Transience, recurrence and speed of diffusions with a non-markovian two-phase “use it or lose it” drift
Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 4, pp. 1198-1212.

Nous étudions la transience/récurrence d’un processus de diffusion non-Markovien à une dimension, consistant en un mouvement brownien avec une dérive non anticipative qui a deux phases - un mode transitoire à +∞ qui est activé quand la diffusion est suffisamment proche du processus de son maximum, et un mode récurrent qui est activé dans le cas contraire. On considère également la vitesse d’une diffusion avec une dérive à deux phases, où la dérive est égale à une certaine constante positive lorsque la diffusion est suffisamment proche du processus de son maximum, et est égale à une certaine constante strictement positive dans le cas contraire.

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 +∞ 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 non-negative constant when the diffusion is sufficiently near its running maximum, and is equal to a certain positive constant otherwise.

DOI : 10.1214/13-AIHP549
Classification : 60J60
Mots-clés : diffusion process, transience, recurrence, non-markovian drift 
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Pinsky, Ross G. Transience, recurrence and speed of diffusions with a non-markovian two-phase “use it or lose it” drift. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 4, pp. 1198-1212. doi : 10.1214/13-AIHP549. https://www.numdam.org/articles/10.1214/13-AIHP549/

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