Capacity Scaling Laws for Boundary-Induced Drift-Diffusion Noise Channels

TL;DR

This paper characterizes the high-power capacity scaling of boundary-induced drift-diffusion noise channels, revealing geometric and entropy-based universal properties and establishing asymptotic optimality of Gaussian signaling.

Contribution

It introduces the NDFHL model, derives exact high-SNR capacity expansions, and uncovers the geometric origin of degrees of freedom and a universal entropy-based capacity characterization.

Findings

Gaussian signaling is asymptotically capacity-achieving.

Capacity pre-log depends only on boundary dimension.

Capacity affected by transport parameters through noise entropy.

Abstract

This paper studies the high-power capacity scaling of additive noise channels whose noise arises from the first-hitting location of a multidimensional drift-diffusion process on an absorbing hyperplane. By identifying the underlying stochastic transport mechanism as a Gaussian variance-mixture, we introduce and analyze the Normally-Drifted First-Hitting Location (NDFHL) family as a geometry-driven model for boundary-induced noise. Under a second-moment constraint, we derive an exact high-SNR capacity expansion and show that the asymptotic upper and lower bounds coincide at the constant level, yielding a vanishing capacity gap. As a consequence, isotropic Gaussian signaling is asymptotically capacity-achieving for all fixed drift strengths, despite the non-Gaussian and semi-heavy-tailed nature of the noise. The pre-log factor is determined solely by the dimension of the receiving boundary, revealing a geometric origin of the channel's degrees of freedom. The refined expansion further uncovers an entropy-dominant universality, whereby all physical parameters of the transport process -- including drift strength, diffusion coefficient, and boundary separation -- affect the capacity only through the differential entropy of the induced noise. Although the NDFHL density does not admit a simple closed form, its entropy is shown to be finite and to vary continuously as the drift vanishes, thereby connecting the finite-variance regime with the singular infinite-variance Cauchy limit. Together, these results provide a unified geometric and information-theoretic characterization of boundary-hitting channels across both regular and singular transport regimes.