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

TL;DR

This work analyzes boundary-hitting additive noise channels arising from drift–diffusion to an absorbing hyperplane. By revealing a latent Gaussian variance-mean mixture with first-passage time $T hicksim IG$, the authors define the Normally-Drifted First-Hitting Location (NDFHL) distribution family, showing it is radially symmetric and infinitely divisible. They derive an exact high-SNR capacity expansion under a second-moment constraint, proving a universal pre-log $p/2$ determined solely by the receiving boundary dimension and establishing that Gaussian signaling is asymptotically optimal. The refined expansion shows the capacity offset depends only on the differential entropy of the boundary-induced noise, yielding an entropy-dominant universal scaling that remains connected to the Cauchy limit as $u o0$. Together, these results provide a geometric-information-theoretic framework for boundary-hitting channels across regular and singular transport regimes, with implications for molecular and nanoscale communication systems.

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.

Figures (2)

• Figure 1: Physical origin of the boundary-induced additive noise $\mathrm{NDFHL}^{(d)}$. A particle released from the origin $\mathbf{0}$ in the half-space $x_1<\lambda$ undergoes drift--diffusion and is absorbed upon first hitting the hyperplane $\mathcal{B}: x_1=\lambda$. The receiver records the first-hitting location $\mathbf{N}\in\mathbb{R}^{d-1}$ at the random hitting time $T$. This first-hitting location serves as the additive noise term in the equivalent channel model, with its statistics determined solely by the boundary geometry and the underlying drift--diffusion dynamics.
• Figure 2: Differential entropy $h(\mathbf{N})$ of the $\mathrm{NDFHL}^{(d)}$ distribution as a function of the normalized drift speed $u$ for dimensions $d=2,3,4$ with $\lambda=1$. Solid markers correspond to numerical integration results, while the dashed horizontal lines indicate the differential entropy of the corresponding isotropic multivariate Cauchy distributions obtained in the limit $u=0$. For all $u>0$, the entropy $h(\mathbf{N})$ is finite and is observed to approach, as $u \downarrow 0$, the Cauchy entropy value $g(p)$, see \ref{['eq:gp_def']}, even though the second moment of the noise diverges in this limit.

Theorems & Definitions (19)

• Remark 1: Parameter convention
• Remark 2: On related distributions and novelty
• Definition 1: $\mathrm{NDFHL}^{(d)}$ Distribution
• Proposition 1: Isotropy
• Proposition 2: Mean
• Lemma 1: Covariance
• Lemma 2: Characteristic Function of $\mathrm{NDFHL}$
• Lemma 3: Convolution Closure
• Corollary 1: Infinite Divisibility
• Lemma 4: Entropy Bounds and Decomposition