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Localization and number of visited valleys for a transient diffusion in random environment. (English) Zbl 1321.60214
Summary: 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 [Stochastic Processes Appl. 119, No. 6, 1765–1791 (2009; Zbl 1169.60318)], and on a precise analysis of exponential functionals of \(W_{\kappa}\) and of \(W_{\kappa}\) Doob-conditioned to stay positive.

60K37 Processes in random environments
60J60 Diffusion processes
60K05 Renewal theory
60F05 Central limit and other weak theorems
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
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