论文标题
马尔可夫跳跃过程中的时间依赖性和时间周期性线性响应的martingale方法
A martingale approach to time-dependent and time-periodic linear response in Markov jump processes
论文作者
论文摘要
我们考虑在一般状态空间上进行的马尔可夫跳跃过程，我们在有限的时间间隔内应用了时间依赖的弱扰动。通过基于Martingale的随机演算，在扰动的适当指数力矩结合下，我们表明，扰动过程几乎不会爆炸，并且我们研究了可观察到的和添加功能的线性响应（LR）。当不受干扰的过程是静止的时，上述LR公式可以根据稳态两次相关函数和固定分布的计算。讨论了用于出生和死亡过程的应用，在狭窄的潜力中随机步行，随机导电场随机行走。然后，我们转到有限状态空间上的马尔可夫跳跃过程，并研究振动稳态（因此，在无限时间范围内）的观察力和添加剂功能的LR，当扰动是时间周期性时。作为一个应用程序，我们为在离散的$ d $维圆环上随机步行的复杂移动矩阵提供了一个公式，可能是异质的跳跃率。
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.
