The Annals of Probability

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

Marina Talet

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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

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

First available in Project Euclid: 19 March 2007

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Mathematical Reviews number (MathSciNet)

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

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


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.

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