Optimal linear responses for Markov chains and stochastically perturbed dynamical systems

TL;DR

This paper develops a theory for optimally perturbing Markov chains and stochastic dynamical systems to maximize their linear response in various properties, with explicit solutions and numerical applications.

Contribution

It introduces a novel framework for selecting optimal perturbations to enhance linear responses in Markov chains and stochastic systems, including explicit solutions and numerical methods.

Findings

Explicit solutions for optimal perturbations in finite-state Markov chains

Numerical application to stochastically perturbed dynamical systems

Framework for maximizing linear response of system properties

Abstract

The linear response of a dynamical system refers to changes to properties of the system when small external perturbations are applied. We consider the little-studied question of selecting an optimal perturbation so as to (i) maximise the linear response of the equilibrium distribution of the system, (ii) maximise the linear response of the expectation of a specified observable, and (iii) maximise the linear response of the rate of convergence of the system to the equilibrium distribution. We also consider the inhomogeneous or time-dependent situation where the governing dynamics is not stationary and one wishes to select a sequence of small perturbations so as to maximise the overall linear response at some terminal time. We develop the theory for finite-state Markov chains, provide explicit solutions for some illustrative examples, and numerically apply our theory to stochastically perturbed dynamical systems, where the Markov chain is replaced by a matrix representation of an approximate annealed transfer operator for the random dynamical system.