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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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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DOI: https://doi.org/10.1007/s004400050113
- Mathematics Subject Classification (1991): 60J65