On the continuity of Lyapunov exponents of random walk in random potential
type: Item
instance of: scholarly article
title: On the continuity of Lyapunov exponents of random walk in random potential
zbMATH Open document ID: 1365.60048
DOI: 10.3150/15-BEJ753
author: Le Thi Thu Hien
published in: Bernoulli
publication date: 2017-01-11T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bj/1475001364
review text: The paper considers a simple random walk on the \(d\)-dimensional \((d\geq 3)\) cubic lattice. Independently of the random walk, a family of i.i.d. random variables taking values in \([0,\infty )\) is given, which are called potentials. Let \(F\) be the distribution function of the potential. Using \(F\), following the results of \textit{M. P. W. Zerner} [Ann. Appl. Probab. 8, No. 1, 246--280 (1998; Zbl 0938.60098)] and \textit{M. Flury} [Stochastic Processes Appl. 117, No. 5, 596--612 (2007; Zbl 1193.60033)], the author introduces \(\alpha _F(x)\) and \(\beta _F(x)\), the corresponding quenched and annealed Lyapunov exponents. He studies the continuity of these exponents with respect to the law of the potential, assuming independence. Let \(\mathcal{D}\) be the set of distribution functions, \(\mathcal{D}_1\) the subset of \(\mathcal{D}\) containing all distribution functions with finite mean and \((F_n)\) a sequence of distribution functions. It is shown that if \(F_n\) tends to \(F\), then \(\lim _{n\to \infty }\alpha _{F_n}(x)=\alpha _F(x)\) in \(\mathcal{D}_1\) and \(\lim _{n\to \infty }\beta _{F_n}(x)=\beta _F(x)\) in \(\mathcal{D}\) for all \(x\in \mathbb{R}^d\), and the convergence is uniform on any compact subset of \(\mathbb{R}^d\).
reviewed by: László Lakatos
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 6673487
zbMATH Keywords: random walk
zbMATH Keywords: random potential
zbMATH Keywords: continuity
zbMATH Keywords: Lyapunov exponents
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6673487