Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, USA.
Phys Rev E. 2017 Jan;95(1-1):012130. doi: 10.1103/PhysRevE.95.012130. Epub 2017 Jan 19.
We generalize the Feynman-Kac formula to analyze the local and occupation times of a Brownian particle moving in a stochastically gated one-dimensional domain. (i) The gated local time is defined as the amount of time spent by the particle in the neighborhood of a point in space where there is some target that only receives resources from (or detects) the particle when the gate is open; the target does not interfere with the motion of the Brownian particle. (ii) The gated occupation time is defined as the amount of time spent by the particle in the positive half of the real line, given that it can only cross the origin when a gate placed at the origin is open; in the closed state the particle is reflected. In both scenarios, the gate randomly switches between the open and closed states according to a two-state Markov process. We derive a stochastic, backward Fokker-Planck equation (FPE) for the moment-generating function of the two types of gated Brownian functional, given a particular realization of the stochastic gate, and analyze the resulting stochastic FPE using a moments method recently developed for diffusion processes in randomly switching environments. In particular, we obtain dynamical equations for the moment-generating function, averaged with respect to realizations of the stochastic gate.
我们将费曼-卡茨公式推广到分析在随机门控一维域中运动的布朗粒子的局部时间和占据时间。（i）门控局部时间定义为粒子在空间中某个点附近花费的时间，在该点处存在一个目标，只有当门打开时目标才会从（或检测到）粒子接收资源；目标不会干扰布朗粒子的运动。（ii）门控占据时间定义为粒子在实线上正半轴花费的时间，前提是当位于原点的门打开时，粒子只能穿过原点；在关闭状态下，粒子被反射。在这两种情况下，门根据二态马尔可夫过程在打开和关闭状态之间随机切换。我们为两种类型的门控布朗函数的矩生成函数推导出一个随机的反向福克-普朗克方程（FPE），给定随机门的特定实现，并使用最近为随机切换环境中的扩散过程开发的矩方法分析得到的随机 FPE。特别地，我们获得了关于矩生成函数的动态方程，该方程是相对于随机门的实现进行平均的。
