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

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.

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