Analysis of an Inhomogeneous Random Walk with Spatial Decay of Transition Probabilities and Parameter Renewal per Excursion

TL;DR

This paper introduces a new one-dimensional inhomogeneous random walk model with spatial decay and renewal structure, analyzing hitting probabilities, occupation times, and maximum depth through rigorous mathematical methods.

Contribution

It combines spatial inhomogeneity with a renewal process per excursion and derives explicit formulas for key probabilistic quantities.

Findings

Derived the hitting probability to an upper boundary.

Provided the probability generating function of the first hitting time.

Calculated the expected occupation time and maximum penetration depth.

Abstract

In this paper, we propose and analyze a novel one-dimensional inhomogeneous random walk model that combines spatial decay of transition probabilities with a temporal renewal structure for each excursion. In this model, the probability of moving to the right from each state creats a spatial inhomogeneity that causes a stronger pull-back toward the origin as the process moves farther away. Furthermore, it features a random environment aspect where the parameter of each transition probability is independently resampled from a uniform distribution at the beginning of each excursion. We rigorously derive the hitting probability to an upper boundary using a scale function. Furthermore, by solving linear difference equations, we provide the probability generating function of the first hitting time, the expected occupation time for each state during an excursion (discrete Green's function), and the distribution and expectation of the maximum penetration depth.