Transience, recurrence and speed of diffusions with a non-Markovian two-phase "use it or lose it" drift

TL;DR

This paper studies a non-Markovian one-dimensional diffusion with a two-phase drift, analyzing its transience, recurrence, and speed, depending on the diffusion's proximity to its maximum, revealing complex behaviors influenced by the drift phases.

Contribution

It introduces a novel non-Markovian diffusion model with a two-phase drift, analyzing its transience, recurrence, and speed based on proximity to the maximum.

Findings

The diffusion exhibits different behaviors depending on the phase of the drift.

The process can be transient or recurrent based on the drift activation.

Speed analysis shows dependence on the proximity to the maximum.

Abstract

We investigate the transience/recurrence of a non-Markovian, one-dimensional diffusion process which consists of a Brownian motion with a non-anticipating drift that has two phases---a transient to $+\infty$ mode which is activated when the diffusion is sufficiently near its running maximum, and a recurrent mode which is activated otherwise. We also consider the speed of a diffusion with a two-phase drift, where the drift is equal to a certain positive constant when the diffusion is sufficiently near its running maximum, and is equal to another positive constant otherwise.