A martingale approach to timedependent and timeperiodic linear response in Markov jump processes
Abstract
We consider a Markov jump process on a general state space to which we apply a timedependent weak perturbation over a finite time interval. By martingalebased 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 twotime 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 timeperiodic. 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2022
 arXiv:
 arXiv:2201.02982
 Bibcode:
 2022arXiv220102982F
 Keywords:

 Mathematics  Probability;
 Condensed Matter  Statistical Mechanics;
 Mathematical Physics
 EPrint:
 40 pages