Critical Slowing Down in Bifurcating Stochastic Partial Differential Equations with Red Noise

The phenomenon of critical slowing down (CSD) has played a key role in the search for reliable precursors of catastrophic regime shifts. This is caused by its presence in a generic class of bifurcating dynamical systems. Simple time-series statistics such as variance or autocorrelation can be taken as proxies for the phenomenon, making their increase a useful early warning signal (EWS) for catastrophic regime shifts. However, the modelling basis justifying the use of these EWSs is usually a finite-dimensional stochastic ordinary differential equation, where a mathematical proof for the aptness is possible. Only recently has the phenomenon of CSD been proven to exist in infinite-dimensional stochastic partial differential equations (SPDEs), which are more appropriate to model real-world spatial systems. In this context, we provide an essential extension of the results for SPDEs under a specific noise forcing, often referred to as red noise. This type of time-correlated noise is omnipresent in many physical systems, such as climate and ecology. We approach the question with a mathematical proof and a numerical analysis for the linearised problem. We find that also under red noise forcing, the aptness of EWSs persists, supporting their employment in a wide range of applications. However, we also find that false or muted warnings are possible if the noise correlations are non-stationary. We thereby extend a previously known complication with respect to red noise and EWSs from finite-dimensional dynamics to the more complex and realistic setting of SPDEs.

💡 Research Summary

The paper investigates whether the phenomenon of critical slowing down (CSD), which underlies many early‑warning signals (EWS) for catastrophic regime shifts, persists in infinite‑dimensional stochastic partial differential equations (SPDEs) when the driving noise exhibits temporal correlation (so‑called red noise). While previous theoretical work on CSD has focused on finite‑dimensional stochastic ordinary differential equations (SODEs) or SPDEs forced by white noise, real‑world systems such as climate and ecological models are often subject to red noise. The authors therefore extend the mathematical foundation of CSD to SPDEs with Ornstein‑Uhlenbeck type red noise, and they complement the analysis with numerical experiments on linearised versions of the equations.

Model framework.
The spatial domain (X_1\subset\mathbb R^N) and its boundary (X_0=\partial X_1) are equipped with Hilbert spaces (H_1=L^2(X_1)) and (H_0=L^2(X_0)). Red noise is introduced via processes (\xi_j(x,t)) solving \