This paper addresses two fundamental problems in diffusive molecular communication: characterizing the first arrival position (FAP) density and determining 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 previously undiscovered weak stability properties intrinsic to vertically-drifted FAP noise distributions.

翻译：本文研究了扩散分子通信中的两个基本问题：表征首次到达位置（FAP）密度并确定FAP信道的信息传输容量。以往关于FAP信道模型的研究，主要采用噪声密度函数来描述，但仅限于特定的空间维度、漂移方向和接收器几何形状。为此，我们提出了一种统一解决方案，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，涵盖任意空间维度、漂移方向和接收器形状。我们证明了若干现有FAP密度公式是该创新表达式的特例。同时，受矢量高斯干扰信道的启发，我们建立了三维垂直漂移FAP信道容量的显式上下界。在推导这些界值的过程中，我们揭示了垂直漂移FAP噪声分布特征函数的显式解析表达式，相比于密度函数，该表达式提供了更紧凑的描述。值得关注的是，这一表达式揭示了垂直漂移FAP噪声分布中此前未被发现的弱稳定性属性。