Kolmogorov Modes and Linear Response of Jump-Diffusion Models

Abstract

We present a generalized linear response theory for mixed jump-diffusion models -- combining Gaussian and L\'evy noise interacting with nonlinear dynamics -- by deriving comprehensive response formulas accounting 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 thus quantify uncertainties in parameterized components (e.g., jump laws) or measure dynamical changes due to drift term perturbations (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, revealing 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 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 energy balance climate model, a spatially-extended model forced by a spatio-temporal $\alpha$-stable 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 climate tipping points. Our results may find applications beyond climate, and are relevant for epidemiology, biology, finance, and quantitative social sciences.