# Stochastic spikes and strong noise limits of stochastic differential equations

## TL;DR

This paper studies the universal behavior of solutions to stochastic differential equations with infinitesimal repulsive perturbations, revealing convergence to spiky trajectories via time rescaling and Poisson processes.

## Contribution

It proves that under specific conditions, solutions exhibit universal spiky behavior when appropriately rescaled, using novel analytical tools like a time change and an effective Skorokhod's lemma.

## Key findings

- Solutions converge to spiky trajectories after rescaling
- The behavior is governed by an auxiliary Poisson process
- New analytical techniques are developed for first passage times

## Abstract

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.