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January 2007 Annealed tail estimates for a Brownian motion in a drifted Brownian potential
Marina Talet
Ann. Probab. 35(1): 32-67 (January 2007). DOI: 10.1214/009117906000000539

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.

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Marina Talet. "Annealed tail estimates for a Brownian motion in a drifted Brownian potential." Ann. Probab. 35 (1) 32 - 67, January 2007. https://doi.org/10.1214/009117906000000539

Information

Published: January 2007
First available in Project Euclid: 19 March 2007

MathSciNet: MR2303943
Digital Object Identifier: 10.1214/009117906000000539

Subjects:
Primary: 60F15 , 60J60

Keywords: Bessel process , Brownian motion in a random potential , drifted Brownian motion , Lamperti’s representation , large deviation

Rights: Copyright © 2007 Institute of Mathematical Statistics

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Vol.35 • No. 1 • January 2007
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