First-Hitting Location Laws as Boundary Observables of Drift-Diffusion Processes

TL;DR

This paper studies how first-hitting location statistics in drift-diffusion processes reveal boundary geometry and dynamics, providing analytical tools and simulations to understand their role as natural probes of stochastic transport.

Contribution

It introduces a unified framework for analyzing first-hitting location laws as boundary observables influenced by geometry and drift, with exact analytical derivations and validation.

Findings

Heavy-tailed boundary fluctuations in diffusion regimes

Drift introduces a length scale that suppresses tails

Directed transport regularizes fluctuations and causes boundary statistic transitions

Abstract

We investigate first-hitting location (FHL) statistics induced by drift-diffusion processes in domains with absorbing boundaries, and examine how such boundary laws give rise to intrinsic information observables. Rather than introducing explicit encoding or decoding mechanisms, information is viewed as emerging directly from the geometry and dynamics of stochastic transport through first-passage events. Treating the FHL as the primary observable, we characterize how geometry and drift jointly shape the induced boundary measure. In diffusion-dominated regimes, the exit law exhibits scale-free, heavy-tailed spatial fluctuations along the boundary, whereas a nonzero drift component introduces an intrinsic length scale that suppresses these tails and reorganizes the exit statistics. Within a generator-based formulation, the FHL arises naturally as a boundary measure induced by an elliptic operator, allowing exact $(d+1)$-dimensional boundary kernels to be derived analytically. Planar absorbing boundaries are examined as benchmark cases and validated via Monte Carlo simulations, illustrating how directed transport thermodynamically regularizes diffusion-driven fluctuations -- quantified by a robust effective width -- and induces qualitative transitions in boundary statistics. Overall, the present work provides a unified structural framework for first-hitting location laws and highlights FHL statistics as natural probes of geometry, drift, and irreversibility in stochastic transport.