Gated first-passage processes, where completion depends on both hitting a target and satisfying additional constraints, are prevalent across various fields. Despite their significance, analytical solutions to basic problems remain unknown, e.g. the detection time of a diffusing particle by a gated interval, disk, or sphere. In this paper, we elucidate the challenges posed by continuous gated first-passage processes and present a renewal framework to overcome them. This framework offers a unified approach for a wide range of problems, including those with single-point, half-line, and interval targets. The latter have so far evaded exact solutions. Our analysis reveals that solutions to gated problems can be obtained directly from the ungated dynamics. This, in turn, reveals universal properties and asymptotic behaviors, shedding light on cryptic intermediate-time regimes and refining the notion of high-crypticity for continuous-space gated processes. Moreover, we extend our formalism to higher dimensions, showcasing its versatility and applicability. Overall, this work provides valuable insights into the dynamics of continuous gated first-passage processes and offers analytical tools for studying them across diverse domains.

中文翻译：

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连续门控首次传代过程


门控的第一通道过程在各个领域都很普遍，其中完成取决于达到目标和满足额外的约束。尽管它们很重要，但基本问题的分析解决方案仍然未知，例如门控间隔、圆盘或球体对扩散粒子的检测时间。在本文中，我们阐明了连续门控首次传代过程带来的挑战，并提出了一个克服这些挑战的更新框架。该框架为各种问题提供了统一的方法，包括单点、半线和区间目标的问题。到目前为止，后者还没有找到精确的解决方案。我们的分析表明，门控问题的解决方案可以直接从非门控动力学中获得。反过来，这揭示了普遍性质和渐近行为，揭示了神秘的中间时间状态，并完善了连续空间门控过程的高度神秘性的概念。此外，我们将形式主义扩展到更高的维度，展示其多功能性和适用性。总体而言，这项工作为连续门控首次传代过程的动态提供了宝贵的见解，并为跨不同领域的研究提供了分析工具。