The stabilizing role of multiplicative noise in non-confining potentials

TL;DR

This paper introduces a framework for analyzing how multiplicative noise affects stochastic differential equations, revealing that increased noise concentrates probability around minima and can induce intermittency, with implications for systems near tipping points.

Contribution

The work provides a novel analytical approach to understanding the effects of multiplicative noise, including on-off intermittency and a new measure for stationarity in stochastic systems.

Findings

Increased multiplicative noise concentrates probability around minima.

Zero in the noise term causes on-off intermittency.

The framework applies to systems with double well potentials.

Abstract

We provide a simple framework for the study of parametric (multiplicative) noise, making use of scale parameters. We show that for a large class of stochastic differential equations increasing the multiplicative noise intensity surprisingly causes the mass of the stationary probability distribution to become increasingly concentrated around the minima of the multiplicative noise term, whilst under quite general conditions exhibiting a kind of intermittent burst like jumps between these minima. If the multiplicative noise term has one zero this causes on-off intermittency. Our framework relies on first term expansions, which become more accurate for larger noise intensities. In this work we show that the full width half maximum in addition to the maximum is appropriate for quantifying the stationary probability distribution (instead of the mean and variance, which are often undefined). We define a corresponding new kind of weak sense stationarity. We consider a double well potential as an example of application, demonstrating relevance to tipping points in noisy systems.