# Localization and number of visited valleys for a transient diffusion in random environment

Abstract : We consider a transient diffusion in a $(-\kappa/2)$-drifted Brownian potential $W_{\kappa}$ with $0<\kappa<1$. We prove its localization at time $t$ in the neighborhood of some random points depending only on the environment, which are the positive $h_t$-minima of the environment, for $h_t$ a bit smaller than $\log t$. We also prove an Aging phenomenon for the diffusion, a renewal theorem for the hitting time of the farthest visited valley, and provide a central limit theorem for the number of valleys visited up to time $t$. The proof relies on a decomposition of the trajectory of $W_{\kappa}$ in the neighborhood of $h_t$-minima, with the help of results of A. Faggionato, and on a precise analysis of exponential functionals of $W_{\kappa}$ and of $W_{\kappa}$ Doob-conditioned to stay positive.
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https://hal.archives-ouvertes.fr/hal-00908626
Contributor : Pierre Andreoletti <>
Submitted on : Monday, March 9, 2015 - 7:59:33 PM
Last modification on : Monday, February 10, 2020 - 6:13:49 PM
Document(s) archivé(s) le : Monday, April 17, 2017 - 5:45:08 AM

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StrLoc_B_202.pdf
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### Identifiers

• HAL Id : hal-00908626, version 3
• ARXIV : 1311.6332

### Citation

Pierre Andreoletti, Alexis Devulder. Localization and number of visited valleys for a transient diffusion in random environment. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2015, pp.1-59. ⟨hal-00908626v3⟩

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