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

TL;DR

This paper develops a framework for identifying optimal perturbations in stochastic dynamical systems with Hilbert-Schmidt operators, maximizing responses like expectation and spectral gap, with explicit solutions for specific noisy maps.

Contribution

It introduces a novel method to find unique optimal perturbations for systems with $L^2$ kernels, including explicit formulas and applications to noisy Pomeau-Manneville and interval exchange maps.

Findings

Explicit formulas for optimal perturbations in specific systems

Maximal response achieved by perturbations of kernels or deterministic maps with noise

Unique solutions to the optimization problems in the developed framework

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.