In problems of reducing complex stochastic systems, for example in molecular dynamics or climate modeling, it is important to understand which properties of the full dynamics are preserved in a reduced state space. Traditional approaches compare moments or probability densities but do not account for temporal correlations. This work follows a line of research where correlation functions and power spectra are used as metrics to assess the quality of reduced models. Related work includes studies on quantum analogs of the Mpemba effect, quantum control, and hidden evolution paths under measurements, but this paper differs in its focus on classical stochastic differential equations and spectral analysis of their generators.
The authors use the theory of Markov semigroups and the spectral theory of strongly continuous semigroups. Ruelle–Pollicott resonances are defined as isolated eigenvalues of the generator of a Markov semigroup in L2 with respect to the invariant measure. For reduced resonances, a family of reduced Markov operators acting on functions of the observed part of the state is introduced. These operators are constructed via disintegration of the invariant measure. It is shown that the pseudogenerator of this family approximates the conditional expectation generator for small or large time lags. For numerical estimation of reduced resonances, the maximum likelihood method is used: a transition matrix between cells of a partition of the observation space is constructed, and the eigenvalues of the matrix are converted into resonances via the formula λ = (log|ζ| + i arg ζ)/τ.
For a stochastic slow–fast system with cubic nonlinearity, the authors numerically estimated reduced resonances in three regimes: with strong time-scale separation (two cases) and without it (one case). In the strong separation cases, reduced resonances almost exactly match the resonances of the reduced system on the slow manifold, and the reconstructed correlation functions and power spectra almost coincide with sample estimates. In the case without time-scale separation, the slow-manifold resonances differ noticeably from the reduced resonances, yet the latter still allow nearly perfect recovery of the autocorrelation function. This indicates that conditional expectation remains a good approximation even under weak scale separation, unlike the slow manifold.
This work provides a rigorous mathematical foundation for using Ruelle–Pollicott resonances in the analysis of stochastic systems from partial observations. Unlike neighboring works that focus on quantum systems or deterministic flows, here the theory is developed for a broad class of stochastic differential equations with additive or multiplicative noise. This opens the way to diagnosing nonlinear oscillations in noisy data and to testing the necessity of accounting for non-Markovian effects in reduced models.
The topic may develop toward applications to high-dimensional climate models, such as the El Niño model, and to problems of stochastic inverse modeling. Also promising is the use of reduced resonances for detecting proximity to bifurcation points in climate data. Developing methods for estimating resonances at intermediate time lags and for nonlocal observables remains an open problem.
The method may influence time series analysis in climatology, oceanography, astrophysics, and molecular dynamics.
The next step could be applying the theory to El Niño observational data and other geophysical datasets, as well as developing adaptive methods for choosing the time lag.
The work connects to the unresolved problem of model reduction: when can non-Markovian effects be neglected, and when are they essential? It also provides a tool for identifying nonlinear oscillations in a stochastic background, which is important in many fields, from asteroseismology to climate dynamics.
Ruelle–Pollicott resonances are named after David Ruelle and Mark Pollicott, who introduced them in the mid-1980s for chaotic deterministic systems. In the stochastic context, they correspond to eigenvalues of the Kolmogorov operator, which describes the evolution of probability distributions.