A martingale approach to time-dependent and time-periodic linear response in Markov jump processes

TL;DR

This paper develops a martingale-based framework to analyze the linear response of Markov jump processes under time-dependent and periodic perturbations, providing explicit formulas and applications to various stochastic models.

Contribution

It introduces a martingale approach to derive linear response formulas for Markov jump processes under time-dependent and periodic perturbations, including explicit applications.

Findings

Linear response formulas are derived using martingale calculus.

Perturbed processes remain non-explosive under certain conditions.

Applications include birth-death processes, random walks, and mobility matrices.

Abstract

We consider a Markov jump process on a general state space to which we apply a time-dependent weak perturbation over a finite time interval. By martingale-based stochastic calculus, under a suitable exponential moment bound for the perturbation we show that the perturbed process does not explode almost surely and we study the linear response (LR) of observables and additive functionals. When the unperturbed process is stationary, the above LR formulas become computable in terms of the steady state two-time correlation function and of the stationary distribution. Applications are discussed for birth and death processes, random walks in a confining potential, random walks in a random conductance field. We then move to a Markov jump process on a finite state space and investigate the LR of observables and additive functionals in the oscillatory steady state (hence, over an infinite time horizon), when the perturbation is time-periodic. As an application we provide a formula for the complex mobility matrix of a random walk on a discrete $d$-dimensional torus, with possibly heterogeneous jump rates.