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First Passage Problem: Asymptotic Corrections due to Discrete Sampling

Lars Fritz

TL;DR

This work shows that first-passage statistics are fundamentally reshaped by discrete, stroboscopic observation. By embedding the observation protocol into the evolution via the one-step operator $K=P\,G_0\,P$ and its projector–resolvent structure, the authors derive a linear scaling with $\rho$ for boundary starts and a quadratic scaling with universal linear and constant corrections for bulk starts, with constants tied to Gaussian kernels and domain geometry. They further extend the framework to random frame intervals, demonstrating that only subleading terms depend on timing variance, and validate the theory with a Nyström discretization that yields high-precision constants. The results reveal that measurement protocol is an intrinsic component of stochastic dynamics, with broad implications for experiments in single-particle tracking and related stochastic timing problems.

Abstract

How long a stochastic process survives before leaving a domain depends not only on its intrinsic dynamics but also on how it is observed. Classical first-passage theory assumes continuous monitoring with absorbing boundaries (``kill-on-touch''). In practice, however, measurements are often taken at discrete times. Between two checks, a trajectory may leave and re-enter the domain without being detected. Under this \emph{stroboscopic} rule (``kill-on-check''), exit statistics change qualitatively. We analyze one-dimensional Brownian motion confined to an interval of length $L$ and observed at frame intervals~$Δt$, with diffusive step scale $σ\sqrt{Δt}$. The dynamics collapse onto a single confinement ratio $ρ=L/(σ\sqrt{Δt})$. For boundary starts we obtain linear scaling of the mean number of frames until exit, while for bulk starts the survival is governed by the spectral gap of a one-step stroboscopic operator, leading to a quadratic law with linear corrections. These results identify the stroboscopic first-passage problem where the observation protocol itself reshapes the statistics of escape.

First Passage Problem: Asymptotic Corrections due to Discrete Sampling

TL;DR

This work shows that first-passage statistics are fundamentally reshaped by discrete, stroboscopic observation. By embedding the observation protocol into the evolution via the one-step operator and its projector–resolvent structure, the authors derive a linear scaling with for boundary starts and a quadratic scaling with universal linear and constant corrections for bulk starts, with constants tied to Gaussian kernels and domain geometry. They further extend the framework to random frame intervals, demonstrating that only subleading terms depend on timing variance, and validate the theory with a Nyström discretization that yields high-precision constants. The results reveal that measurement protocol is an intrinsic component of stochastic dynamics, with broad implications for experiments in single-particle tracking and related stochastic timing problems.

Abstract

How long a stochastic process survives before leaving a domain depends not only on its intrinsic dynamics but also on how it is observed. Classical first-passage theory assumes continuous monitoring with absorbing boundaries (``kill-on-touch''). In practice, however, measurements are often taken at discrete times. Between two checks, a trajectory may leave and re-enter the domain without being detected. Under this \emph{stroboscopic} rule (``kill-on-check''), exit statistics change qualitatively. We analyze one-dimensional Brownian motion confined to an interval of length and observed at frame intervals~, with diffusive step scale . The dynamics collapse onto a single confinement ratio . For boundary starts we obtain linear scaling of the mean number of frames until exit, while for bulk starts the survival is governed by the spectral gap of a one-step stroboscopic operator, leading to a quadratic law with linear corrections. These results identify the stroboscopic first-passage problem where the observation protocol itself reshapes the statistics of escape.

Paper Structure

This paper contains 44 sections, 84 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The setup: Diffusion equation, the Green function, and the Wiener path integral
  3. Diffusion equation as conservation law
  4. Green function and free propagation
  5. Spectral representation and mode filtering
  6. Wiener path integral and additivity of the action
  7. Complementarity and outlook
  8. Survival: continuous vs. stroboscopic
  9. Continuous monitoring (Dirichlet absorption)
  10. Stroboscopic monitoring (discrete absorption)
  11. Undetected excursions and relaxation of the boundary condition
  12. Continuous benchmark via Dirichlet resolvent
  13. Dirichlet kernel and Poisson problem
  14. Explicit solution and physical interpretation
  15. Role as baseline for discrete monitoring
  16. ...and 29 more sections

Figures (4)

  • Figure 1: Schematic of stroboscopic monitoring. A Brownian trajectory (gray line) is sampled only at discrete observation times $t_n = n\Delta t$ (vertical markers). Between frames the motion is continuous, allowing undetected excursions—segments that leave and re-enter the confining domain before the next check. Under this "kill-on-check" rule, absorption is enforced only at sampling instants, in contrast to continuous (Dirichlet) monitoring where escape is detected immediately.
  • Figure 2: Apparent scaling exponents in finite $\rho$-windows for the bulk-start law $\mathbb{E}[\tau](\rho;\tfrac{1}{2})=\tfrac{1}{4}\rho^2+0.58\rho+0.57$ (solid). The reference $\tfrac{1}{4}\rho^2$ ($\alpha=2$) is shown dashed. Log--log fits over limited windows yield effective exponents $\alpha_{\mathrm{eff}}\!\approx\!1.87$ on $[10,30]$ and $\alpha_{\mathrm{eff}}\!\approx\!1.96$ on $[30,100]$. The subleading linear term $0.58\rho$ thus creates transient $\alpha_{\mathrm{eff}}<2$ over realistic ranges, a mechanism relevant for interpreting discretely sampled trajectories.
  • Figure 3: Boundary start ($y_0=0$). Nyström data for $M(\rho;0)$ (markers), fit $A\rho+B+C/\rho$ (solid), and asymptotic prediction $\rho/\sqrt{2}+(\lvert\zeta(\tfrac{1}{2})\rvert/\sqrt{\pi}-1)$ (dashed).
  • Figure 4: Bulk start ($y_0=\tfrac{1}{2}$). Nyström data for $M(\rho;\tfrac{1}{2})$ (markers), quadratic fit $a\rho^2+b\rho+c$ (solid), and asymptotic prediction $\tfrac{1}{4}\rho^2$ (dashed).