Urn-related random walk with drift $\rho x^{\alpha} / t^{\beta}$

TL;DR

This paper investigates a one-dimensional random walk with a drift influenced by time and position, revealing a phase transition between recurrence and transience, and connecting it to urn models and Lamperti's walk.

Contribution

It introduces a novel class of urn-related random walks with position and time-dependent drift and analyzes their recurrence properties.

Findings

Identifies a phase transition for recurrence vs. transience

Establishes connections with Friedman's urn and Lamperti's walk

Provides conditions for recurrence and transience

Abstract

We study a one-dimensional random walk whose expected drift depends both on time and the position of a particle. We establish a non-trivial phase transition for the recurrence vs. transience of the walk, and show some interesting applications to Friedman's urn, as well as showing the connection with Lamperti's walk with asymptotically zero drift.