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Journal of Theoretical Probability

, Volume 32, Issue 1, pp 90–105 | Cite as

On a Coupling of Solutions to the Interface Stochastic Differential Equation on a Star Graph

  • Hatem HajriEmail author
  • Marc Arnaudon
Article
  • 683 Downloads

Abstract

Inspired by Tsirelson’s proof of the non-Brownian character of Walsh Brownian motion filtration on three or more rays, we prove some results on a particular coupling of solutions to the interface stochastic differential equation on a star graph, recently introduced in Hajri and Raimond (Stoch Process Appl 126:33–65, 2016). This coupling consists of two solutions which are independent given the driving Brownian motion. As a consequence, we deduce that if the star graph contains three or more rays, the argument of the solution at a fixed time is independent of the driving Brownian motion.

Keywords

Tsirelson theorem Non-Brownian filtration Walsh Brownian motion Stochastic equations on graphs 

Mathematics Subject Classification (2010)

60H05 60H10 60J25 60G48 

Notes

Acknowledgements

We thank the reviewer for his/her thorough review and highly appreciate the comments and suggestions which significantly improved two versions of the paper. In particular, the reviewer suggested the present construction of the perturbation process instead of a stochastic flow-based construction given in the first version which used the semigroup \(Q^r\) (13).

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Institut VEDECOMVersaillesFrance
  2. 2.Institut de Mathématiques de Bordeaux UMR 5251TalenceFrance

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