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

Günter Last, Wenpin Tang, Hermann Thorisson

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1 Citation (Scopus)


We consider two jointly stationary and ergodic random measures ξ and η on the real line R with equal intensities. An allocation is an equivariant random mapping from R to R. We give sufficient and partially necessary conditions for the existence of allocations transporting ξ to η. An important ingredient of our approach is a transport kernel balancing ξ and η, 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 −∞, 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.

Original languageEnglish
Pages (from-to)2286-2303
Number of pages18
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Issue number4
Publication statusPublished - Nov 2018

Bibliographical note

Publisher Copyright:
© Association des Publications de l'Institut Henri Poincaré, 2018

Other keywords

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


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