Chekroun, Mickaël David, Alexis Tantet, Henk A. Dijkstra, and J. David Neelin. Submitted. “Ruelle-Pollicott Resonances of Stochastic Systems in Reduced State Space. Part I: Theory”. arXiv's link Abstract

A theory of Ruelle-Pollicott (RP) resonances for stochastic systems is introduced. These resonances are defined as the eigenvalues of the generator (Kolmogorov operator) of a given stochastic system. By relying on the theory of Markov semigroups, decomposition formulas of correlation functions and power spectral densities (PSDs) in terms of RP resonances are then derived. These formulas describe, for a broad class of stochastic differential equations (SDEs), how the RP resonances characterize the decay of correlations as well as the signal's oscillatory components manifested by peaks in the PSD. It is then shown that a notion reduced RP resonances can be rigorously defined, as soon as the dynamics is partially observed within a reduced state space V. These reduced resonances are obtained from the spectral elements of reduced Markov operators acting on functions of the state space V , and can be estimated from series. When the sampling rate (in time) at which the observations are collected is either sufficiently small or large, it is shown that the reduced RP resonances approximate the RP resonances of the generator of the conditional expectation in V , i.e. the optimal reduced system in V obtained by averaging out the contribution of the unobserved variables. The approach is illustrated on a stochastic slow-fast system for which it is shown that the reduced RP resonances allow for a good reconstruction of the correlation functions and PSDs, even when the timescale separation is weak. The companions articles, Part II[TCND19a] and Part III [TCND19b], deal with further practical aspects of the theory presented in this contribution. One important byproduct consists of the diagnosis usefulness of stochastic dynamics that RP resonances offer. This is illustrated in the case of a stochastic Hopf bifurcation in Part II. There, it is shown that such a bifurcation has a clear signature in terms of the geometric organization of the RP resonances in the left half plane. This analysis provides thus an unambiguous signature of nonlinear oscillations contained in a noisy signal and that can be extracted from time series. By relying then on the theory of reduced RP resonances presented in this contribution, Part III addresses then the question of detection and characterization of such oscillations in a high-dimensional stochastic system, namely the Cane-Zebiak model of El Ni{ñ}o-Southern Oscillation subject to noise modeling fast atmospheric fluctuations.