Joint Time-Position Statistics and Fisher Information in Drift-Diffusion Molecular Channels

This letter presents a closed-form characterization of the joint distribution of first arrival time (FAT) and first arrival position (FAP) in diffusion-based molecular communication (MC) systems with drift. Prior studies have investigated FAT modeling via inverse Gaussian distributions [1] and applied FAT statistics for parameter estimation and synchronization tasks [2], [3], while more recent work has characterized FAP for spatial channel analysis [4]. In contrast, we derive an explicit joint probability density function (PDF) under constant drift and isotropic diffusion in arbitrary spatial dimensions. Our result reveals a nontrivial coupling between arrival time and lateral position, generalizing known inverse Gaussian models. We further compute the Fisher information matrix (FIM) with respect to key channel parameters, showing that the joint observation enables estimation of lateral drift and improves sensitivity to the diffusion coefficient -- capabilities not achievable with time-only or position-only models. This joint framework enhances the modeling and inference capabilities for molecular communication channels where spatial randomness itself carries non-negligible information.