Optimal Linear Response for Markov Hilbert–Schmidt Integral Operators and Stochastic Dynamical Systems

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