Modified discrete random walk with absorption

TL;DR

This paper analyzes a modified discrete random walk with absorption, deriving key probabilities and expected times, especially focusing on a version with different transition probabilities at a barrier.

Contribution

It introduces a modified random walk model with asymmetric probabilities at a barrier, extending previous models and providing analytical expressions for various absorption-related metrics.

Findings

Derived expected number of arrivals and absorption probabilities.

Calculated expected time before absorption in the modified walk.

Provided analytical formulas for the modified random walk with barriers.

Abstract

We obtain expected number of arrivals, probability of arrival, absorption probabilities and expected time before absorption for a modified discrete random walk on the (sub)set of integers. In a [pqrs] random walk the particle can move one step forward or backward, stay for a moment in the same state or it can be absorbed immediately in the current state. M[pqrs] is a modified version, where probabilities on both sides of a multiple function barrier M are of different [pqrs] type.