Abstract
A theory of Ruelle–Pollicott (RP) resonances for stochastic differential systems is presented. 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. They inform us about the spectral elements of some coarsegrained version of the SDE generator. When the timelag at which the transitions are collected from partial observations in V, is either sufficiently small or large, it is shown that the reduced RP resonances approximate the (weak) 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 slowfast 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 [114] and Part III [113], 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 provide. 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 manifestation in terms of a geometric organization of the RP resonances along discrete parabolas in the left half plane. Such geometric features formed by (reduced) RP resonances are extractable from time series and allow thus for providing an unambiguous “signature” of nonlinear oscillations embedded within a stochastic background. By relying then on the theory of reduced RP resonances presented in this contribution, Part III addresses the question of detection and characterization of such oscillations in a highdimensional stochastic system, namely the Cane–Zebiak model of El NiñoSouthern Oscillation subject to noise modeling fast atmospheric fluctuations.
This is a preview of subscription content, access via your institution.
Notes
 1.
In practice however it is often observed that the reduced RP resonances still provide useful information for “intermediate” timelags; see Part III [113].
 2.
We refer to [12] for a mathematical analysis of the related JinNeelin model.
 3.
For instance any semigroup \({\mathcal {T}}\) such that \(\Vert T(t)\Vert _{ess}\le M \exp {(\epsilon t^{\alpha })}\), with \(\epsilon >0\) and \(0<\alpha , M <1\).
 4.
Furthermore if the process is nonexplosive then \(c\equiv 0.\) This excludes the cases for which the underlying Markov process leaving at time 0 from x in \({\mathbb {R}}^d\) escapes to infinity at some finite time \(t > 0\). This article is not concerned with explosive stochastic processes.
 5.
While we recall that in such a case, the RP resonances are the isolated eigenvalues of finite multiplicity, lying within a strip \(\gamma < \mathrm{Re \,}(z) \le 0\); see Panel (a) of Fig. 1.
 6.
Variation of this theorem is used in the study of spectral gaps for deterministic maps and is known as Rokhlin’s disintegration theorem; see [55].
 7.
i.e. up to an exceptional set of null measure with respect to \({\mathfrak {m}}\).
 8.
Note that \(f_n\) defined by (3.32) implies that \(\varPi _j f_n\circ h=\varPi _j \varphi _n\) belongs to D(K) by construction, and thus \(\varPi _j f_n\) belongs to \(D({\mathcal {G}})\) for every \(1\le j\le q\), since the RHS of (3.16) is also the domain of \({\mathcal {G}}\) as (3.16) is independent on t.
 9.
We refer to [109] for useful error bounds regarding the dominant eigenvalues for certain types of coarsegraining maps.
 10.
\(\omega \) labelling the noise realization.
 11.
Recall that a \(C^2\) function U is called a Lyapunov function \(U(x)\ge 1\) and \(\lim _{x\rightarrow \infty } U(x)=\infty \), ensuring thus that the level sets \(\{U\le \alpha \}\) are compact.
 12.
A probability kernel \( {\mathfrak {T}}_t\) allows for representing the Markov semigroup \(P_t\) as \(P_t f(x)=\int {\mathfrak {T}}_t(x,\, \text {d}y) f(y)\); e.g. [8, Prop. 1.2.3]. Having a smooth kernel means that \({\mathfrak {T}}_t(x,\, \text {d}y)={\mathfrak {p}}_t(x,y)\, \text {d}y\) with \({\mathfrak {p}}_t\) infinitely differentiable, i.e. smooth.
References
 1.
Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Springer, New York (2008)
 2.
Arnold, L., Kliemann, W.: On unique ergodicity for degenerate diffusions. Stochastics 21(1), 41–61 (1987)
 3.
Arnold, L.: Random Dynamical Systems. Springer Monographs in Mathematics. Springer, Berlin (1998)
 4.
Agrachev, A.A., Sarychev, A.V.: NavierStokes equations: controllability by means of low modes forcing. J. Math. Fluid Mech. 7(1), 108–152 (2005)
 5.
Berner, J., Achatz, U., Batté, L., Bengtsson, L., Cámara, A., Christensen, H.M., Colangeli, M., Coleman, D.R.B., Crommelin, D., Dolaptchiev, S.I., Franzke, C.L.E., Friederichs, P., Imkeller, P., Järvinen, H., Juricke, S., Kitsios, V., Lott, F., Lucarini, V., Mahajan, S., Palmer, T.N., Penland, C., Sakradzija, M., von Storch, J.S., Weisheimer, A., Weniger, M., Williams, P.D., Yano, J.I.: Stochastic parameterization: toward a new view of weather and climate models. Bull. Am. Met. Soc. 98(3), 565–588 (2017)
 6.
Baladi, V.: Positive Transfer Operators and Decay of Correlations. World Scientific, Singapore (2000)
 7.
Baladi, V., Eckmann, J.P., Ruelle, D.: Resonances for intermittent systems. Nonlinearity 2(1), 119 (1989)
 8.
Bakry, D., Gentil, I., Ledoux, M.: Analysis and Geometry of Markov Diffusion Operators, vol. 348. Springer, New York (2013)
 9.
Bittracher, A., Hartmann, C., Junge, O., Koltai, P.: Pseudo generators for underresolved molecular dynamics. Eur. Phys. J. Special Top. 224(12), 2463–2490 (2015)
 10.
Bittracher, A., Koltai, P., Junge, O.: Pseudo generators of spatial transfer operators. SIAM J. Appl. Dyn. Syst. 14(3), 1478–1517 (2015)
 11.
Baladi, V., Kuna, T., Lucarini, V.: Linear and fractional response for the SRB measure of smooth hyperbolic attractors and discontinuous observables. Nonlinearity 30(3), 1204 (2017)
 12.
Cao, Y., Chekroun, M.D., Huang, A., Temam, R.: Mathematical analysis of the JinNeelin model of El NiñoSouthern Oscillation. Chin. Ann. Math. B 40(1), 1–38 (2019)
 13.
Cerrai, S.: SecondOrder PDE’s in Finite and Infinite Dimension: A Probabilistic Approach, vol. 1762. Springer, New York (2001)
 14.
Cessac, B.: Does the complex susceptibility of the Hénon map have a pole in the upperhalf plane? A numerical investigation. Nonlinearity 20(12), 2883 (2007)
 15.
Chorin, A.J., Hald, O.H.: Stochastic Tools in Mathematics and Science, Surveys and Tutorials in the Applied Mathematical Sciences, vol. 147. Springer, New York (2006)
 16.
Chekroun, M.D., Kondrashov, D.: Dataadaptive harmonic spectra and multilayer StuartLandau models. Chaos 27(9), 093110 (2017)
 17.
Chekroun, M.D., Liu, H., McWilliams, J.C.: The emergence of fast oscillations in a reduced primitive equation model and its implications for closure theories. Comput. Fluids 151, 3–22 (2017)
 18.
Chekroun, M.D., Liu, H., McWilliams, J.C.: Variational approach to closure of nonlinear dynamical systems: autonomous case. J. Stat. Phys. (2019). https://doi.org/10.1007/s10955019024582
 19.
Chekroun, M.D., Lamb, J.S.W., Pangerl, C.J., Rasmussen, M.: A Girsanov approach to slow parameterizing manifolds in the presence of noise, arXiv preprint, arXiv:1903.08598 (2019)
 20.
Chekroun, M.D., Liu, H., Wang, S.: Approximation of Stochastic Invariant Manifolds: Stochastic Manifolds for Nonlinear SPDEs I. Springer Briefs in Mathematics. Springer, New York (2015)
 21.
Chekroun, M.D., Liu, H., Wang, S.: Stochastic Parameterizing Manifolds and NonMarkovian Reduced Equations: Stochastic Manifolds for Nonlinear SPDEs II. Springer Briefs in Mathematics. Springer, New York (2015)
 22.
Chekroun, M.D., Neelin, J.D., Kondrashov, D., McWilliams, J.C., Ghil, M.: Rough parameter dependence in climate models: the role of RuellePollicott resonances. Proc. Natl. Acad. Sci. 111(5), 1684–1690 (2014)
 23.
Chang, J.T., Pollard, D.: Conditioning as disintegration. Stat. Neerl. 51(3), 287–317 (1997)
 24.
Chekroun, M.D., Simonnet, E., Ghil, M.: Stochastic climate dynamics: random attractors and timedependent invariant measures. Physica D 240(21), 1685–1700 (2011)
 25.
Crommelin, D., VandenEijnden, E.: Fitting time series by continuoustime Markov chains: a quadratic programming approach. J. Comput. Phys. 217(2), 782–805 (2006)
 26.
Crommelin, D., VandenEijnden, E.: Databased inference of generators for Markov jump processes using convex optimization. Multisc. Model. Simul. 7(4), 1751–1778 (2009)
 27.
Crommelin, D., VandenEijnden, E.: Diffusion estimation from multiscale data by operator eigenpairs. Multisc. Model. Simul. 9(4), 1588–1623 (2011)
 28.
Dacorogna, B.: Introduction to the Calculus of Variations, vol. 13. World Scientific, Singapore (2004)
 29.
Davies, E.B.: Spectral Theory and Differential Operators, vol. 42. Cambridge University Press, Cambridge (1996)
 30.
Davies, E.B.: Linear Operators and Their Spectra, vol. 106. Cambridge University Press, Cambridge (2007)
 31.
Douc, R., Fort, G., Guillin, A.: Subgeometric rates of convergence of fergodic strong Markov processes. Stoch. Process. Appl. 119(3), 897–923 (2009)
 32.
Dellnitz, M., Froyland, G., Junge, O.: The Algorithms Behind Gaio—Set Oriented Numerical Methods for Dynamical Systems, Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, pp. 145–174. Springer (2001)
 33.
Dragoni, F., Kontis, V., Zegarliński, B.: Ergodicity of Markov semigroups with Hörmander type generators in infinite dimensions. Potential Anal. 1–29 (2012)
 34.
Duong, M.H., Lamacz, A., Peletier, M.A., Schlichting, A., Sharma, U.: Quantification of coarsegraining error in Langevin and overdamped Langevin dynamics. Nonlinearity 31(10), 4517 (2018)
 35.
Dellacherie, C., Meyer, P.A.: Probabilities and Potential, vol. 29. NorthHolland Publishing Co., Amsterdam (1978)
 36.
Doob, J.L.: Asymptotic properties of Markoff transition probabilities. Trans. Am. Math. Soc. 63(3), 393–421 (1948)
 37.
Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge (1996)
 38.
Dyatlov, S., Zworski, M.: Stochastic stability of PollicottRuelle resonances. Nonlinearity 28(10), 3511 (2015)
 39.
Eckmann, J.P., Hairer, M.: Spectral properties of hypoelliptic operators. Commun. Math. Phys. 235(2), 233–253 (2003)
 40.
Engel, K.J., Nagel, R.: Oneparameter semigroups for linear evolution equations, vol. 194. Springer, New York (2000)
 41.
Engel, K.J., Nagel, R.: A short course on operator semigroups. Springer, New York (2006)
 42.
Forgoston, E., Billings, L., Schwartz, I.B.: Accurate noise projection for reduced stochastic epidemic models. Chaos 19(4), 043110 (2009)
 43.
Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31(1), 53–98 (1979)
 44.
Flandoli, F., Gubinelli, M., Priola, E.: Flow of diffeomorphisms for SDEs with unbounded Holder continuous drift. Bull. Sci. Math. 134(4), 405–422 (2010)
 45.
Froyland, G., Junge, O., Koltai, P.: Estimating longterm behavior of flows without trajectory integration: the infinitesimal generator approach. SIAM J. Numer. Anal. 51(1), 223–247 (2013)
 46.
Fenichel, N., Moser, J.K.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21(3), 193–226 (1971)
 47.
Froyland, G.: Computerassisted bounds for the rate of decay of correlations. Commun. Math. Phys. 189(1), 237–257 (1997)
 48.
Froyland, G.: Extracting dynamical behavior via markov models. Nonlinear dynamics and statistics, pp. 281–321. Springer (2001)
 49.
Forgoston, E., Schwartz, I.B.: Escape rates in a stochastic environment with multiple scales. SIAM J. Appl. Dyn. Syst. 8(3), 1190–1217 (2009)
 50.
Ghil, M., Allen, M.R., Dettinger, M.D., Ide, K., Kondrashov, D., Mann, M.E., Robertson, A.W., Saunders, A., Tian, Y., Varadi, F., et al.: Advanced spectral methods for climatic time series. Rev. Geophys. 40(1), 1003 (2002)
 51.
Gaspard, P.: Trace formula for noisy flows. J. Stat. Phys. 106(1–2), 57–96 (2002)
 52.
Gottwald, G.A., Crommelin, D.T., Franzke, C.L.E.: Stochastic climate theory. In: Franzke, C.L.E., O’Kane, T.J. (eds.) Nonlinear and Stochastic Climate Dynamics, pp. 209–240. Cambridge University Press, Cambridge (2017)
 53.
Givon, D., Kupferman, R., Stuart, A.: Extracting macroscopic dynamics: model problems and algorithms. Nonlinearity 17(6), R55 (2004)
 54.
Gruberbauer, M., Kallinger, T., Weiss, W.W., Guenther, D.B.: On the detection of Lorentzian profiles in a power spectrum: a Bayesian approach using ignorance priors. Astronomy Astrophys. 506(2), 1043–1053 (2009)
 55.
Galatolo, S., Lucena, R.: Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenzlike maps. Disc. Cont. Dyn. Syst. A 40(3), 1309 (2020)
 56.
Giulietti, P., Liverani, C., Pollicott, M.: Anosov flows and dynamical zeta functions. Ann. Math. 687–773 (2013)
 57.
Goldys, B., Maslowski, B.: Exponential ergodicity for stochastic reactiondiffusion equations. In: Stochastic Partial Differential Equations and ApplicationsVII, Lect. Notes Pure Appl. Math., vol. 245, Chapman Hall/CRC, Boca Raton, FL, 2006, pp. 115–131 (2005)
 58.
Ganidis, H., Roynette, B., Simonot, F.: Convergence rate of some semigroups to their invariant probability. Stoch. Process. Appl. 79(2), 243–263 (1999)
 59.
Guionnet, A., Zegarlinksi, B.: Lectures on logarithmic Sobolev inequalities, pp. 1–134. Springer, Séminaire de Probabilités XXXVI (2003)
 60.
Hairer, M.: An introduction to stochastic PDEs, arXiv preprint arXiv:0907.4178 (2009)
 61.
Hairer, M., Mattingly, J.C., Scheutzow, M.: Asymptotic coupling and a general form of Harris theorem with applications to stochastic delay equations. Probab. Theory Relat. Fields 149(1–2), 223–259 (2011)
 62.
Hérau, F., Nier, F.: Isotropic hypoellipticity and trend to equilibrium for the FokkerPlanck equation with a highdegree potential. Arch. Ration. Mech. Anal. 171(2), 151–218 (2004)
 63.
Hörmander, L.: Hypoelliptic second order differential equations. Acta Mathematica 119(1), 147–171 (1967)
 64.
Hairer, M., Stuart, A.M., Vollmer, S.J.: Spectral gaps for a MetropolisHastings algorithm in infinite dimensions. Ann. Appl. Probab. 24(6), 2455–2490 (2014)
 65.
Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, vol. 24. Elsevier, New York (2014)
 66.
Jurdjevic, V., Kupka, I.: Polynomial control systems. Math. Ann. 272(3), 361–368 (1985)
 67.
Jones, C.K.R.T.: Geometric Singular Perturbation Theory. Springer, Berlin (1995)
 68.
Kondrashov, D., Chekroun, M.D., Berloff, P.: Multiscale stuartlandau emulators: application to winddriven ocean gyres. Fluids 3, 21 (2018)
 69.
Kondrashov, D., Chekroun, M.D., Ghil, M.: Datadriven nonMarkovian closure models. Physica D 297, 33–55 (2015)
 70.
Kondrashov, D., Chekroun, M.D., Ghil, M.: Dataadaptive harmonic decomposition and prediction of Arctic sea ice extent. Dyn. Stat. Clim. Syst. 3(1), 1–23 (2018)
 71.
Kondrashov, D., Chekroun, M.D., Yuan, X., Ghil, M.: Dataadaptive harmonic decomposition and stochastic modeling of Arctic sea ice. In: Tsonis, A. (ed.) Advances in Nonlinear Geosciences, pp. 179–205. Springer, New York (2018)
 72.
Khasminskii, R.Z.: Ergodic properties of recurrent diffusion processes and stabilization of the solution to the Cauchy problem for parabolic equations. Theory Prob. Appl. 5(2), 179–196 (1960)
 73.
Kliemann, W.: Recurrence and invariant measures for degenerate diffusions. Ann. Probab. 158, 690–707 (1987)
 74.
Kallinger, T., Mosser, B., Hekker, S., Huber, D., Stello, D., Mathur, S., Basu, S., Bedding, T.R., Chaplin, W.J., De Ridder, J., et al.: Asteroseismology of red giants from the first four months of Kepler data: fundamental stellar parameters. Astronomy Astrophys. 522, A1 (2010)
 75.
Kuehn, C.: Multiple Time Scale Dynamics, Applied Mathematical Sciences, vol. 191. Springer, New York (2015)
 76.
Lorenzi, L., Bertoldi, M.: Analytical Methods for Markov Semigroups. CRC Press, Taylor & Francis Group, Boca Raton (2006)
 77.
Legoll, F., Lelièvre, T.: Effective dynamics using conditional expectations. Nonlinearity 23(9), 2131 (2010)
 78.
Legoll, F., Lelièvre, T., Olla, S.: Pathwise estimates for an effective dynamics. Stoch. Process. Appl. 127(9), 2841–2863 (2017)
 79.
Lehoucq, R.B., Sorensen, D.C., Yang, C.: ARPACK Users’ Guide: Solution of Large Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, pp. xv + 137 (1997)
 80.
Lucarini, V.: Revising and extending the linear response theory for statistical mechanical systems: evaluating observables as predictors and predictands. J. Stat. Phys. 173(6), 1698–1721 (2018)
 81.
Lu, J., VandenEijnden, E.: Exact dynamical coarsegraining without timescale separation. J. Chem. Phys. 141(4), 044109 (2014)
 82.
Melbourne, I., Gottwald, G.A.: Power spectra for deterministic chaotic dynamical systems. Nonlinearity 21(1), 179 (2007)
 83.
Metafune, G., Pallara, D., Priola, E.: Spectrum of OrnsteinUhlenbeck operators in \({L}^p\) spaces with respect to invariant measures. J. Funct. Anal. 196(1), 40–60 (2002)
 84.
Metafune, G., Pallara, D., Wacker, M.: Compactness properties of Feller semigroups. Studia Math. 153(2), 179–206 (2002)
 85.
Meyn, S.P., Tweedie, R.L.: Stability of Markovian processes II: continuoustime processes and sampled chains. Adv. Appl. Probab. 25, 487–517 (1993)
 86.
Meyn, S.P., Tweedie, R.L.: Stability of Markovian processes III: FosterLyapunov criteria for continuoustime processes. Adv. Appl. Probab. 25, 518–548 (1993)
 87.
Majda, A.J., Tong, X.T.: Ergodicity of truncated stochastic Navier Stokes with deterministic forcing and dispersion. J. Nonlinear Sci. 26(5), 1483–1506 (2016)
 88.
Majda, A.J., Timofeyev, I., VandenEijnden, E.: A mathematical framework for stochastic climate models. Commun. Pure Appl. Math. 54, 891–974 (2001)
 89.
Noack, B.R., Afanasiev, K., Morzyński, M., Tadmor, G., Thiele, F.: A hierarchy of lowdimensional models for the transient and posttransient cylinder wake. J. Fluid Mech. 497, 335–363 (2003)
 90.
Norris, J.: Simplified Malliavin calculus, Séminaire de Probabilités XX 1984/85, Springer, pp. 101–130 (1986)
 91.
Nipp, K., Stoffer, D.: Invariant Manifolds in Discrete and Continuous Dynamical Systems, vol. 21. European Mathematical Society (2013)
 92.
Nonnenmacher, S., Zworski, M.: Decay of correlations for normally hyperbolic trapping. Inventiones mathematicae 200(2), 345–438 (2015)
 93.
Ottobre, M., Pavliotis, G.A., PravdaStarov, K.: Exponential return to equilibrium for hypoelliptic quadratic systems. J. Funct. Anal. 262(9), 4000–4039 (2012)
 94.
Pazy, A.: Semigroups of Linear Operators and Application to Partial Differential Equations. Springer, New York (1983)
 95.
Pollicott, M.: Meromorphic extensions of generalised zeta functions. Inventiones Mathematicae 85(1), 147–164 (1986)
 96.
Penland, C., Sardeshmukh, P.D.: The optimal growth of tropical sea surface temperature anomalies. J. Clim. 8(8), 1999–2024 (1995)
 97.
Pavliotis, G., Stuart, A.: Multiscale Methods: Averaging and Homogenization. Springer, New York (2008)
 98.
ReyBellet, L.: Ergodic properties of Markov processes, pp. 1–39. Springer, Open Quantum Systems II (2006)
 99.
Romito, M.: Ergodicity of the finite dimensional approximation of the 3D NavierStokes equations forced by a degenerate noise. J. Stat. Phys. 114(1–2), 155–177 (2004)
 100.
Romito, M.: A geometric cascade for the spectral approximation of the NavierStokes equations. In: Probability and Partial Differential Equations in Modern Applied Mathematics. Springer, pp. 197–212 (2005)
 101.
Ruelle, D.: Locating resonances for axiom a dynamical systems. J. Stat. Phys. 44(3–4), 281–292 (1986)
 102.
Ruelle, D.: Differentiating the absolutely continuous invariant measure of an interval map f with respect to f. Commun. Math. Phys. 258(2), 445–453 (2005)
 103.
Ruelle, D.: A review of linear response theory for general differentiable dynamical systems. Nonlinearity 22(4), 855 (2009)
 104.
Rogers, L.G., Williams, D.: Diffusions, Markov Processes and Martingales: Volume 1, Foundations, vol. 2. Cambridge University Press, Cambridge (2000)
 105.
Seidler, J.: Ergodic behaviour of stochastic parabolic equations. Czechoslovak Math. J. 47(2), 277–316 (1997)
 106.
Schütte, Ch., Fischer, A., Huisinga, W., Deuflhard, P.: A direct approach to conformational dynamics based on hybrid Monte Carlo. J. Comput. Phys. 151(1), 146–168 (1999)
 107.
Schütte, Ch., Huisinga, W.: On conformational dynamics induced by Langevin processes. In: Proceedings of the International Conference on Differential Equations, vol. 1, p. 7. World Scientific (1999)
 108.
Schütte, Ch., Huisinga, W., Deuflhard, P.: Transfer operator approach to conformational dynamics in biomolecular systems. In: Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems. Springer, pp. 191–223 (2001)
 109.
Schütte, Ch., Sarich, M.: Metastability and Markov State Models in Molecular Dynamics, vol. 24. American Mathematical Soc, Providence, RI (2013)
 110.
Stettner, L.: Remarks on Ergodic conditions for Markov processes on polish spaces. Bull. Polish Acad. Sci. 42(2), 103–114 (1994)
 111.
Stroock, D.W., Varadhan, S.R.S.: On the support of diffusion processes with applications to the strong maximum principle. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 3: Probability Theory (Berkeley, Calif.), pp. 333–359. University of California Press (1972)
 112.
Tam, C.K.W.: Supersonic jet noise. Annu. Rev. Fluid Mech. 27(1), 17–43 (1995)
 113.
Tantet, A., Chekroun, M.D., Neelin, J.D., Dijkstra, H.A.: RuellePollicott resonances of stochastic systems in reduced state space. Part III: Application to El NiñoSouthern Oscillation. J. Stat. Phys. (2019). https://doi.org/10.1007/s10955019024448
 114.
Tantet, A., Chekroun, M.D., Neelin, J.D., Dijkstra, H.A.: RuellePollicott resonances of stochastic systems in reduced state space. Part II: Stochastic Hopf Bifurcation. J. Stat. Phys. (Accepted) (2020)
 115.
Tantet, A., Lucarini, V., Lunkeit, F., Dijkstra, H.A.: Crisis of the chaotic attractor of a climate model: a transfer operator approach. Nonlinearity 31(5), 2221 (2018)
 116.
Tantet, A., van der Burgt, F.R., Dijkstra, H.A.: An early warning indicator for atmospheric blocking events using transfer operators. Chaos 25(3), 036406 (2015)
 117.
Ulam, S.M.: Problems in Modern Mathematics, science edn. Wiley, New York (1964)
 118.
van Neerven, J.: The Asymptotic Behaviour of Semigroups of Linear Operators, vol. 88. Birkhäuser, Basel (2012)
 119.
Wouters, J., Lucarini, V.: Disentangling multilevel systems: averaging, correlations and memory. J. Stat. Mech. 2012, P03003 (2012)
 120.
Wouters, J., Lucarini, V.: Multilevel dynamical systems: connecting the Ruelle response theory and the MoriZwanzig approach. J. Stat. Phys. 151(5), 850–860 (2013)
 121.
Weinan, E., VandenEijnden, E.: Metastability, conformation dynamics, and transition pathways in complex systems. In: Multiscale Modelling and Simulation, pp. 35–68. Springer (2004)
 122.
K. Yosida: Functional Analysis. Reprint of the sixth: edition. Classics in Mathematics, Springer, Berlin 11(1995), 501 (1980)
 123.
Zhang, W., Hartmann, C., Schütte, Ch.: Effective dynamics along given reaction coordinates, and reaction rate theory. Faraday Discuss. 195, 365–394 (2017)
 124.
Zworski, M.: Mathematical study of scattering resonances. Bull. Math. Sci. 7(1), 1–85 (2017)
Acknowledgements
The authors would like to thank the reviewers for their very useful and constructive comments. This work has been partially supported by the European Research Council under the European Union’s Horizon 2020 research and innovation program (grant Agreement No. 810370 (MDC)), by the Office of Naval Research (ONR) Multidisciplinary University Research Initiative (MURI) grant N000141612073 (MDC), by the National Science Foundation grants OCE1658357 (MDC), DMS1616981(MDC), AGS1540518 and AGS1936810 (JDN), by the LINC Project (No. 289447) funded by EC’s MarieCurie ITN (FP7PEOPLE2011ITN) program (AT and HD) and by the Utrecht University Center for Water, Climate and Ecosystems (AT).
Author information
Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Elements of Stochastic Analysis
Elements of Stochastic Analysis
In this appendix we present a short survey of elements of stochastic analysis used in the Main Text. The main objective is to introduce the key concepts and tools of stochastic analysis for stochastic differential equations (SDEs), to a wider audience in the geosciences and macroscopic physics.
Markov Semigroups
Two approaches dominate the analysis of stochastic dynamics. We are here concerned with the approach rooted in Stochastic Analysis which, contrary to the random dynamical system (RDS) approach [3, 20, 24], does not substitute a deterministic (nonlinear) flow S(t) by a stochastic flow \(S(t,\omega )\) acting^{Footnote 10} on the state space \({\mathcal {X}}\) but rather by a family of linear operators \(P_t\), acting on a space of observables of the state space, i.e. on functions of \({\mathcal {X}}\). A typical choice of observables is given by \({\mathcal {C}}_b({\mathcal {X}})\), the space of bounded and continuous functions on \({\mathcal {X}}\). In what follows \({\mathcal {X}}\) is a finitedimensional Polish space.
More precisely, this family \(P_t\) reflects the (averaged) action of the stochastic flow at the level of functions and is given as the mapping which to each function \(\phi \) in \({\mathcal {C}}_b({\mathcal {X}})\) associates the function:
In (A.1), the function \(\phi \) is the aforementioned observable. Its physical meaning could be, for instance, the potential vorticity or the temperature of a fluid at a given location or averaged over a volume. The RHS of (A.1) involves averaging over the realizations \(\omega \), i.e. expectation. For deterministic flow it reduces to \(P_t\phi (x)=\phi (S(t)x)\) and is known as the Koopman operator. Note that \(P_t\) such as defined in (A.1) is not limited to stochastic flow, more generally \(P_t \phi (x) ={\mathbb {E}}( \phi (X_t^x))\) where \(X_t^x\) denotes a stochastic process that solves Eq. (2.1) (as associated with \(P_t\)) and emanates from x in \({\mathcal {X}}\).
Under general assumptions on F and D, the stochastic process \(X_t\) solving Eq. (2.1) is Markovian (i.e. the future is determined only by the present value of the process) which translates at the level of \(P_t\) into the following semigroup property
A breakdown of (A.2) indicates thus that the underlying stochastic process is nonMarkovian.
It is noteworthy to mention that even when \(P_t\) satisfies (A.2), it does not ensure that \(P_t\) is a strongly continuous semigroup [94] on \({\mathcal {C}}_b({\mathcal {X}})\). Nevertheless, \((P_t)_{t\ge 0}\) is extendable to a strongly continuous semigroup in \(L^2_{\mu }\) as soon as \(\mu \) is an invariant measure of the Markov semigroup; see Theorem 4 below. The spectral theory of such semigroups [40] is at the core of the description of mixing properties in \(L^2_{\mu }\), such as presented in Sect. 2.2 in the Main Text.
Ergodic Invariant Measures and the Strong Feller–Irreducibility Approach
The Fokker–Planck equation (2.3) may support several weak stationary solutions. An important question, is thus the identification of stationary measures that describe the asymptotic statistical behavior of the solutions of Eq. (2.1), in a typical fashion. The notion of ergodic invariant measures plays a central role in that respect, and relies on the following important characterization of ergodic measures for (stochastically continuous) Markov semigroups [37, Theorem 3.2.4].
Definition 1
An invariant measure is ergodic if one of the following three equivalent statements holds:

(i)
For any \(f \in L^2_\mu ({\mathcal {X}})\), if \(P_t f =f\), almost surely w.r.t \(\mu \) (\(\mu \)a.s.) for all \(t\ge 0\), then f is constant \(\mu \)a.s.

(ii)
For any Borel set \(\varGamma \) of \({\mathcal {X}}\), if \(P_t \mathbb {1}_{\varGamma } =\mathbb {1}_{\varGamma }\)\(\mu \)a.s. for all \(t\ge 0\), then \(\mu (\varGamma )=0\) or 1.

(iii)
For any \(f \in L^2_\mu ({\mathcal {X}})\), \(\frac{1}{T} \int _0^T P_s f \, \text {d}s \underset{T\rightarrow \infty }{\longrightarrow }\int f \, \text {d}\mu \) in \(L^2_\mu ({\mathcal {X}}).\)
In practice, an efficient approach to show the existence of an ergodic measure consists of showing the existence of a unique invariant measure, since in this case such an invariant measure is necessarily ergodic [37, Theorem 3.2.6]. Various powerful approaches exist to deal with the existence of a unique invariant measure. The next section discusses the classical approach based on the theory of strong Feller Markov semigroups and irreducibility.
The main interest of the strong Feller–Irreducibility approach lies in its usefulness for checking the conditions of the Doob–Khasminskii Theorem [36, 37, 72], the latter ensuring the existence of at most one ergodic invariant measure. This strategy requires the proof of certain smoothing properties of the associated Markov semigroup, and to show that any point can be (in probability) reached at any time instant by the process regardless of initial data. This property is known as irreducibility. It means that \(P_t \mathbb {1}_{U} (x)> 0\) for all x in \({\mathcal {X}}\), every \(t>0\), and all nonempty open sets U of \({\mathcal {X}}\), which is equivalent to say that
for any z in \({\mathcal {X}}\), \(\epsilon >0\) and \(t>0\); see [13, p. 67]. In other words the irreducibility condition expresses the idea that any neighborhood of any point z in \({\mathcal {X}}\), is reachable at each time, with a positive probability.
Remarkably, the irreducibility is usually inferred from the controllability of the associated control system \({\dot{x}}=F(x) + D(X) u(t)\); see [19] for a simple illustration. This approach is wellknown and based on the support theorem of Stroock and Varadhan [?] (see also [65, Theorem 8.1]) that shows that several properties of the SDEs can be studied and expressed in terms of the control theory of ordinary differential equations (ODEs); see [37, Secns. 7.3 and 7.4] for the case of additive (nondegenerate) noise and [2, 73] for the more general case of nonlinear degenerate noise, i.e. in the case where the noise acts only on part of the system’s equations, corresponding to ker\((Q)\ne \{0\}\).
The strong Feller property means that the Markov semigroup maps bounded measurable functions into bounded continuous functions. This property, related to a regularizing effect of the Markov semigroup \((P_t)_{t\ge 0}\), is a consequence of the hypoellipticity of the Kolmogrorov operator\({\mathcal {K}}\) defined on smooth functions \(\psi \) (of class \(C^2\)) as follows when \({\mathcal {X}}={\mathbb {R}}^d\):
where
Here \(\text {Tr}\) denotes the trace of a matrix. Note that hypoelliptic operators include those that are uniformly elliptic for which the Weyl’s smoothing lemma applies; e.g. [28, Theorem 4.7]. Hypoellipticity allows nevertheless for dealing with the case of degenerate noise, which is important in applications.
A very efficient criteria for hypoellipticity is given by Hörmander’s theorem [63, 90]; see also [24, Appendix C1] for a discussion on the related Hörmander’s bracket condition and its implications to the existence of other types of meaningful measures for SDEs, namely the Sinaï–Ruelle–Bowen (SRB) random measures. We refer also to Part II [114], for an instructive verification of the Hörmander’s condition in the case of the Hopf normal form subject to additive noise.
From a geophysical perspective, it is noteworthy to mention that the strong Feller–Irreducibility approach allows for dealing with a broad class of truncations of fluid dynamics models that would be perturbed by noise, possibly degenerate. For instance, in the case of truncations of 2D or 3D Navier–Stokes equations, the strong Feller–Irreducibility approach has been shown to be applicable even for an additive noise that forces only very few modes [4, 99]. The delicate point of the analysis is the verification of the controllability (and thus irreducibility) of the associated control system, by techniques typically adapted from [66] or rooted in chronological calculus as in [4]. Whatever the approach, the analysis requires the appropriate translation into geometrical terms of the cascade of energy in which the nonlinear terms transmit the forcing from the few modes to all the others [100]. We mentioned however [87] for an example of a stochastic dynamical system which has the square of the Euclidean norm as the Lyapunov function, is hypoelliptic with nonzero noise forcing, and that yet fails to be reachable or ergodic.
Markov Semigroups and Mixing
We recall here standard results about Markov semigroups. It states that any Markov semigroup that is strong Feller and irreducible and for which an invariant measure exists (which is thus unique) is not only ergodic but also strongly mixing for the total variation norm of measures. Given two probability measures \(\mu _1\) and \(\mu _2\) on \({\mathcal {X}}\), we recall that the latter is defined as [60, Eq. (3.1)]
where \({\mathcal {B}}_b({\mathcal {X}})\) denotes the set of Borel measurable and bounded functions on \({\mathcal {X}}.\)
Theorem 4
Let \(\mu \) be an invariant measure of a Markov semigroup \((P_t)_{t\ge 0}\). For any \(p\ge 1\) and \(t\ge 0\), \(P_t\) is extendable to a linear bounded operator on \(L^p_\mu ({\mathcal {X}})\) still denoted by \(P_t\). Moreover

(i)
\(\Vert P_t\Vert _{{\mathcal {L}}(L^p_\mu ({\mathcal {X}}))} \le 1\)

(ii)
\(P_t\) is strongly continuous semigroup in \(L^p_\mu ({\mathcal {X}})\).
If furthermore \((P_t)_{t\ge 0}\) is strong Feller and irreducible, then \(\mu \) is ergodic (and unique) and for any x in \({\mathcal {X}}\) and g in \(L^1_{\mu }\)
where \(X_t^x\) denotes the stochastic process solving the SDE associated with \(P_t\).
In this case, the invariant measure \(\mu \) is also strongly mixing in the sense that for any measure \(\nu \) on \({\mathcal {X}}\), we have:
For the definition of a strongly continuous semigroup also known as \(C_0\)semigroup we refer to [40, p. 36]. For an introduction to semigroup theory we refer to [41, 118].
Proof
We prove first (i). The proof is standard and can be found e.g. in [59, Prop. 1.14] but is reproduced here for the reader’s convenience. Let g be in \({\mathcal {C}}_b({\mathcal {X}})\). By the Hölder inequality, we have
If we now integrate both sides of this inequality with respect to \(\mu \), we obtain
the latter equality resulting from the invariance of \(\mu \). Since \({\mathcal {C}}_b({\mathcal {X}})\) is dense in \(L^p_\mu ({\mathcal {X}})\), the inequality (A.10) can be extended to any function in \(L^p_\mu ({\mathcal {X}})\), and thus \((P_t)_{t\ge 0}\) can be uniquely extended to a contraction semigroup in \(L^p_\mu ({\mathcal {X}})\), and property (i) is proved.
Let us show now that \((P_t)_{t\ge 0}\) is strongly continuous in \(L^p_\mu ({\mathcal {X}})\). Since \((P_t)_{t\ge 0}\) is a Markov semigroup, for any g in \({\mathcal {C}}_b({\mathcal {X}})\) and x in \({\mathcal {X}}\), we have that the mapping \(t \mapsto P_t g (x)\) is continuous. Therefore by the dominated convergence theorem
The density of \({\mathcal {C}}_b({\mathcal {X}})\) in \(L^p_\mu ({\mathcal {X}})\) allows us to conclude that this convergence holds when g is in \(L^p_\mu ({\mathcal {X}})\).
The ergodicity of \(\mu \) results from the aforementioned Doob’s theorem. The timeaverage property (A.7) and the mixing property (A.8) can be obtained as a consequence of e.g. [105, Cor. 2.3]; see also [110, Cor. 1]. \(\square \)
Generator of a Markov Semigroup
Recall that the generator A of any strongly continuous semigroup \((T(t))_{t\ge 0}\) on a Hilbert space \({\mathcal {H}}\) is defined as the operator \(A:D(A)\subset {\mathcal {H}} \rightarrow {\mathcal {H}}\), such that
defined for every \(\varphi \) in the domain
As any generator of a contraction semigroup, given an invariant measure \(\mu \), the generator K of the contraction semigroup \((P_t)_{t\ge 0}\) in \(L^2_{\mu }\) (Theorem 4(i)) is dissipative, which is equivalent to say, since \(L^2_{\mu }\) is a Hilbert space, that
where D(K) denotes the domain of K; see e.g. [40, Prop. II.3.23]. The domain D(K) is furthermore dense in \(L^2_{\mu }\) and K is a closed operator; see [94, Cor. 2.5 p. 5]. The isolated part of the spectrum of K provides the Ruelle–Pollicott resonances; see Sect. 2.2.
Return to Equilibrium and Spectral Gap
We present here some useful results concerning (i) the exponential return to equilibrium for strong Feller and irreducible Markov semigroups, and (ii) spectral gap in the spectrum of the Markov semigroup generator K; see Theorems 5 and 6 below. Theorem 5 deals with semigroups that become quasicompact after a finite time, and Theorem 6 addresses the exponential \(L^2\)convergence and lower bound of the spectral gap. For Theorem 5, the approach is based on Lyapunov functions such as formulated in [98]. We propose a slightly different presentation for which we provide the main elements of the proof. We refer to [31] for an efficient (and beautiful) generalization of such Lyapunovtype criteria allowing for subexponential convergence towards the equilibrium.
Recall that the essential spectral radius\(\mathbf{r }_{ess}(T)\) of a linear bounded operator T on a Banach space \({\mathcal {E}}\) satisfies [40, p. 249] the Hadamard formula
where
We have then the following convergence result.
Theorem 5
Let \({\mathcal {P}}=(P_t)_{t\ge 0}\) be a strong Feller and irreducible Markov semigroup in \(L^2_\mu ({\mathbb {R}}^d)\) (\({\mathcal {X}}={\mathbb {R}}^d\)) generated by an SDE given by Eq. (2.1) for which F and G are locally Lipschitz. Assume that there exists a Lyapunov function^{Footnote 11}U and a compact set \({\mathfrak {A}}\) for which there exist \(a >0\), \(0<\kappa <1\) and \(b<\infty \), such that
where \({\mathcal {K}}\) is the Kolmogorov differential operator generating the Markov process associated with \({\mathcal {P}}\). Then for all \(t>t_0\), \(P_t\) becomes quasicompact, i.e.
where the essential spectral radius is taken for \(P_t\) as acting on \({\mathcal {E}}={\mathcal {F}}_{U}\) given by
and endowed with the norm
Furthermore \((P_t)_{t\ge 0}\) has a unique invariant measure \(\mu \), and the inequality (A.18) ensures that there exist \(C>0\) and \(\lambda >0\) such that for all f in \({\mathcal {F}}_{U}\),
The proof of this result is found in Appendix A.6.
Remark 5
The assumption (A.17b) is sometimes verified from moment estimates in practice. For instance if there exist \(k_0>0\) and \(k_1>0\) such that
then for any \(t\ge \frac{1}{k_1} \log (\frac{1}{4 k_0})\), we have \( {\mathbb {E}} (X_t^x +1)\le \frac{1}{2} (x+1)\frac{1}{4} x+c+\frac{1}{2}\), which leads to
for all \(r>4(c+\frac{1}{2}),\) and thus (A.17b) holds with \(U(x)=x+1.\)
More generally, if
then \(\frac{\, \text {d}}{\, \text {d}t} P_t U (x)={ P_t {\mathcal {K}} U} (x)\le \alpha P_t U(x) +\beta \), leading to
and similarly (A.17b) holds. In addition, (A.24) implies (A.17a). Note that (A.24) and (A.25) are quite standard; see e.g. [33, Lemma 2.11].
Finally, note also that finding a Lyapunov function may be easier than proving inequalities of the form (A.22). For instance, if there is a Lyapunov function which grows polynomially like \(\Vert p\Vert ^q\), then one knows that the process has moments of order q; see [85, 86].
Finally, lower bounds of the spectral gap in \(L^2_{\mu }\) may be derived for a broad class of SDEs. Recall that the generator K has a spectral gap in \(L^2_{\mu }\) if there exists \(\delta >0\) such that
The largest \(\delta >0\) with this property is denoted by \(\text {gap}(K)\), namely
The following result is a consequence in finite dimension of more general convergence results [57, Theorems 2.5 and 2.6]. Since \((P_t)_{t\ge 0}\) is a \(\hbox {C}_0\)semigroup in \(L^2_{\mu }\), the theory of asymptotic behavior of a semigroup with a strictly dominant, algebraically simple eigenvalue (e.g. [118, Theorem. 3.6.2]) implies the spectral gap property stated in the following.
Theorem 6
Assume that \((P_t)_{t\ge 0}\) is strong Feller and irreducible. Assume furthermore that the following ultimate bound holds for the associated stochastic process \(X_t^x\), i.e. there exist \(c,k,\alpha >0\) such that
Then there exists a unique invariant measure \(\mu \) for which the Uuniform ergodicity (A.21) holds with \(U(x)=1+x^2\), as well as the following exponential \(L^2\)convergence
with C and \(\lambda \) positive constants independent of \(\varphi \); the latter rate of convergence being the same as that of (A.21). Furthermore, one has the following lower bound for the \(L^2_{\mu }\)spectrum of the generator K:
We will see in Part II [114] of this threepart article that Theorem 6 has important practical consequences. In particular it shows for a broad class of controllable ODEs, perturbed by a white noise process for which the Kolmogorov operator is hypoelliptic, that an \(L_{\mu }^2\)spectral gap is naturally induced by the noise whereas in absence of the latter the gap may be zero, leading thus to a form of mixing enhancement by the noise. We finally mention [64] for other conditions, ensuring an \(L^2_\mu \)gap based on spectral gaps in Wasserstein distances, verifiable in practice by following the approach of [61].
Proof of Theorem 5
Proof It is standard from the theory of Lyapunov functions that the existence of a unique invariant measure \(\mu \) is ensured by the condition (A.17a) together with the irreducibility and strong Feller properties. The rest of the proof is thus concerned with (A.18) and the exponential convergence (A.21).
Step 1 First, note that the Itô formula gives
which leads (since \({\mathcal {K}} U \le a U\)) to
and therefore \(P_t\) is extendable to a linear operator on \({\mathcal {F}}_U\) (defined in (A.19)) with norm \(\Vert P_t\Vert \le e^{a t}\).
The second inequality in (A.17) ensures that for any \(t>t_0\),
By definition, a Markov semigroup is monotone, thus one may iterate (A.33) to obtain (by using \(P_t \mathbb {1}_{{\mathbb {R}}^d}=\mathbb {1}_{{\mathbb {R}}^d}\)),
Consider now an arbitrary compact set \({\mathfrak {B}}\) in \({\mathbb {R}}^d\) and f in \({\mathcal {F}}_{U}\), we have the bound
where we have used the basic inequality (A.9) (with \(p=1\)). This last inequality with (A.34) leads to
Since \(\lim _{x\rightarrow \infty } U(x)=\infty \), given \(\epsilon >0\) and \(n>1\) one may thus choose a compact set \({\mathfrak {B}}_n\) such that
which leads to
Step 2 We show now that the linear operator
is compact for any compact set \({\mathfrak {B}}\) of \({\mathbb {R}}^d\). This is equivalent to showing that for any sequence \(g_k\) in \({\mathcal {F}}_U\) such that \(\Vert g_k\Vert _U\le 1\), one can extract a subsequence such that \(\varLambda g_k\) is convergent in \({\mathcal {F}}_U\). Since \(P_t\) is strongly Feller and \(\mathbb {1}_{{\mathfrak {B}}} g_k\) is bounded for each k, then \(P_{t} \mathbb {1}_{{\mathfrak {B}}} g_k\) belongs to \({\mathcal {C}}_b({\mathfrak {B}})\), by definition. Thus the sequence \((\varLambda g_k)\) lies in \({\mathcal {C}}({\mathfrak {B}})\).
We have
which shows that \(\{\varLambda g_k\}\) is equibounded.
Furthermore, since \(P_t\) is strong Feller, it has a smooth kernel^{Footnote 12} and we have for all x and \(x'\) in \({\mathfrak {B}}\)
which shows that \(\{\varLambda g_k\}\) is equicontinuous.
Thus, the Ascoli–Arzelà theorem [122, p. 85] applies and guarantees that a subsequence from \(\varLambda g_k\) converges in \({\mathcal {C}}({\mathfrak {B}})\) to g. Now since \(U\ge 1\), the same extraction from \(\varLambda g_k\) converges to \(g\mathbb {1}_{{\mathfrak {B}}}\) in \({\mathcal {F}}_U\). We conclude that \(\mathbb {1}_{{\mathfrak {B}}} \,P_{t} \mathbb {1}_{{\mathfrak {B}}}\) is a compact mapping for any compact set \({\mathfrak {B}}\) of \({\mathbb {R}}^d\).
Step 3 Let \({\mathfrak {B}}_n\) be a sequence of compact sets satisfying (A.37), and let us consider the compact operators (from Step 2) \({\mathfrak {C}}_n\) defined by \(\mathbb {1}_{{\mathfrak {B}}_n} \,P_{n t} \mathbb {1}_{{\mathfrak {B}}_n}\). We have then
By applying to \(P_t\) the Hadamard formula recalled in (A.15), we have thus for \(t>t_0\)
for all \(\epsilon >0,\) and we deduce (A.18).
The exponential convergence is then ensured by showing that there is no other eigenvalue than 1 on the unit disk (or outside the unit disk) and that 1 is a simple eigenvalue; see [98].
\(\square \)
Rights and permissions
About this article
Cite this article
Chekroun, M.D., Tantet, A., Dijkstra, H.A. et al. Ruelle–Pollicott Resonances of Stochastic Systems in Reduced State Space. Part I: Theory. J Stat Phys 179, 1366–1402 (2020). https://doi.org/10.1007/s1095502002535x
Received:
Accepted:
Published:
Issue Date:
Keywords
 Ruelle–Pollicott resonances
 Conditional expectation
 Correlation functions
 Kolmogorov operator
 Markov semigroups