First Passage Problem: Asymptotic Corrections due to Discrete Sampling

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~$\Delta t$, with diffusive step scale $\sigma\sqrt{\Delta t}$. The dynamics collapse onto a single confinement ratio $\rho=L/(\sigma\sqrt{\Delta t})$. 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.