Kolmogorov modes and linear response of jump-diffusion models

We present a generalization of linear response theory for mixed jump-diffusion models-which combine both Gaussian and Lévy noise forcings that interact with the nonlinear dynamics-by deriving a comprehensive response formulas that accounts for perturbations to both the drift term and the jumps law. This class of models is particularly relevant for parameterizing the effects of unresolved scales in complex systems. Our formulas are thus particularly relevant to quantify uncertainties in either what needs to be parameterized (e.g. the jumps law), or to measure dynamical changes due to perturbations of the drift term (e.g. parameter variations). By generalizing the concepts of Kolmogorov operators and Green's functions, we obtain new forms of fluctuation-dissipation relations. The resulting response is decomposed into contributions from the eigenmodes of the Kolmogorov operator, providing a fresh look into the intimate relationship between a system's natural and forced variability. We demonstrate the theory's predictive power with two distinct climate-centric applications. First, we apply our framework to a paradigmatic El Niño-Southern Oscillation (ENSO) model subject to state-dependent jumps and additive white noise, showing how the theory accurately predicts the system's response to perturbations and how Kolmogorov modes can be used to diagnose its complex time variability. In a second, more challenging application, we use our linear response theory to perform accurate climate change projections in the Ghil-Sellers (GS) energy balance climate model, which is a spatially-extended model forced here by a spatio-temporal αstable process. This work provides a comprehensive approach to climate modeling and prediction that enriches Hasselmann's program, with implications for understanding climate sensitivity, detection and attribution of climate change, and assessing the risk of climate tipping points. Our results may find applications beyond the realm of climate, and seem of relevance for epidemiology, biology, finance, and quantitative social sciences, among others.