  Motivated by studies of indirect measurements in quantum mechanics, we
investigate stochastic differential equations with a fixed point subject to an
additional infinitesimal repulsive perturbation. We conjecture, and prove for
an important class, that the solutions exhibit a universal behavior when time
is rescaled appropriately: by fine-tuning of the time scale with the
infinitesimal repulsive perturbation, the trajectories converge in a precise
sense to spiky trajectories that can be reconstructed from an auxiliary
time-homogeneous Poisson process. Our results are based on two main tools. The
first is a time change followed by an application of Skorokhod's lemma. We
prove an effective approximate version of this lemma of independent interest.
The second is an analysis of first passage times, which shows a deep interplay
between scale functions and invariant measures. We conclude with some
speculations of possible applications of the same techniques in other areas.
