The Annals of Probability

Annealed tail estimates for a Brownian motion in a drifted Brownian potential

Marina Talet

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Abstract

We study Brownian motion in a drifted Brownian potential. Kawazu and Tanaka [J. Math. Soc. Japan 49 (1997) 189–211] exhibited two speed regimes for this process, depending on the drift. They supplemented these laws of large numbers by central limit theorems, which were recently completed by Hu, Shi and Yor [Trans. Amer. Math. Soc. 351 (1999) 3915–3934] using stochastic calculus. We studied large deviations [Ann. Probab. 29 (2001) 1173–1204], showing among other results that the rate function in the annealed setting, that is, after averaging over the potential, has a flat piece in the ballistic regime. In this paper we focus on this subexponential regime, proving that the probability of deviating below the almost sure speed has a polynomial rate of decay, and computing the exponent in this power law. This provides the continuous-time analogue of what Dembo, Peres and Zeitouni proved for the transient random walk in random environment [Comm. Math. Phys. 181 (1996) 667–683]. Our method takes a completely different route, making use of Lamperti’s representation together with an iteration scheme.

Article information

Source
Ann. Probab., Volume 35, Number 1 (2007), 32-67.

Dates
First available in Project Euclid: 19 March 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1174324123

Digital Object Identifier
doi:10.1214/009117906000000539

Mathematical Reviews number (MathSciNet)
MR2303943

Subjects
Primary: 60J60: Diffusion processes [See also 58J65] 60F15: Strong theorems

Keywords
Large deviation Brownian motion in a random potential Lamperti’s representation drifted Brownian motion Bessel process

Citation

Talet, Marina. Annealed tail estimates for a Brownian motion in a drifted Brownian potential. Ann. Probab. 35 (2007), no. 1, 32--67. doi:10.1214/009117906000000539. https://projecteuclid.org/euclid.aop/1174324123


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