Continuous Gated First-Passage Processes

TL;DR

This paper introduces a renewal framework for continuous gated first-passage processes, providing analytical solutions and revealing universal properties across various geometries and dimensions.

Contribution

It presents a unified renewal approach to solve complex gated first-passage problems, including those with previously unsolved geometries, and extends the analysis to higher dimensions.

Findings

Solutions can be derived directly from ungated dynamics.

Reveals universal properties and asymptotic behaviors.

Extends formalism to higher-dimensional spaces.

Abstract

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.