多层介质中单颗粒扩散的更新方程
Renewal equations for single-particle diffusion in multi-layered media
摘要：
在本文中，我们建立了一个概率模型的单粒子扩散在一维多层介质中，通过构建一个多层版本的所谓抢购出布朗运动（BM）。后者将连续轮的反射BM缝合在一起，每个反射BM被限制在单个层中。当层一端的本地时间超过一个独立的、指数分布的随机变量时，每一轮反射的BM被杀死。(The局部时间指定反射的布朗粒子在边界的邻域中花费的时间量。然后，粒子立即以相等的概率在同一层或边界另一侧的层中恢复反射的BM，并且迭代该过程。我们通过构建用于多层弹出BM的最后更新方程来进行，该方程将全概率密度与每层中部分反射的BM的概率密度相关联。然后，我们展示了如何转移矩阵可以用来解决拉普拉斯变换更新方程，并证明更新方程和相应的多层扩散方程是等价的。我们说明了理论分析的第一次通过时间（FPT）的问题，在域的外部边界逃逸。最后，我们使用更新的方法，结合概括的抢购出BM的基础上，基于反射器的表面吸收的方法;每一轮的反射BM现在被杀死根据非指数分布的每个本地时间阈值。这是通过考虑对应的第一更新方程来实现的，该第一更新方程将全概率密度与用于杀死每轮反射BM的FPT密度相关联。我们表明，对于某些配置，非指数杀伤导致一个有效的时间依赖性的渗透率是归一化的，但重尾。
英文摘要：
In this paper we develop a probabilistic model of single-particle diffusion in 1D multi-layered media by constructing a multi-layered version of so-called snapping out Brownian motion (BM). The latter sews together successive rounds of reflected BM, each of which is restricted to a single layer. Each round of reflected BM is killed when the local time at one end of the layer exceeds an independent, exponentially distributed random variable. (The local time specifies the amount of time a reflected Brownian particle spends in a neighborhood of a boundary.) The particle then immediately resumes reflected BM in the same layer or the layer on the other side of the boundary with equal probability, and the process is iterated We proceed by constructing a last renewal equation for multi-layered snapping out BM that relates the full probability density to the probability densities of partially reflected BM in each layer. We then show how transfer matrices can be used to solve the Laplace transformed renewal equation, and prove that the renewal equation and corresponding multi-layer diffusion equation are equivalent. We illustrate the theory by analyzing the first passage time (FPT) problem for escape at the exterior boundaries of the domain. Finally, we use the renewal approach to incorporate a generalization of snapping out BM based on the encounter-based method for surface absorption; each round of reflected BM is now killed according to a non-exponential distribution for each local time threshold. This is achieved by considering a corresponding first renewal equation that relates the full probability density to the FPT densities for killing each round of reflected BM. We show that for certain configurations, non-exponential killing leads to an effective time-dependent permeability that is normalizable but heavy-tailed.
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