Optimal linear response for Markov Hilbert-Schmidt integral operators and stochastic dynamical systems

Abstract

We consider optimal control problems for discrete-time random dynamical systems, finding unique perturbations that provoke maximal responses of statistical properties of the system. We treat systems whose transfer operator has an $L^2$ kernel, and we consider the problems of finding (i) the infinitesimal perturbation maximising the expectation of a given observable and (ii) the infinitesimal perturbation maximising the spectral gap, and hence the exponential mixing rate of the system. Our perturbations are either (a) perturbations of the kernel or (b) perturbations of a deterministic map subjected to additive noise. We develop a general setting in which these optimisation problems have a unique solution and construct explicit formulae for the unique optimal perturbations. We apply our results to a Pomeau-Manneville map and an interval exchange map, both subjected to additive noise, to explicitly compute the perturbations provoking maximal responses.

References in corpus (14)

• The Physics of Climate Variability and Climate Change
• A simple framework to justify linear response theory
• An analytic framework for identifying finite-time coherent sets in time-dependent dynamical systems
• Linear response, or else
• Controlling the statistical properties of expanding maps
• Quadratic response of random and deterministic dynamical systems
• Spectral degeneracy and escape dynamics for intermittent maps with a hole
• Optimal linear responses for Markov chains and stochastically perturbed dynamical systems
• Fréchet differentiable drift dependence of Perron--Frobenius and Koopman operators for non-deterministic dynamics
• Regularity of characteristic exponents and linear response for transfer operator cocycles
• Cutting and shuffling with diffusion: Evidence for cut-offs in interval exchange maps
• Steady states, fluctuation-dissipation theorems and homogenization for diffusions in a random environment with finite range of dependence
• Einstein relation and linear response in one-dimensional Mott variable-range hopping
• Quantitative statistical stability and speed of convergence to equilibrium for partially hyperbolic skew products

Cited by in corpus (7)

• Theoretical tools for understanding the climate crisis from Hasselmann's program and beyond
• Self consistent transfer operators. Invariant measures, convergence to equilibrium, linear response and control of the statistical properties
• Recursive divergence formulas for perturbing unstable transfer operators and physical measures
• Fast differentiation of hyperbolic chaos
• Kolmogorov Modes and Linear Response of Jump-Diffusion Models
• Bridging the Gap between Koopmanism and Response Theory: Using Natural Variability to Predict Forced Response
• Invariant Measures in Time-Delay Coordinates for Unique Dynamical System Identification