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2015 Localization and number of visited valleys for a transient diffusion in random environment
Pierre Andreoletti, Alexis Devulder
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Electron. J. Probab. 20: 1-58 (2015). DOI: 10.1214/EJP.v20-3173

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 adecomposition 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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Pierre Andreoletti. Alexis Devulder. "Localization and number of visited valleys for a transient diffusion in random environment." Electron. J. Probab. 20 1 - 58, 2015. https://doi.org/10.1214/EJP.v20-3173

Information

Accepted: 27 May 2015; Published: 2015
First available in Project Euclid: 4 June 2016

zbMATH: 1321.60214
MathSciNet: MR3354616
Digital Object Identifier: 10.1214/EJP.v20-3173

Subjects:
Secondary: 60J60, 60K05, 60K37, 82D30

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