Perturbed Brownian motions

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).