportant $\xi$ sur $\eta$. Un ingrédient important de notre approche est un noyau de transport équilibrant $\xi$ et $\eta$, sous la condition que ces mesures aléatoires sont mutuellement singulières. Dans la deuxième partie de cet article, on applique ce résultat à la décomposition des trajectoires d’un mouvement brownien symétrique en trois parties indépendantes: un mouvement brownien renversé dans le temps sur $(-\infty,0]$, une excursion distribuée selon une mesure conditionnelle d’Itó, et un mouvement brownien après cette excursion. Un résultat analogue est valable pour la measure d’excursion de Bismut.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 4 (2018), 2286-2303.

Received: 16 August 2016
Revised: 3 August 2017
Accepted: 25 October 2017
First available in Project Euclid: 18 October 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G57: Random measures 60G55: Point processes
Secondary: 60G60: Random fields

Stationary random measure Point process Allocation Invariant transport Palm measure Shift-coupling Brownian motion Excursion theory


Last, Günter; Tang, Wenpin; Thorisson, Hermann. Transporting random measures on the line and embedding excursions into Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 4, 2286--2303. doi:10.1214/17-AIHP871.

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