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. http://www.numdam.org/articles/10.1214/13-AIHP549/

[1] M. Cranston and T. Mountford. The strong law of large numbers for a Brownian polymer. Ann. Probab. 24 (1996) 1300-1323. | MR | Zbl

[2] B. Davis. Weak limits of perturbed random walks and the equation Y t =B t +αsupY s :s≤t+βinfY s :s≤t. Ann. Probab. 24 (1996) 2007-2023. | MR | Zbl

[3] B. Davis. Brownian motion and random walk perturbed at extrema. Probab. Theory Related Fields 113 (1999) 501-518. | MR | Zbl

[4] R. Durrett and L. C. G. Rogers. Asymptotic behavior of Brownian polymers. Probab. Theory Related Fields 92 (1992) 337-349. | MR | Zbl

[5] I | MR | Zbl

[6] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes, 2nd edition. North-Holland, Amsterdam, 1989. | MR | Zbl

[7] E. Kosygina and M. Zerner. Excited random walks: Results, methods, open problems. Bull. Inst. Math. Acad. Sin. (N.S.) 8 (2013) 105-157. | MR

[8] T. Mountford and P. Tarrès. An asymptotic result for Brownian polymers. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008) 29-46. | Numdam | MR | Zbl

[9] R. Pemantle. A survey of random processes with reinforcement. Probab. Surv. 4 (2007) 1-79. | MR | Zbl

[10] R. G. Pinsky. Positive Harmonic Functions and Diffusion. Cambridge Studies in Advanced Mathematics 45. Cambridge Univ. Press, Cambridge, 1995. | MR | Zbl

[11] R. G. Pinsky. One-dimensional diffusions that eventually stop down-crossing. Bull. Lond. Math. Soc. 42 (2010) 634-638. | MR | Zbl

[12] O. Raimond and B. Schapira. Excited Brownian motions as limits of excited random walks. Probab. Theory Related Fields 154 (2012) 875-909. | MR | Zbl

[13] O. Zeitouni. Random walks in random environment. In Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1837. Springer, Berlin, 2004. | MR | Zbl

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