Kolmogorov modes and linear response of jump-diffusion models.

Area of Science:

• Complex Systems Dynamics
• Nonlinear Stochastic Processes
• Climate Modeling

Background:

• Linear response theory (LRT) is crucial for understanding system dynamics under perturbations.
• Mixed jump-diffusion models, combining Gaussian and Lévy noise, are vital for parameterizing unresolved scales in complex systems.
• Existing LRT frameworks often struggle with the complexities of interacting noise forcings and nonlinear dynamics.

Purpose of the Study:

• To generalize linear response theory (LRT) for mixed jump-diffusion models.
• To derive comprehensive response formulas accounting for perturbations to drift and jump laws.
• To provide a unified framework for quantifying uncertainties and measuring dynamical changes in complex systems.

Main Methods:

• Generalization of Kolmogorov operators and Green's functions for mixed jump-diffusion processes.
• Derivation of novel fluctuation-dissipation relations.
• Decomposition of system response into contributions from Kolmogorov operator eigenmodes.

Main Results:

• A generalized LRT framework applicable to nonlinear dynamics with interacting Gaussian and Lévy noise.
• New formulas for quantifying uncertainties in parameterizations and assessing dynamical changes.
• Demonstrated predictive power in El Niño-Southern Oscillation and energy balance climate models.

Conclusions:

• The generalized LRT offers a powerful tool for analyzing complex systems with mixed noise.
• The framework enhances climate modeling, prediction, and understanding of climate sensitivity and tipping points.
• Potential applications extend to epidemiology, biology, finance, and quantitative social sciences.