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

2013/11/25 by Pierre Andreoletti, Andreoletti, Pierre, Alexis Devulder +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR

paper · pdf · doi:10.48550/arxiv.1311.6332

55 pages, 2 figures

openalex publication_date 2013/11/25 · arxiv created 2015/03/09 · arxiv updated 2015/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a transient diffusion in a (-κ/2)-drifted Brownian potential W_κ with 0\textlessκ\textless1. 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_κ 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_κ and of W_κ Doob-conditioned to stay positive.

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