Hitting the blinking target under stochastic resetting

TL;DR

This paper analyzes the distribution of first hitting times for a stochastic process with a target that switches between active and inactive states, incorporating stochastic resetting and revealing non-Markovian effects.

Contribution

It provides closed-form formulas for hitting times with a switching target and extends these results to include stochastic resetting, highlighting non-Markovian behavior.

Findings

Closed-form formulas for hitting times with switching targets

Derivation of hitting time distribution under stochastic resetting

Identification of non-Markovian memory effects due to resetting

Abstract

The first hitting times of a stochastic process, i.e., the first time a process reaches a particular level, are of significant interest across various scientific disciplines, including biology, chemistry, and economics. We modify the standard setup by allowing the target to spontaneously switch between two states, either active or inactive, and investigate the distribution of first hitting times accrued while the target is active. For this setup, we provide closed formulas for the distribution of the first hitting time. Additionally, we can introduce stochastic resetting to the underlying process and, utilizing our results, derive the formulas for the first time the active target is hit by the process under stochastic resetting. Interestingly, we show that resetting in this setup still leaves some memory; the system is no longer Markovian, which prevents a straightforward application of standard techniques. The analytical results are accompanied by computer simulations of Langevin dynamics.