Three examples of Brownian flows on ℝ

Yves Le Jan, Olivier Raimond

Research output: Contribution to journalArticlepeer-review

Abstract

We show that the only flow solving the stochastic differential equation (SDE) on R (Formula presented.), where W+ and W- are two independent white noises, is a coalescing flow we will denote by Pdbl±. tHE FLOW Pdbl± is aWiener solution of the SDE. Moreover, K+ = (Formula presented.) is the unique solution (it is also a Wiener solution) of the SDE (Formula presented) for s < t, x ∈ ℝ and f a twice continuously differentiable function. A third flow Pdbl+ can be constructed out of the n-point motions of K+ This flow is coalescing and its n-point motion is given by the n-point motions of K+ up to the first coalescing time, with the condition that when two points meet, they stay together. We note finally that K+ = (Formula presented.).

Original languageEnglish (US)
Pages (from-to)1323-1346
Number of pages24
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Volume50
Issue number4
DOIs
StatePublished - Nov 1 2014

Keywords

  • Arratia flow or Brownian web
  • Brownian motion with oblique reflection on a wedge
  • Coalescing flow
  • Stochastic flows

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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