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Response and sensitivity using Markov chains. (English) Zbl 1446.82065

Summary: Dynamical systems are often subject to forcing or changes in their governing parameters and it is of interest to study how this affects their statistical properties. A prominent real-life example of this class of problems is the investigation of climate response to perturbations. In this respect, it is crucial to determine what the linear response of a system is as a quantification of sensitivity. Alongside previous work, here we use the transfer operator formalism to study the response and sensitivity of a dynamical system undergoing perturbations. By projecting the transfer operator onto a suitable finite dimensional vector space, one is able to obtain matrix representations which determine finite Markov processes. Further, using perturbation theory for Markov matrices, it is possible to determine the linear and nonlinear response of the system given a prescribed forcing. Here, we suggest a methodology which puts the scope on the evolution law of densities (the Liouville/Fokker-Planck equation), allowing to effectively calculate the sensitivity and response of two representative dynamical systems.

MSC:

82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
35Q84 Fokker-Planck equations
86A05 Hydrology, hydrography, oceanography
82C35 Irreversible thermodynamics, including Onsager-Machlup theory
60J22 Computational methods in Markov chains

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