Information Limits of Geometry-Induced Non-Gaussian Laws: From Hitting Distributions to Variance Mixtures

Abstract

In this talk, I present a geometric and statistical perspective on a class of non-Gaussian probability laws induced by boundary-hitting observables of drift–diffusion processes. Consider a multidimensional Brownian motion with constant drift evolving in a half-space and absorbed upon first hitting a hyperplane. Instead of recording the hitting time, we observe the transverse hitting location on the boundary. This observable induces a non-Gaussian probability distribution whose structure is entirely determined by the underlying transport geometry. I show that the resulting probability law admits a Gaussian variance-mixture representation, leading to a geometry-driven family that interpolates continuously between finite-variance regimes and the singular Cauchy limit as drift vanishes. This representation enables an exact high-SNR capacity expansion under a second-moment constraint. From an information-theoretic viewpoint, the channel exhibits three striking features:
(1) the pre-log factor depends solely on the dimension of the receiving boundary;
(2) the asymptotic upper and lower bounds coincide at the constant level, yielding a vanishing capacity gap; and
(3) all physical parameters of the transport process influence capacity only through the differential entropy of the induced noise.
Beyond its communication interpretation, this framework connects Gaussian variance mixtures, hitting distributions, and geometric observables of stochastic processes, suggesting a broader statistical structure underlying boundary-induced randomness.

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