Spiking and collapsing in large noise limits of SDEs

TL;DR

This paper investigates the behavior of one-dimensional SDEs under strong noise, revealing a phenomenon where solutions collapse to jump processes with spikes, providing rigorous proofs for these limits.

Contribution

It introduces a comprehensive analysis of the strong noise limit in 1D SDEs, including the first rigorous proof of the collapsing and spike phenomena.

Findings

Solutions converge to jump processes with spikes under large noise

The limiting behavior is rigorously characterized for a broad class of diffusions

The results connect noise-induced collapse to metastability phenomena

Abstract

We analyze the strong noise limit of one-dimensional stochastic differential equations (SDEs). Our initial motivation comes from continuous measurements of open quantum systems. In this context, Bauer, Bernard and Tilloy pointed out an intriguing behavior. As the noise grows larger, the solutions exhibit locally a collapsing, that is to say, converge to pure jump processes very reminiscent of a metastability phenomenon. But surprisingly the limiting jump process is decorated by a spike process. We give a precise meaning to the convergence and completely prove these statements for a large class of one-dimensional diffusions, thanks to a robust strategy of proof.