The one-dimensional telegraphic process with noninstantaneous stochastic resetting

TL;DR

This paper studies a one-dimensional stochastic process where a particle moves at constant speed, randomly reverses direction, and is reset to the start with a deterministic return time, analyzing its stationary state and first-passage times.

Contribution

It introduces a model with deterministic return dynamics in stochastic resetting, revealing conditions for stationarity and providing explicit first-passage time formulas.

Findings

The process reaches a stationary state independent of the return phase under certain conditions.

Explicit formulas for mean first-hitting times are derived.

Numerical simulations support the analytical results.

Abstract

In this paper we consider the one-dimensional dynamical evolution of a particle traveling at constant speed and performing, at a given rate, random reversals of the velocity direction. The particle is subject to stochastic resetting, meaning that at random times it is forced to return to the starting point. Here we consider a return mechanism governed by a deterministic law of motion, so that the time cost required to return is correlated to the position occupied at the time of the reset. We show that in such conditions the process reaches a stationary state which, for some kinds of deterministic return dynamics, is independent of the return phase. Furthermore, we investigate the first-passage properties of the system and provide explicit formulas for the mean first-hitting time. Our findings are supported by numerical simulations.