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

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