This paper addresses two fundamental problems in diffusive molecular communication: characterizing the first arrival position (FAP) density and bounding the information transmission capacity of FAP channels. Previous studies on FAP channel models, mostly captured by the density function of noise, have been limited to specific spatial dimensions, drift directions, and receiver geometries. In response, we propose a unified solution for identifying the FAP density in molecular communication systems with fully-absorbing receivers. Leveraging stochastic analysis tools, we derive a concise expression with universal applicability, covering any spatial dimension, drift direction, and receiver shape. We demonstrate that several existing FAP density formulas are special cases of this innovative expression. Concurrently, we establish explicit upper and lower bounds on the capacity of three-dimensional, vertically-drifted FAP channels, drawing inspiration from vector Gaussian interference channels. In the course of deriving these bounds, we unravel an explicit analytical expression for the characteristic function of vertically-drifted FAP noise distributions, providing a more compact characterization compared to the density function. Notably, this expression sheds light on a previously undiscovered weak stability property intrinsic to vertically-drifted FAP noise distributions.

翻译：本文针对扩散分子通信中的两个基本问题：首达位置（FAP）密度的特性化，以及FAP通道信息传输容量的界定。先前关于FAP通道模型的研究，大多通过噪声密度函数描述，且仅限于特定空间维度、漂移方向和接收器几何形状。为此，我们提出一种统一解决方案，用于识别具有全吸收接收器的分子通信系统中的FAP密度。借助随机分析工具，我们推导出一个具有普适性的简洁表达式，可涵盖任意空间维度、漂移方向及接收器形状。我们证明，若干现有FAP密度公式是该创新表达式的特例。同时，受矢量高斯干扰通道启发，我们为三维垂直漂移FAP通道的容量建立了显式的上下界。在推导这些界的过程中，我们揭示了垂直漂移FAP噪声分布特征函数的一个显式解析表达式，相较于密度函数提供了更紧凑的特性描述。值得注意的是，该表达式揭示了垂直漂移FAP噪声分布中先前未被发现的弱稳定性性质。