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

TL;DR

The paper treats first-hitting location (FHL) as a boundary observable for drift–diffusion in domains with absorbing boundaries and develops a generator–Green-function framework that links FHL statistics to the geometry and drift via the boundary kernel $\mathcal{K}(\boldsymbol{x},\boldsymbol{y}) = -\partial_{n(\boldsymbol{y})} \mathcal{G}(\boldsymbol{x},\boldsymbol{y})$. By solving elliptic boundary-value problems for the Dirichlet Green function, explicit closed-form kernels are derived in the absorbing half-space: in 2D (absorbing line) and 3D (absorbing plane), revealing how drift suppresses diffusion-induced fluctuations and introduces an intrinsic length scale $\ell_u = \sigma^2/\|\boldsymbol{v}\| = 1/u$. A unified, dimensionally consistent conjectured form for $d$-dimensional boundaries is presented, supported by the exact 1D and 2D cases, and the framework is used to analyze asymptotic structure and information observables such as the differential entropy of the exit location. The results establish FHL laws as natural probes of geometry, drift, and irreversibility in stochastic transport and provide a foundation for future work on curved boundaries and time-dependent drift, with potential implications for boundary detector design and information-theoretic characterizations of nonequilibrium transport.

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 explicit boundary kernels to be derived analytically in canonical geometries. Planar absorbing boundaries in different ambient dimensions are examined as benchmark cases, illustrating how directed transport regularizes diffusion-driven fluctuations 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.

Figures (4)

• Figure 1: Schematic illustration of a FHL process in an absorbing half-space. A drift--diffusion trajectory is initialized at a normal distance $\lambda$ from the absorbing boundary and evolves in the bulk until it is terminated upon first contact with the boundary. The spatial location of this exit event defines the primary observable of interest. The figure provides an interpretive visualization of how stochastic transport maps initial conditions to boundary exit measures.
• Figure 2: Schematic illustration of a drift--diffusion process in a half-space with an absorbing boundary (shown for $d=3$). A trajectory initialized at $\bm{x}_0\in\Omega$ evolves in the bulk under a constant drift $\bm{v}$ and isotropic diffusion, and terminates upon first reaching the absorbing boundary $\partial\Omega$ located at a normal distance $\lambda$. The corresponding exit time $\tau_\Omega$ and boundary location $\mathbf{X}_{\tau_\Omega}$ define the induced boundary law. The figure highlights the geometric setting and variables underlying the first-hitting location framework analyzed in the text.
• Figure 3: Boundary hitting distributions in the drift-free regime ($\boldsymbol{u}=0$), shown for two- and three-dimensional boundaries. In the absence of drift, the first-hitting location law is isotropic and exhibits scale-free algebraic decay, corresponding to a Cauchy-type boundary measure doob2001classical without an intrinsic length scale.
• Figure 4: Boundary hitting distributions in the presence of drift ($\boldsymbol{u}\neq 0$), shown for two- and three-dimensional boundaries. Drift introduces an intrinsic length scale and exponentially suppresses large boundary displacements, leading to localized and anisotropic hitting profiles consistent with the drift-regularized analytical kernels.