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

Understand

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.