Telegraph process with elastic boundary at the origin

TL;DR

This paper analyzes a one-dimensional telegraph process with an elastic boundary at the origin, deriving distributions of renewal cycles and absorption times, considering both starting points at zero and elsewhere.

Contribution

It provides new analytical results for the telegraph process with an elastic boundary, including distributions of key process metrics under exponential switching times.

Findings

Derived distribution of renewal cycles

Obtained absorption time distribution

Analyzed process starting from different initial states

Abstract

We investigate the one-dimensional telegraph random process in the presence of an elastic boundary at the origin. This process describes a finite-velocity random motion that alternates between two possible directions of motion (positive or negative). When the particle hits the origin, it is either absorbed, with probability $\alpha$, or reflected upwards, with probability $1-\alpha$. In the case of exponentially distributed random times between consecutive changes of direction, we obtain the distribution of the renewal cycles and of the absorption time at the origin. This investigation is performed both in the case of motion starting from the origin and non-zero initial state. We also study the probability law of the process within a renewal cycle.