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 Lévy 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 $α$-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.

Figures (10)

• Figure 1: Jin model's limit cycle (black curve) and a few of its isochrons (blue curves). Recall that an isochron gives the locus of all points sharing the same asymptotic phase when converging towards the limit cycle guckenheimer1975isochrons. The model parameters are those used in jin1997equatorial and listed in Table \ref{['Table_model_param']} with $\delta=0.6$. Throughout this study, the parameter $\delta$ is used as our bifurcation parameter since it relates to the ocean-atmosphere coupling parameter $\mu$ used by Jin in jin1997equatorial ($\mu=\frac{2}{3}(1+\delta)$ in this study, see Table \ref{['Table_model_param']}).
• Figure 2: A realization of the comb noise $f(t)$ used in Eq. \ref{['Eq_F']}.
• Figure 3: Stochastic strange attractor. Here, the snapshots attractors shown in panels A, B, and C are computed from the stochastic Jin model (Eq. \ref{['Eq_Jin_stoch']}). Their time instants at which they are computed out of $10^6$ data points, are marked by the vertical dashed lines shown in panels D and F. These latter panels show two solutions emanating from two distinct initial conditions (blue and red curves), but driven by the same noise path, in both the $h$- and $T$-variable. One observes here on-off synchronization phenomenon illustrating a well-known signature of stochastic chaos csg11. The corresponding locations of these time series on the snapshot attractors shown in panels A, B, and C, are marked by blue and red dots. The model's parameters values used in these computations are those given in Table \ref{['Table_model_param']} with $\delta=0.5$ while the noise parameters are given in Table \ref{['Table_noise_param']} (Case B).
• Figure 4: Jin model's vector field dependence on $\delta$. A few streamlines of the deterministic Jin model shown for different values of $\delta$. The unstable steady states are shown by red dots. A bifurcation of such steady states occurs between $\delta=0.6$ and $\delta=0.7$, while the stable limit cycle remains.
• Figure 5: RP resonances and decay rates. Here, shown for the Kolmogorov operator $K_\sigma$(Eq. \ref{['Eq_Kolmo']}) in a purely diffusive case, $D=0$; Case A in Table \ref{['Table_noise_param']}. The RP resonances share features (parabola and triangular shapes) exhibited by an Hopf normal form in presence of small additive white noise Tantet_al_Hopf. The light-blue line in the right panel corresponds to the non-decaying eigenvalue $\lambda_1=0$. The vertical blue line in the left panel indicates the imaginary axis.

Theorems & Definitions (3)

• Remark 5.1
• Remark 5.2
• Remark 5.3