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

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.

💡 Research Summary

The paper investigates the high‑power (high‑SNR) capacity scaling of additive noise channels whose noise originates from the first‑hitting location of a multidimensional drift‑diffusion process on an absorbing hyperplane. The authors recognize that the probability density function (PDF) of the previously studied vertically‑drifted first‑arrival‑position (VDF‑AP) noise involves modified Bessel functions and is analytically intractable for capacity analysis. To overcome this, they shift focus from the PDF to the underlying stochastic generation mechanism and model the noise as a zero‑mean Gaussian variance‑mixture (GVM). Specifically, the first‑hitting time T follows an inverse‑Gaussian (IG) distribution with mean ν=λ/μ and shape κ=λ²/σ², while conditional on T the transverse displacement N is Gaussian with covariance σ²T·I_{d‑1}. Mixing over T yields the Normally‑Drifted First‑Hitting Location (NDFHL) family, parameterized by dimension d, boundary distance λ, and normalized drift speed u=μ/σ².

The characteristic function of NDFHL admits a compact closed form: Φ_N(ω)=exp{−λ√{‖ω‖²+u²}+λu}. As u→0 this reduces to the characteristic function of a multivariate Cauchy distribution, establishing a continuous bridge between the finite‑variance regime (u>0) and the singular infinite‑variance Cauchy limit. Moreover, because the IG distribution is infinitely divisible, the NDFHL family inherits infinite divisibility, reflecting the additive nature of layered transport processes.

Under a second‑moment input constraint E