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Perturbed Brownian motions
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  • Published: July 1997

Perturbed Brownian motions

  • Mihael Perman1 &
  • Wendelin Werner2 

Probability Theory and Related Fields volume 108, pages 357–383 (1997)Cite this article

Summary.

We study `perturbed Brownian motions', that can be, loosely speaking, described as follows: they behave exactly as linear Brownian motion except when they hit their past maximum or/and maximum where they get an extra `push'. We define with no restrictions on the perturbation parameters a process which has this property and show that its law is unique within a certain `natural class' of processes. In the case where both perturbations (at the maximum and at the minimum) are self-repelling, we show that in fact, more is true: Such a process can almost surely be constructed from Brownian paths by a one-to-one measurable transformation. This generalizes some results of Carmona-Petit-Yor and Davis. We also derive some fine properties of perturbed Brownian motions (Hausdorff dimension of points of monotonicity for example).

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Authors and Affiliations

  1. University of Ljubljana, Institute for Mathematics, Physics and Mechanics, Jadranska 19, 1111 Ljubljana, Slovenia e-mail: mihael.perman@uni-lj.si, , , , ,

    Mihael Perman

  2. C.N.R.S., Laboratoire de Mathématiques, E.N.S., 45 rue d'Ulm, F-75230 Paris cedex 05, France e-mail: wwerner@dmi.ens.fr, , , , , , FR

    Wendelin Werner

Authors
  1. Mihael Perman
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  2. Wendelin Werner
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Additional information

Received: 17 May 1996 / In revised form: 21 January 1997

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Perman, M., Werner, W. Perturbed Brownian motions. Probab Theory Relat Fields 108, 357–383 (1997). https://doi.org/10.1007/s004400050113

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  • Issue Date: July 1997

  • DOI: https://doi.org/10.1007/s004400050113

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  • Mathematics Subject Classification (1991): 60J65
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