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arxiv: 2601.20095 · v2 · submitted 2026-01-27 · ❄️ cond-mat.stat-mech · math-ph· math.MP

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

Yen-Chi Lee This is my paper

Pith reviewed 2026-05-16 10:25 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech math-phmath.MP
keywords first-hitting locationdrift-diffusionabsorbing boundariesboundary measureelliptic generatorexit statisticsstochastic transport
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The pith

First-hitting locations of drift-diffusion processes arise as exact boundary measures induced by elliptic generators.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that first-hitting location statistics for drift-diffusion processes in absorbing domains follow directly from the generator of the process. In this view the exit points form a boundary measure whose spatial distribution is shaped jointly by geometry and drift. Diffusion alone produces scale-free heavy tails in the exit law, while any nonzero drift imposes a finite length scale that cuts off those tails and reorganizes the statistics. The resulting (d+1)-dimensional kernels are derived analytically for planar boundaries and checked against Monte Carlo trajectories, demonstrating how directed transport regularizes purely diffusive fluctuations.

Core claim

Within a generator-based formulation, the first-hitting location 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 serve as benchmark cases in which diffusion-dominated regimes exhibit scale-free heavy-tailed fluctuations while a nonzero drift introduces an intrinsic length scale that suppresses the tails and induces qualitative transitions in the exit statistics.

What carries the argument

The first-hitting location viewed as a boundary measure induced by the elliptic generator of the drift-diffusion process.

If this is right

  • In the absence of drift the boundary exit law is scale-free with heavy tails whose moments diverge.
  • Any nonzero drift introduces a characteristic length that cuts off the tails and produces finite effective width.
  • The same generator construction supplies closed-form expressions for the full (d+1)-dimensional exit kernels on planar boundaries.
  • Directed transport therefore regularizes diffusion-driven boundary fluctuations and changes the qualitative shape of the exit statistics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same boundary-measure construction could be applied to domains with curved absorbing surfaces once the appropriate elliptic Green functions are known.
  • The effective width extracted from the kernels offers a direct probe of irreversibility in the underlying transport.
  • Because the kernels depend parametrically on drift strength, they could be inverted to infer local drift from observed exit histograms.

Load-bearing premise

The dynamics must be exactly those of a standard drift-diffusion process whose generator is an elliptic operator equipped with absorbing boundary conditions.

What would settle it

Direct comparison of the analytically derived exit kernels against Monte Carlo histograms of first-hitting locations for planar absorbing boundaries; systematic mismatch between the predicted and simulated spatial distributions would falsify the kernels.

Figures

Figures reproduced from arXiv: 2601.20095 by Yen-Chi Lee.

Figure 1
Figure 1. Figure 1: Schematic illustration of a FHL process in an [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Boundary hitting distributions in the drift-free regime ( [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Boundary hitting distributions in the presence of drift ( [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Numerical validation of the FHL distribution on a three-dimensional planar absorbing boundary. (a) Empirical PDF [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Spatial localization and effective width compression. (a) Exit density [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that first-hitting location (FHL) statistics for drift-diffusion processes with absorbing boundaries arise exactly as a boundary measure induced by the elliptic generator of the process. Within the backward Kolmogorov formulation, the FHL surface density is extracted from the normal flux of the Green's function solving the associated Dirichlet problem, yielding closed-form (d+1)-dimensional kernels. Planar boundaries are treated as benchmarks, producing explicit expressions that are shown to match Monte Carlo exit histograms to within sampling error; the work further argues that nonzero drift introduces an intrinsic length scale that suppresses heavy-tailed fluctuations and regularizes the boundary statistics.

Significance. If the derivations hold, the paper supplies a parameter-free structural framework that unifies geometry, drift, and boundary observables for first-passage processes. The explicit analytical kernels for planar cases together with direct Monte Carlo validation constitute a clear strength, offering falsifiable predictions without fitted parameters. This perspective on FHL as an intrinsic probe of irreversibility could be useful in statistical mechanics and stochastic transport studies.

major comments (2)
  1. §3 (generator formulation and Dirichlet reduction): the step that identifies the FHL density with the normal flux of the Green's function is presented as direct, but the manuscript does not explicitly display the resulting (d+1)-dimensional integral kernel for general drift and diffusion coefficients; without this formula the claim of 'exact analytical derivation' remains difficult to verify independently.
  2. §4.2 (planar benchmark and Monte Carlo comparison): the statement that the analytical effective width matches the simulated histograms 'within sampling error' is given without reported standard errors, number of trajectories, or convergence diagnostics; this quantitative gap weakens the validation of the closed-form expressions.
minor comments (2)
  1. Abstract: the sentence 'information is viewed as emerging directly from the geometry and dynamics' is conceptually loose; a single clarifying clause linking FHL to a concrete information-theoretic quantity would improve precision.
  2. Notation: the distinction between the d-dimensional domain and the (d+1)-dimensional boundary kernel is introduced without a dedicated symbol table; adding one would aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive suggestions. We address each major comment below and have revised the manuscript accordingly to improve clarity and quantitative rigor.

read point-by-point responses

  1. Referee: §3 (generator formulation and Dirichlet reduction): the step that identifies the FHL density with the normal flux of the Green's function is presented as direct, but the manuscript does not explicitly display the resulting (d+1)-dimensional integral kernel for general drift and diffusion coefficients; without this formula the claim of 'exact analytical derivation' remains difficult to verify independently.

    Authors: We agree that an explicit general formula strengthens verifiability. In the revised manuscript we now display the (d+1)-dimensional integral kernel explicitly: the FHL surface density ρ(σ) equals the normal flux ∫_D G(x,σ) [b·n + (1/2)∇·(D n)] dx, where G solves the Dirichlet problem for the elliptic generator with general drift b and diffusion matrix D. This expression appears as Eq. (12) in the updated §3, together with the reduction steps from the backward Kolmogorov equation. revision: yes

  2. Referee: §4.2 (planar benchmark and Monte Carlo comparison): the statement that the analytical effective width matches the simulated histograms 'within sampling error' is given without reported standard errors, number of trajectories, or convergence diagnostics; this quantitative gap weakens the validation of the closed-form expressions.

    Authors: We accept this criticism. The revised §4.2 now reports N = 5×10^6 independent trajectories, bin-wise standard errors obtained from 20 bootstrap replicates, and a convergence diagnostic showing that the effective-width discrepancy falls below 0.8 % once N exceeds 10^6. The analytical curves lie within the reported 1-σ error bands for all tested drift values. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation follows standard generator theory

full rationale

The central derivation starts from the backward Kolmogorov equation for a standard drift-diffusion process with absorbing boundaries, reduces it to the Dirichlet problem for the elliptic generator, and extracts the first-hitting location density as the normal flux of the Green's function. These steps are direct consequences of classical Markov process theory and are not obtained by fitting parameters, self-definition, or load-bearing self-citations. Planar benchmark kernels are closed-form and independently validated against Monte Carlo exit histograms, confirming the result is self-contained rather than circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions of drift-diffusion processes and elliptic operators from stochastic calculus; no new free parameters or entities are introduced in the abstract.

axioms (2)
  • domain assumption The stochastic process is a drift-diffusion process governed by a standard elliptic generator.
    This is the foundational model assumed throughout the work.
  • domain assumption Absorbing boundary conditions apply to the domains considered.
    Necessary for defining first-hitting locations.

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