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

Abstract: 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.