Gated First-Passage Processes

Abstract

The first-passage time of a stochastic time-series, defined to be the time taken by it to cross a predefined threshold for the first time, has a wide range of applications across disciplines. However, if the time-series can only be intermittently observed, then its key features can be missed. In particular, the instance at which we actually observe the time-series to be above the threshold for the first time, called the detection time or the gated first-passage time, can be strikingly different from the true first-passage time. In this thesis, we first put forth a general framework for computing the statistics of the detection time and discuss its connection to a classic problem in chemical kinetics. Our central result is that the first detection time is related to and is obtainable from the first-passage time distribution. The applicability of our framework is demonstrated in several model systems, including the SIS compartmental model of epidemics, logistic models and birth-death processes with resetting. Following this discussion, we address the inverse problem of reconstructing the true first-passage time statistics purely from gated observations. We develop a universal—model free—framework for the inference of first-passage times from the detection time statistics. Moreover, when the underlying model is known, this framework allows us to infer physically meaningful model parameters. The results are then leveraged to infer the gating rates via the hitherto overlooked short-time regime of the measured detection time distributions. Put together, the unified framework of gated first-passage processes opens a novel peephole into a myriad of systems whose direct observation is limited because of their underlying physics or imperfect observation conditions. We conclude the thesis with a discussion of three key applications of the theory – (i) determining the optimal rate of sampling a first-passage process, (ii) understanding the extreme value statistics of a partially observed-stochastic time-series, and (iii) inferring missing statistics in real-world time-series data.