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

Understand

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.