AI 中文总结
研究一维扩散粒子在δ杀伤陷阱下随机重置的存活行为，推导拉普拉斯空间精确解，揭示无漂移时递归代数存活与有漂移时瞬态存活，以及重置恢复指数吸收的三种存活区域。
AI 中文摘要
我们研究了一个一维扩散粒子，在存在一个不完美的、局域化的目标（可以吸收（杀伤）粒子，由δ函数杀伤率建模）的情况下，对其初始位置进行随机重置。核心问题是随机重置如何与漂移引起的瞬态性以及目标的有限反应性竞争。我们推导了非归一化传播子和存活概率的精确拉普拉斯空间表达式，适用于任意扩散系数$D$、重置率$r$和杀伤强度$k$，首先在无漂移情况下，然后在恒定漂移下。长时间行为分为不同的区域。没有重置时，无偏扩散表现出递归的代数存活定律$S(t)\sim t^{-1/2}$，而任何非零漂移使得运动相对于目标是瞬态的，并留下非零的最终存活概率。相比之下，任何$r>0$和$k>0$都会反复更新与目标的遭遇并恢复最终吸收，产生指数存活定律$S(t)\sim A e^{-\theta t}$。衰减率$\theta$由拉普拉斯变换的主导（最右）极点给出，存活密度在归一化后收敛到一个显式的准稳态轮廓。无漂移和完美吸收极限恢复了标准的重置结果。这些公式提供了递归控制、瞬态控制和更新控制的存活之间交叉的统一描述。
英文摘要
We study a one-dimensional diffusive particle subject to stochastic resetting to its initial position, in the presence of an imperfect, localized target that can absorb (kill) the particle, modeled by a delta-function killing rate. The central question is how stochastic resetting competes with drift-induced transience and the target's finite reactivity. Exact Laplace-space expressions are derived for the non-normalized propagator and survival probability, for arbitrary diffusion coefficient $D$, resetting rate $r$, and killing strength $k$, first in the absence of drift and then under a constant drift. The long-time behavior separates into distinct regimes. Without resetting, unbiased diffusion exhibits the recurrent algebraic survival law $S(t)\sim t^{-1/2}$, whereas any nonzero drift renders the motion transient with respect to the target and leaves a nonzero ultimate survival probability. By contrast, any $r>0$ and $k>0$ repeatedly renews encounters with the target and restores eventual absorption, yielding an exponential survival law $S(t)\sim A e^{-θt}$. The decay rate $θ$ is given by the dominant (rightmost) pole of the Laplace transform, and the surviving density converges after normalization to an explicit quasi-stationary profile. The driftless and perfectly absorbing limits recover standard resetting results. These formulas provide a unified description of the crossover between recurrence-controlled, transience-controlled, and renewal-controlled survival.
Comments27 pages, 4 figures
Journal refPhys. Rev. E 114, 044105 (2026)
