Stochastic Spikes and Strong Noise Limits of Stochastic Differential Equations

Motivated by studies of indirect measurements in quantum mechanics, we investigate the effect, in the strong noise limit, of adding an infinitesimal repulsive perturbation to a stochastic differential equation with an attractive fixed point. We conjecture, and prove for an important subclass, 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.