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

• 成果类型: Article
• 署名作者: Talet, Marina
• 署名单位: Aix-Marseille Universite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117906000000539
• 发表日期: 2007
• 页码: 32-67
• 关键词: dimensional random-walk; large deviations; Diffusion process; DECOMPOSITION; CONVERGENCE; environment

摘要

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. Amen Math. Soc. 351 (1999) 3915-3934] using stochastic calculus. We studied large deviations [Ann. Probab. 29 (2001) 11731204], 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.